From 413e642c85182d1941d05fb0415aad4ebd09c63d Mon Sep 17 00:00:00 2001 From: Script Raccoon Date: Fri, 15 May 2026 17:36:18 +0200 Subject: [PATCH] write property instead of property_id in yaml files --- databases/catdat/data/categories/0.yaml | 14 ++--- databases/catdat/data/categories/1.yaml | 8 +-- databases/catdat/data/categories/2.yaml | 12 ++-- databases/catdat/data/categories/Ab.yaml | 12 ++-- databases/catdat/data/categories/Ab_fg.yaml | 22 ++++---- databases/catdat/data/categories/Alg(R).yaml | 32 +++++------ databases/catdat/data/categories/B.yaml | 22 ++++---- databases/catdat/data/categories/BG_c.yaml | 14 ++--- databases/catdat/data/categories/BG_f.yaml | 14 ++--- databases/catdat/data/categories/BN.yaml | 22 ++++---- databases/catdat/data/categories/BOn.yaml | 34 +++++------ databases/catdat/data/categories/Ban.yaml | 30 +++++----- databases/catdat/data/categories/CAlg(R).yaml | 28 +++++----- databases/catdat/data/categories/CMon.yaml | 26 ++++----- databases/catdat/data/categories/CRing.yaml | 28 +++++----- databases/catdat/data/categories/Cat.yaml | 30 +++++----- .../catdat/data/categories/CompHaus.yaml | 40 ++++++------- databases/catdat/data/categories/Delta.yaml | 38 ++++++------- databases/catdat/data/categories/FI.yaml | 36 ++++++------ databases/catdat/data/categories/FS.yaml | 40 ++++++------- databases/catdat/data/categories/FinAb.yaml | 22 ++++---- databases/catdat/data/categories/FinGrp.yaml | 38 ++++++------- databases/catdat/data/categories/FinOrd.yaml | 40 ++++++------- databases/catdat/data/categories/FinSet.yaml | 24 ++++---- databases/catdat/data/categories/Fld.yaml | 34 +++++------ databases/catdat/data/categories/FreeAb.yaml | 28 +++++----- databases/catdat/data/categories/Grp.yaml | 34 +++++------ databases/catdat/data/categories/Grp_c.yaml | 46 +++++++-------- databases/catdat/data/categories/Haus.yaml | 36 ++++++------ databases/catdat/data/categories/J2.yaml | 10 ++-- databases/catdat/data/categories/LRS.yaml | 22 ++++---- databases/catdat/data/categories/M-Set.yaml | 12 ++-- databases/catdat/data/categories/Man.yaml | 38 ++++++------- databases/catdat/data/categories/Meas.yaml | 34 +++++------ databases/catdat/data/categories/Met.yaml | 52 ++++++++--------- databases/catdat/data/categories/Met_c.yaml | 36 ++++++------ databases/catdat/data/categories/Met_oo.yaml | 30 +++++----- databases/catdat/data/categories/Mon.yaml | 34 +++++------ databases/catdat/data/categories/N.yaml | 18 +++--- databases/catdat/data/categories/N_oo.yaml | 20 +++---- databases/catdat/data/categories/On.yaml | 22 ++++---- databases/catdat/data/categories/PMet.yaml | 48 ++++++++-------- databases/catdat/data/categories/Pos.yaml | 32 +++++------ databases/catdat/data/categories/Prost.yaml | 32 +++++------ databases/catdat/data/categories/R-Mod.yaml | 12 ++-- .../catdat/data/categories/R-Mod_div.yaml | 10 ++-- databases/catdat/data/categories/Rel.yaml | 30 +++++----- databases/catdat/data/categories/Ring.yaml | 32 +++++------ databases/catdat/data/categories/Rng.yaml | 32 +++++------ databases/catdat/data/categories/Sch.yaml | 24 ++++---- databases/catdat/data/categories/SemiGrp.yaml | 32 +++++------ databases/catdat/data/categories/Set.yaml | 14 ++--- databases/catdat/data/categories/Set_c.yaml | 36 ++++++------ databases/catdat/data/categories/Set_f.yaml | 42 +++++++------- databases/catdat/data/categories/Set_op.yaml | 2 +- .../catdat/data/categories/Set_pointed.yaml | 32 +++++------ databases/catdat/data/categories/Setne.yaml | 40 ++++++------- databases/catdat/data/categories/SetxSet.yaml | 12 ++-- databases/catdat/data/categories/Sh(X).yaml | 8 +-- .../catdat/data/categories/Sh(X,Ab).yaml | 8 +-- databases/catdat/data/categories/Sp.yaml | 22 ++++---- databases/catdat/data/categories/Top.yaml | 44 +++++++-------- .../catdat/data/categories/Top_pointed.yaml | 56 +++++++++---------- databases/catdat/data/categories/TorsAb.yaml | 22 ++++---- .../catdat/data/categories/TorsFreeAb.yaml | 22 ++++---- databases/catdat/data/categories/Vect.yaml | 10 ++-- databases/catdat/data/categories/Z.yaml | 36 ++++++------ databases/catdat/data/categories/Z_div.yaml | 18 +++--- .../catdat/data/categories/real_interval.yaml | 18 +++--- databases/catdat/data/categories/sSet.yaml | 14 ++--- .../walking_commutative_square.yaml | 16 +++--- .../categories/walking_composable_pair.yaml | 14 ++--- .../categories/walking_coreflexive_pair.yaml | 34 +++++------ .../catdat/data/categories/walking_fork.yaml | 26 ++++----- .../data/categories/walking_idempotent.yaml | 20 +++---- .../data/categories/walking_isomorphism.yaml | 8 +-- .../data/categories/walking_morphism.yaml | 14 ++--- .../catdat/data/categories/walking_pair.yaml | 22 ++++---- .../catdat/data/categories/walking_span.yaml | 18 +++--- .../data/categories/walking_splitting.yaml | 24 ++++---- .../catdat/data/category-properties/CIP.yaml | 2 +- .../catdat/data/category-properties/CSP.yaml | 2 +- .../category-properties/Cauchy complete.yaml | 2 +- .../data/category-properties/Malcev.yaml | 2 +- .../data/category-properties/abelian.yaml | 2 +- .../data/category-properties/accessible.yaml | 2 +- .../data/category-properties/additive.yaml | 2 +- .../data/category-properties/balanced.yaml | 2 +- .../category-properties/binary copowers.yaml | 2 +- .../binary coproducts.yaml | 2 +- .../category-properties/binary powers.yaml | 2 +- .../category-properties/binary products.yaml | 2 +- .../data/category-properties/biproducts.yaml | 2 +- .../category-properties/cartesian closed.yaml | 2 +- .../cartesian filtered colimits.yaml | 2 +- .../data/category-properties/co-Malcev.yaml | 2 +- .../category-properties/coaccessible.yaml | 2 +- .../cocartesian coclosed.yaml | 2 +- .../cocartesian cofiltered limits.yaml | 2 +- .../data/category-properties/cocomplete.yaml | 2 +- .../category-properties/codistributive.yaml | 2 +- .../category-properties/coequalizers.yaml | 2 +- .../data/category-properties/coextensive.yaml | 2 +- .../cofiltered limits.yaml | 2 +- .../cofiltered-limit-stable epimorphisms.yaml | 2 +- .../data/category-properties/cofiltered.yaml | 2 +- .../category-properties/cogenerating set.yaml | 2 +- .../data/category-properties/cogenerator.yaml | 2 +- .../data/category-properties/cokernels.yaml | 2 +- .../data/category-properties/complete.yaml | 2 +- .../connected colimits.yaml | 2 +- .../category-properties/connected limits.yaml | 2 +- .../data/category-properties/connected.yaml | 2 +- .../data/category-properties/conormal.yaml | 2 +- .../data/category-properties/copowers.yaml | 2 +- .../data/category-properties/coproducts.yaml | 2 +- .../coquotients of cocongruences.yaml | 2 +- .../data/category-properties/core-thin.yaml | 2 +- .../coreflexive equalizers.yaml | 2 +- .../data/category-properties/coregular.yaml | 2 +- .../category-properties/cosifted limits.yaml | 2 +- .../data/category-properties/cosifted.yaml | 2 +- .../data/category-properties/counital.yaml | 2 +- .../countable copowers.yaml | 2 +- .../countable coproducts.yaml | 2 +- .../category-properties/countable powers.yaml | 2 +- .../countable products.yaml | 2 +- .../data/category-properties/countable.yaml | 2 +- .../countably codistributive.yaml | 2 +- .../countably distributive.yaml | 2 +- .../data/category-properties/direct.yaml | 2 +- .../directed colimits.yaml | 2 +- .../category-properties/directed limits.yaml | 2 +- .../data/category-properties/discrete.yaml | 2 +- .../disjoint coproducts.yaml | 2 +- .../disjoint finite coproducts.yaml | 2 +- .../disjoint finite products.yaml | 2 +- .../disjoint products.yaml | 2 +- .../category-properties/distributive.yaml | 2 +- .../effective cocongruences.yaml | 2 +- .../effective congruences.yaml | 2 +- .../data/category-properties/epi-regular.yaml | 2 +- .../data/category-properties/equalizers.yaml | 2 +- .../essentially countable.yaml | 2 +- .../essentially discrete.yaml | 2 +- .../essentially finite.yaml | 2 +- .../essentially small.yaml | 2 +- .../exact cofiltered limits.yaml | 2 +- .../exact filtered colimits.yaml | 2 +- .../data/category-properties/extensive.yaml | 2 +- .../filtered colimits.yaml | 2 +- ...filtered-colimit-stable monomorphisms.yaml | 2 +- .../data/category-properties/filtered.yaml | 2 +- .../category-properties/finite copowers.yaml | 2 +- .../finite coproducts.yaml | 2 +- .../category-properties/finite powers.yaml | 2 +- .../category-properties/finite products.yaml | 2 +- .../data/category-properties/finite.yaml | 2 +- .../finitely cocomplete.yaml | 2 +- .../finitely complete.yaml | 2 +- .../data/category-properties/gaunt.yaml | 2 +- .../category-properties/generating set.yaml | 2 +- .../data/category-properties/generator.yaml | 2 +- .../data/category-properties/groupoid.yaml | 2 +- .../infinitary codistributive.yaml | 2 +- .../infinitary coextensive.yaml | 2 +- .../infinitary distributive.yaml | 2 +- .../infinitary extensive.yaml | 2 +- .../data/category-properties/inhabited.yaml | 2 +- .../category-properties/initial object.yaml | 2 +- .../data/category-properties/inverse.yaml | 2 +- .../data/category-properties/kernels.yaml | 2 +- .../left cancellative.yaml | 2 +- .../locally cartesian closed.yaml | 2 +- .../locally cocartesian coclosed.yaml | 2 +- .../locally copresentable.yaml | 2 +- .../locally essentially small.yaml | 2 +- .../category-properties/locally finite.yaml | 2 +- .../locally presentable.yaml | 2 +- .../category-properties/locally small.yaml | 2 +- .../category-properties/mono-regular.yaml | 2 +- .../category-properties/multi-cocomplete.yaml | 2 +- .../category-properties/multi-complete.yaml | 2 +- .../multi-initial object.yaml | 2 +- .../multi-terminal object.yaml | 2 +- .../data/category-properties/normal.yaml | 2 +- .../data/category-properties/one-way.yaml | 2 +- .../data/category-properties/pointed.yaml | 2 +- .../data/category-properties/powers.yaml | 2 +- .../data/category-properties/preadditive.yaml | 2 +- .../data/category-properties/products.yaml | 2 +- .../data/category-properties/pullbacks.yaml | 2 +- .../data/category-properties/pushouts.yaml | 2 +- .../quotient object classifier.yaml | 2 +- .../category-properties/quotient-trivial.yaml | 2 +- .../quotients of congruences.yaml | 2 +- .../reflexive coequalizers.yaml | 2 +- .../regular quotient object classifier.yaml | 2 +- .../regular subobject classifier.yaml | 2 +- .../data/category-properties/regular.yaml | 2 +- .../right cancellative.yaml | 2 +- .../data/category-properties/self-dual.yaml | 2 +- .../semi-strongly connected.yaml | 2 +- .../sequential colimits.yaml | 2 +- .../sequential limits.yaml | 2 +- .../category-properties/sifted colimits.yaml | 2 +- .../data/category-properties/sifted.yaml | 2 +- .../data/category-properties/skeletal.yaml | 2 +- .../data/category-properties/small.yaml | 2 +- .../category-properties/split abelian.yaml | 2 +- .../strict initial object.yaml | 2 +- .../strict terminal object.yaml | 2 +- .../strongly connected.yaml | 2 +- .../subobject classifier.yaml | 2 +- .../subobject-trivial.yaml | 2 +- .../category-properties/terminal object.yaml | 2 +- .../catdat/data/category-properties/thin.yaml | 2 +- .../data/category-properties/trivial.yaml | 2 +- .../data/category-properties/unital.yaml | 2 +- .../category-properties/well-copowered.yaml | 2 +- .../category-properties/well-powered.yaml | 2 +- .../category-properties/wide pullbacks.yaml | 2 +- .../category-properties/wide pushouts.yaml | 2 +- .../category-properties/zero morphisms.yaml | 2 +- .../data/functor-properties/cocontinuous.yaml | 2 +- .../coequalizer-preserving.yaml | 2 +- .../data/functor-properties/cofinitary.yaml | 2 +- .../data/functor-properties/comonadic.yaml | 2 +- .../data/functor-properties/conservative.yaml | 2 +- .../data/functor-properties/continuous.yaml | 2 +- .../coproduct-preserving.yaml | 2 +- .../epimorphism-preserving.yaml | 2 +- .../equalizer-preserving.yaml | 2 +- .../data/functor-properties/equivalence.yaml | 2 +- .../essentially surjective.yaml | 2 +- .../catdat/data/functor-properties/exact.yaml | 2 +- .../data/functor-properties/faithful.yaml | 2 +- .../data/functor-properties/finitary.yaml | 2 +- .../finite-coproduct-preserving.yaml | 2 +- .../finite-product-preserving.yaml | 2 +- .../catdat/data/functor-properties/full.yaml | 2 +- .../initial-object-preserving.yaml | 2 +- .../data/functor-properties/left adjoint.yaml | 2 +- .../data/functor-properties/left exact.yaml | 2 +- .../data/functor-properties/monadic.yaml | 2 +- .../monomorphism-preserving.yaml | 2 +- .../product-preserving.yaml | 2 +- .../functor-properties/representable.yaml | 2 +- .../functor-properties/right adjoint.yaml | 2 +- .../data/functor-properties/right exact.yaml | 2 +- .../terminal-object-preserving.yaml | 2 +- .../catdat/data/functors/abelianization.yaml | 16 +++--- .../catdat/data/functors/forget_vector.yaml | 16 +++--- .../catdat/data/functors/free_group.yaml | 16 +++--- databases/catdat/data/functors/id_Set.yaml | 4 +- .../functors/power_set_contravariant.yaml | 16 +++--- .../data/functors/power_set_covariant.yaml | 22 ++++---- databases/catdat/scripts/seed.ts | 16 +++--- databases/catdat/scripts/seed.types.ts | 10 ++-- 259 files changed, 1263 insertions(+), 1263 deletions(-) diff --git a/databases/catdat/data/categories/0.yaml b/databases/catdat/data/categories/0.yaml index a1eaef92..ee405955 100644 --- a/databases/catdat/data/categories/0.yaml +++ b/databases/catdat/data/categories/0.yaml @@ -14,26 +14,26 @@ related_categories: - '1' satisfied_properties: - - property_id: discrete + - property: discrete reason: This is trivial. - - property_id: binary products + - property: binary products reason: This is vacuously true. - - property_id: finite + - property: finite reason: This is trivial. - - property_id: small + - property: small reason: This is trivial. - - property_id: preadditive + - property: preadditive reason: This is vacuously true. - - property_id: multi-algebraic + - property: multi-algebraic reason: The terminal category $\1$ becomes an FPC-sketch by selecting the unique empty cone and cocone. Then, a $\Set$-valued model of this sketch is a functor $\1 \to \Set$ sending the unique object to a terminal and initial object, which never exists. Hence, $\0$ is the category of models of this FPC-sketch. unsatisfied_properties: - - property_id: inhabited + - property: inhabited reason: This is trivial. special_objects: {} diff --git a/databases/catdat/data/categories/1.yaml b/databases/catdat/data/categories/1.yaml index fdc04070..59e95233 100644 --- a/databases/catdat/data/categories/1.yaml +++ b/databases/catdat/data/categories/1.yaml @@ -16,16 +16,16 @@ related_categories: - '2' satisfied_properties: - - property_id: trivial + - property: trivial reason: This is trivial. - - property_id: finite + - property: finite reason: This is trivial. - - property_id: small + - property: small reason: This is trivial. - - property_id: discrete + - property: discrete reason: This is trivial. unsatisfied_properties: [] diff --git a/databases/catdat/data/categories/2.yaml b/databases/catdat/data/categories/2.yaml index 45ee532a..bb93b87b 100644 --- a/databases/catdat/data/categories/2.yaml +++ b/databases/catdat/data/categories/2.yaml @@ -14,23 +14,23 @@ related_categories: - '1' satisfied_properties: - - property_id: discrete + - property: discrete reason: This is trivial. - - property_id: finite + - property: finite reason: This is trivial. - - property_id: small + - property: small reason: This is trivial. - - property_id: inhabited + - property: inhabited reason: This is trivial. - - property_id: multi-algebraic + - property: multi-algebraic reason: There is an FPC-sketch whose $\Set$-model is precisely a pair $(X,Y)$ of sets such that the coproduct $X+Y$ is a singleton. Any $\Set$-model of such a sketch is isomorphic to either $(\varnothing, 1)$ or $(1, \varnothing)$, hence the category of models is equivalent to $\2$. unsatisfied_properties: - - property_id: connected + - property: connected reason: The objects $0$, $1$ have no zig-zag path between them. special_objects: {} diff --git a/databases/catdat/data/categories/Ab.yaml b/databases/catdat/data/categories/Ab.yaml index e422d281..95d041f3 100644 --- a/databases/catdat/data/categories/Ab.yaml +++ b/databases/catdat/data/categories/Ab.yaml @@ -20,23 +20,23 @@ related_categories: - TorsFreeAb satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Ab \to \Set$ and $\Set$ is locally small. - - property_id: abelian + - property: abelian reason: This is standard, see Mac Lane, Ch. VIII. - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic theory of a commutative group. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: split abelian + - property: split abelian reason: The short exact sequence $0 \xrightarrow{} \IZ \xrightarrow{p} \IZ \xrightarrow{} \IZ/p \xrightarrow{} 0$ does not split. - - property_id: CSP + - property: CSP reason: The canonical homomorphism $\bigoplus_{n \geq 0} \IZ \to \prod_{n \geq 0} \IZ$ is not surjective, hence no epimorphism. special_objects: diff --git a/databases/catdat/data/categories/Ab_fg.yaml b/databases/catdat/data/categories/Ab_fg.yaml index ff8925c8..1f9473a0 100644 --- a/databases/catdat/data/categories/Ab_fg.yaml +++ b/databases/catdat/data/categories/Ab_fg.yaml @@ -14,38 +14,38 @@ related_categories: - FinAb satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\FinAb \to \Set$ and $\Set$ is locally small. - - property_id: abelian + - property: abelian reason: This follows from the fact for abelian groups and the fact that subgroups of finitely generated abelian groups are also finitely generated. - - property_id: generator + - property: generator reason: The group $\IZ$ is a generator since it represents the forgetful functor to $\Set$. - - property_id: essentially countable + - property: essentially countable reason: Every finitely generated abelian group is isomorphic to a group of the form $\IZ^n / U$, where $n \in \IN$ and $U$ is a subgroup of $\IZ^n$. Since $\IZ^n$ is Noetherian as a $\IZ$-module, $U$ is finitely generated, hence the category $\Ab_\fg$ has only countably many objects up to isomorphism. Furthermore, for any objects $A \cong \IZ^n / U$ and $B \cong \IZ^m / T$, the hom-set $\Hom(A,B)$ is countable. Indeed, precomposition with the quotient map yields an injection $\Hom(A,B) \hookrightarrow \Hom(\IZ^n, B) \cong B^n$, and $B^n$ is countable. - - property_id: ℵ₁-accessible + - property: ℵ₁-accessible reason: The inclusion $\Ab_{\fg} \hookrightarrow \Ab$ is closed under $\aleph_1$-filtered colimits by MO/400763. In particular, $\Ab_{\fg}$ has $\aleph_1$-filtered colimits. Since $\Ab_{\fg}$ is essentially small, there is a set $G$ such that every f.g. abelian group is isomorphic to one in $G$. So trivially it is also a $\aleph_1$-filtered colimit of such objects (take the constant diagram). Finally, every object is $\Ab_{\fg} = \Ab_{\fp}$ is finitely presentable in $\Ab$ and hence also in $\Ab_{\fg}$, a fortiori $\aleph_1$-presentable. unsatisfied_properties: - - property_id: small + - property: small reason: Even the collection of trivial groups is not small. - - property_id: locally finite + - property: locally finite reason: The group $\Hom(\IZ,\IZ) \cong \IZ$ is not finite. - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: countable + - property: countable reason: This is trivial. - - property_id: split abelian + - property: split abelian reason: The short exact sequence $0 \xrightarrow{} \IZ \xrightarrow{p} \IZ \xrightarrow{} \IZ/p \xrightarrow{} 0$ does not split. - - property_id: cogenerator + - property: cogenerator reason: Let $Q$ be a finitely generated abelian group. By their well-known classification, we have $Q = F \oplus T$ for a free abelian group $F$ and a finite abelian group $T$. Let $p$ be a prime number which does not divide the order of $T$. Then $\Hom(\IZ/p, Q) = 0$, but $\IZ/p \neq 0$. Therefore, $Q$ is no cogenerator. special_objects: diff --git a/databases/catdat/data/categories/Alg(R).yaml b/databases/catdat/data/categories/Alg(R).yaml index db52b6e8..1da3593b 100644 --- a/databases/catdat/data/categories/Alg(R).yaml +++ b/databases/catdat/data/categories/Alg(R).yaml @@ -15,56 +15,56 @@ related_categories: - Ring satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Alg(R) \to \Set$ and $\Set$ is locally small. - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic theory of an $R$-algebra. - - property_id: strict terminal object + - property: strict terminal object reason: 'If $f : 0 \to A$ is an algebra homomorphism, then $A$ satisfies $1=f(1)=f(0)=0$, so that $A=0$.' - - property_id: disjoint finite products + - property: disjoint finite products reason: One can take the same proof as for $\Ring$. - - property_id: Malcev + - property: Malcev reason: This follows in the same way as for $\Grp$, see also Example 2.2.5 in Malcev, protomodular, homological and semi-abelian categories. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: Take a prime ideal $P \subseteq R$ and consider the $R$-algebra $A := R/P$ (which is an integral domain). Then the inclusion $A \hookrightarrow Q(A)$ is a counterexample. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: This is because already the full subcategory $\CAlg(R)$ of commutative algebras is not semi-strongly connected. - - property_id: cogenerating set + - property: cogenerating set reason: 'We apply this lemma to the collection of $R$-algebras which are fields: If $F$ is an $R$-algebra that is also a field and $A$ is a non-trivial $R$-algebra, any algebra homomorphism $F \to A$ is injective. For every infinite cardinal $\kappa$ the field of rational functions in $\kappa$ variables over some residue field of $R$ has cardinality $\geq \kappa$ and a non-trivial automorphism (swap two variables).' - - property_id: codistributive + - property: codistributive reason: 'If $\sqcup$ denotes the coproduct of $R$-algebras (see MSE/625874 for their description) and $A$ is an $R$-algebra, the canonical morphism $A \sqcup R^2 \to (A \sqcup R)^2 = A^2$ is usually no isomorphism. For example, for $A = R[X]$ the coproduct on the LHS is not commutative, it has the algebra presentation $\langle X,E : E^2=E \rangle$.' - - property_id: co-Malcev + - property: co-Malcev reason: 'See MO/509552: Consider the forgetful functor $U : \Alg(R) \to \Set$ and the relation $S \subseteq U^2$ defined by $S(A) := \{(a,b) \in U(A)^2 : ab = a^2\}$. Both are representable: $U$ by $R[X]$ and $S$ by $R \langle X,Y \rangle / \langle XY-X^2 \rangle$. It is clear that $S$ is reflexive, but not symmetric.' - - property_id: coregular + - property: coregular reason: 'We just need to tweak the proof for $\Ring$. Since $R \neq 0$, there is an infinite field $K$ with a homomorphism $R \to K$. Since $K$ is infinite, we may choose some $\lambda \in K \setminus \{0,1\}$. Let $B := M_2(K)$ and $A := K \times K$. Then $A \to B$, $(x,y) \mapsto \diag(x,y)$ is a regular monomorphism: A direct calculation shows that a matrix is diagonal iff it commutes with $M := \bigl(\begin{smallmatrix} 1 & 0 \\ 0 & \lambda \end{smallmatrix}\bigr)$, so that $A \to B$ is the equalizer of the identity $B \to B$ and the conjugation $B \to B$, $X \mapsto M X M^{-1}$. Consider the homomorphism $A \to K$, $(a,b) \mapsto a$. We claim that $K \to K \sqcup_A B$ is not a monomorphism, because in fact, the pushout $K \sqcup_A B$ is zero: Since $A \to K$ is surjective with kernel $0 \times K$, the pushout is $B/\langle 0 \times K \rangle$, which is $0$ because $B$ is simple (proof) or via a direct calculation with elementary matrices.' - - property_id: regular quotient object classifier + - property: regular quotient object classifier reason: We may copy the proof for $\CRing$ (since the proof there did not use that $P$ is commutative). Alternatively, any regular quotient object classifier in $\Alg(R)$ would produce one in $\CAlg(R)$ by this lemma (dualized). - - property_id: cocartesian cofiltered limits + - property: cocartesian cofiltered limits reason: >- Consider the ring $A = R[X]$ and the sequence of rings $B_n = R[Y]/(Y^{n+1})$ with projections $B_{n+1} \to B_n$, whose limit is $R[[Y]]$. Every element in the coproduct of rings $R[X] \sqcup R[[Y]]$ has a finite "free product" length. Now consider the elements $$w_n = (1 + XY) (1+XY^2) \cdots (1+X Y^n) \in A \sqcup B_n.$$ Because of $w_n \equiv w_{n-1} \bmod Y^n$ these form an element $w \in \lim_n (A \sqcup B_n)$. Expanding $w_n$, the longest term is $XY XY^2 \cdots X Y^n$ of "free product" length $2n$, which is unbounded. - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: We already know that $\CAlg(R)$ does not have this property. Now apply the contrapositive of the dual of this lemma to the forgetful functor $\CAlg(R) \to \Alg(R)$. It preserves epimorphisms by MSE/5133488. - - property_id: effective cocongruences + - property: effective cocongruences reason: 'The counterexample is similar to the one for $\Ring$: Let $X := R[p] / (p^2-p)$ with cocongruence $E := R \langle p, q \rangle / (p^2-p, q^2-q, pq-q, qp-p)$.' special_objects: diff --git a/databases/catdat/data/categories/B.yaml b/databases/catdat/data/categories/B.yaml index 8134d540..0d063553 100644 --- a/databases/catdat/data/categories/B.yaml +++ b/databases/catdat/data/categories/B.yaml @@ -15,38 +15,38 @@ related_categories: - FS satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\IB \to \Set$ and $\Set$ is locally small. - - property_id: locally finite + - property: locally finite reason: There is a faithful functor $\IB \to \FinSet$ and $\FinSet$ is locally finite. - - property_id: inhabited + - property: inhabited reason: This is trivial. - - property_id: groupoid + - property: groupoid reason: This is trivial. - - property_id: essentially countable + - property: essentially countable reason: Every finite set is isomorphic to some $\{1,\dotsc,n\}$ for some $n \in \IN$. unsatisfied_properties: - - property_id: small + - property: small reason: Even the collection of singletons is not small. - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: generator + - property: generator reason: This is trivial. - - property_id: essentially finite + - property: essentially finite reason: This is trivial. - - property_id: countable + - property: countable reason: This is trivial. - - property_id: connected + - property: connected reason: For every $n \geq 0$ there is a connected component of sets of size $n$. special_objects: {} diff --git a/databases/catdat/data/categories/BG_c.yaml b/databases/catdat/data/categories/BG_c.yaml index a1ddd7b7..5c605a5f 100644 --- a/databases/catdat/data/categories/BG_c.yaml +++ b/databases/catdat/data/categories/BG_c.yaml @@ -16,26 +16,26 @@ related_categories: - BN satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: groupoid + - property: groupoid reason: This is trivial. - - property_id: connected + - property: connected reason: This is trivial. - - property_id: skeletal + - property: skeletal reason: There is just one object. - - property_id: generator + - property: generator reason: The unique object is a generator for trivial reasons. - - property_id: countable + - property: countable reason: This is because $G$ is countable. unsatisfied_properties: - - property_id: locally finite + - property: locally finite reason: This is because we choose $G$ to be infinite. special_objects: {} diff --git a/databases/catdat/data/categories/BG_f.yaml b/databases/catdat/data/categories/BG_f.yaml index 53508d69..74d3f75f 100644 --- a/databases/catdat/data/categories/BG_f.yaml +++ b/databases/catdat/data/categories/BG_f.yaml @@ -17,26 +17,26 @@ related_categories: - BN satisfied_properties: - - property_id: finite + - property: finite reason: This is trivial. - - property_id: small + - property: small reason: This is trivial. - - property_id: groupoid + - property: groupoid reason: This is trivial. - - property_id: connected + - property: connected reason: This is trivial. - - property_id: skeletal + - property: skeletal reason: There is just one object. - - property_id: generator + - property: generator reason: The unique object is a generator for trivial reasons. unsatisfied_properties: - - property_id: trivial + - property: trivial reason: This is trivial. special_objects: {} diff --git a/databases/catdat/data/categories/BN.yaml b/databases/catdat/data/categories/BN.yaml index f5b7a565..5e895b3b 100644 --- a/databases/catdat/data/categories/BN.yaml +++ b/databases/catdat/data/categories/BN.yaml @@ -16,38 +16,38 @@ related_categories: - BOn satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: countable + - property: countable reason: This is trivial. - - property_id: strongly connected + - property: strongly connected reason: This is trivial. - - property_id: self-dual + - property: self-dual reason: The identity is a self-duality since the addition is commutative. - - property_id: generator + - property: generator reason: The unique object is a generator for trivial reasons. - - property_id: left cancellative + - property: left cancellative reason: This is because addition of natural numbers is cancellative. - - property_id: gaunt + - property: gaunt reason: This is because $0$ is the only natural number with an additive inverse. - - property_id: locally cartesian closed + - property: locally cartesian closed reason: The slice category $B\IN / *$ is isomorphic to the poset $(\IN,\geq)$ (not to $(\IN,\leq)$). This category is thin and and semi-strongly connected, hence cartesian closed. unsatisfied_properties: - - property_id: one-way + - property: one-way reason: This is trivial. - - property_id: locally finite + - property: locally finite reason: This is trivial. - - property_id: sequential limits + - property: sequential limits reason: Assume that the sequence $\cdots \xrightarrow{1} \bullet \xrightarrow{1} \bullet \xrightarrow{1} \bullet$ has a limit. This is a (universal) sequence of natural numbers $n_0,n_1,\dotsc$ satisfying $n_i = n_{i+1} + 1$. But then $n_i = n_0 - i$, and in particular $n_{n_0 + 1} = - 1$, a contradiction. special_objects: {} diff --git a/databases/catdat/data/categories/BOn.yaml b/databases/catdat/data/categories/BOn.yaml index 6e88a0bf..ea77a211 100644 --- a/databases/catdat/data/categories/BOn.yaml +++ b/databases/catdat/data/categories/BOn.yaml @@ -14,56 +14,56 @@ related_categories: - BN satisfied_properties: - - property_id: generator + - property: generator reason: There is just one object. - - property_id: cogenerator + - property: cogenerator reason: There is just one object. - - property_id: strongly connected + - property: strongly connected reason: This is trivial. - - property_id: gaunt + - property: gaunt reason: This is because $0$ is the only ordinal number with an additive inverse. - - property_id: left cancellative + - property: left cancellative reason: It is well-known that ordinal addition satisfies $\alpha + \beta = \alpha + \gamma \implies \beta = \gamma$. - - property_id: well-copowered + - property: well-copowered reason: This follows from the description of epimorphisms as finite ordinals, see MO/5029605. - - property_id: equalizers + - property: equalizers reason: See MSE/5029668. - - property_id: cofiltered limits + - property: cofiltered limits reason: See MSE/5129138. - - property_id: locally cartesian closed + - property: locally cartesian closed reason: The slice category $B\On / *$ is isomorphic to the poset $(\On,\geq)$ (not to $(\On,\leq)$). This category is thin and and semi-strongly connected, hence cartesian closed. unsatisfied_properties: - - property_id: initial object + - property: initial object reason: This is trivial. - - property_id: one-way + - property: one-way reason: This is trivial. - - property_id: locally essentially small + - property: locally essentially small reason: This is because $\On$ is large. - - property_id: balanced + - property: balanced reason: Every finite ordinal is both a mono- and an epimorphism (see below), but only $0$ is an isomorphism. - - property_id: well-powered + - property: well-powered reason: This is because all ordinals are monomorphisms (see below) and they do not form a set. - - property_id: sequential colimits + - property: sequential colimits reason: Assume that the sequence $\bullet \xrightarrow{1} \bullet \xrightarrow{1} \cdots$ has a colimit. This mounts to a (universal) sequence of ordinals $\alpha_n$ with $\alpha_n = \alpha_{n+1} + 1$. But then $\alpha_{n+1} < \alpha_n$, contradicting the fact that $\alpha_0$ is well-ordered. - - property_id: pushouts + - property: pushouts reason: Assume that $1,\omega$ have a pushout. This is a (universal) pair of ordinals $\alpha,\beta$ with $\alpha + 1 = \beta + \omega$. But $\beta + \omega$ is a limit ordinal, while $\alpha + 1$ is not. - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: The epimorphisms are the finite ordinals (see below), but the limit of the sequential diagram $\cdots \xrightarrow{1} * \xrightarrow{1} *$ is the ordinal $\omega$ by MSE/5129138. special_objects: {} diff --git a/databases/catdat/data/categories/Ban.yaml b/databases/catdat/data/categories/Ban.yaml index 3ef06ebe..e9772630 100644 --- a/databases/catdat/data/categories/Ban.yaml +++ b/databases/catdat/data/categories/Ban.yaml @@ -13,51 +13,51 @@ related_categories: - Met satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Ban \to \Set$ and $\Set$ is locally small. - - property_id: pointed + - property: pointed reason: The trivial Banach space $\{0\}$ is a zero object. check_redundancy: false - - property_id: cogenerator + - property: cogenerator reason: The Hahn-Banach theorem implies that $\IC$ is a cogenerator. - - property_id: CIP + - property: CIP reason: This is immediate from the concrete description of coproducts and products. - - property_id: locally ℵ₁-presentable + - property: locally ℵ₁-presentable reason: Example 1.48 in Adamek-Rosicky. - - property_id: cartesian filtered colimits + - property: cartesian filtered colimits reason: If $X$ is a Banach space and $(Y_i)$ is a filtered diagram of Banach spaces, the canonical map $\colim_i (X \times Y_i) \to X \times \colim_i Y_i$ is the completion of the canonical map in the category of normed vector spaces with non-expansive linear maps. Now the claim follows directly from $\Met$. - - property_id: cocartesian cofiltered limits + - property: cocartesian cofiltered limits reason: 'If $X$ is a Banach space and $(Y_i)$ is a cofiltered diagram of Banach spaces, the canonical map $X \oplus \lim_i Y_i \to \lim_i (X \oplus Y_i)$ is an isomorphism: Since the forgetful functor $\Ban \to \Vect$ preserves finite coproducts and all limits, and $\Vect$ has the claimed property (see here), the canonical map is bijective. It remains to show that it is isometric. For $(x,y) \in X \oplus \lim_i Y_i$ the norm in the domain is $|x| + \sup_i |y_i|$, and the norm in the codomain is $\sup_i (|x| + |y_i|)$, and these clearly agree.' unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: The linear map $\IC \to \IC$, $x \mapsto x/2$ is a counterexample. It is bijective, hence a mono- and epimorphism, but not isometric and therefore no isomorphism. - - property_id: CSP + - property: CSP reason: By using the concrete description of products and coproducts, for the constant family $X_n = \IC$ the canonical morphism $\coprod_n X_n \to \prod_n X_n$ becomes the canonical inclusion map $\ell^1 \hookrightarrow \ell^\infty$. This is not an epimorphism (i.e., has no dense image) since the closure of the image is precisely $c_0$. So for example, $(1,1,\dotsc)$ is not contained in the closure of the image. - - property_id: regular subobject classifier + - property: regular subobject classifier reason: If $\Omega$ is a regular subobject classifier, then by the classification of regular monomorphisms, $\Hom(X,\Omega)$ is isomorphic to the set of closed subspaces of $X$ for any Banach space $X$. For $X = \IC$ this implies that there are exactly two vectors in $\Omega$ with norm $\leq 1$, which is absurd. (For $\Omega = 0$ there is just one, and for $\Omega \neq 0$ there are infinitely many.) - - property_id: unital + - property: unital reason: See MSE/5033161. - - property_id: co-Malcev + - property: co-Malcev reason: See the comments to MO/509552. - - property_id: filtered-colimit-stable monomorphisms + - property: filtered-colimit-stable monomorphisms reason: 'The proof is similar to $\Met$. For $n \geq 1$ let $V_n$ be the Banach space with underlying vector space $\IC$ and the norm $|x|_n := \frac{1}{n} |x|$. For $n \leq m$ the identity map provides a morphism $V_n \to V_m$, which is clearly a monomorphism (also an epimorphism by the way, but an isomorphism iff $n=m$). Let $V$ be the colimit of all $V_n$ in the category of semi-normed vector spaces. It is constructed as the colimit in the category of vector spaces with the semi-norm $|x| := \inf \{|x|_m : n \leq m \}$ for $x \in V_n$. So clearly, the semi-norm is zero. Hence, the colimit in the category of normed vector spaces is $0$. The colimit in the category of Banach spaces is its completion, also $0$. Thus, the monomorphisms $V_1 \to V_n$ become $V_1 \to 0$ in the colimit.' - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: 'We show that epimorphisms are not stable under sequential limits. Let $X_n = Y_n = \IC$ for all $n \geq 0$. The transition morphism $Y_{n+1} \to Y_n$ is the identity, and the transition morphism $X_{n+1} \to X_n$ is $x \mapsto x/2$. The morphisms $X_n \to Y_n$, $x \mapsto x/2^n$ are compatible with the transitions, and they are surjective, hence epimorphisms. Now we check $\lim_n X_n = 0$: An element $(x_n) \in \lim_n X_n$ is a family of complex numbers satisfying $x_n = x_{n+1}/2$ and $\sup_n |x_n| < \infty$. But then $x_n = 2^n x_0$ and this can only be bounded when $x_0=0$. Hence, $0 = \lim_n X_n \to \lim_n Y_n = \IC$ is no epimorphism.' special_objects: diff --git a/databases/catdat/data/categories/CAlg(R).yaml b/databases/catdat/data/categories/CAlg(R).yaml index c8411f09..101ecd8a 100644 --- a/databases/catdat/data/categories/CAlg(R).yaml +++ b/databases/catdat/data/categories/CAlg(R).yaml @@ -15,48 +15,48 @@ related_categories: - R-Mod satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\CAlg(R) \to \Set$ and $\Set$ is locally small. - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic theory of a commutative ring. - - property_id: strict terminal object + - property: strict terminal object reason: 'If $f : 0 \to R$ is a homomorphism, then $R$ satisfies $1=f(1)=f(0)=0$, so that $R=0$.' check_redundancy: false - - property_id: Malcev + - property: Malcev reason: This follows in the same way as for $\Grp$, see also Example 2.2.5 in Malcev, protomodular, homological and semi-abelian categories. - - property_id: coextensive + - property: coextensive reason: One can use the same proof as for $\CRing$. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: Take a prime ideal $P \subseteq R$ and consider the commutative $R$-algebra $A := R/P$ (which is an integral domain). Then the inclusion $A \hookrightarrow Q(A)$ is a counterexample. - - property_id: cogenerating set + - property: cogenerating set reason: 'We apply this lemma to the collection of commutative $R$-algebras which are fields: If $F$ is a commutative $R$-algebra that is also a field and $A$ is a non-trivial commutative $R$-algebra, any algebra homomorphism $F \to A$ is injective. For every infinite cardinal $\kappa$ the field of rational functions in $\kappa$ variables over some residue field of $R$ has cardinality $\geq \kappa$ and a non-trivial automorphism (swap two variables).' - - property_id: countably codistributive + - property: countably codistributive reason: 'The canonical homomorphism $A \otimes_R R^{\IN} \to A^{\IN}$ is given by $a \otimes (r_n)_n \mapsto (r_n a)_n$ and does not have to be surjective: Since $R \neq 0$, there is a commutative $R$-algebra $K$ which is a field. Now take $A := K[X]$ and consider the sequence $(X^n)_{n} \in A^{\IN}$.' - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Choose a maximal ideal $\mathfrak{m}$ of $R$, so $K := R/\mathfrak{m}$ is a field. If $\CAlg(R)$ is semi-strongly connected, then also $\CAlg(K)$ is semi-strongly connected. This has been disproven in MSE/5129689. - - property_id: coregular + - property: coregular reason: See MSE/3745302. - - property_id: co-Malcev + - property: co-Malcev reason: 'See MO/509552: Consider the forgetful functor $U : \CAlg(R) \to \Set$ and the relation $S \subseteq U^2$ defined by $S(A) := \{(a,b) \in U(A)^2 : ab = a^2\}$. Both are representable: $U$ by $R[X]$ and $S$ by $R[X,Y] / \langle XY-X^2 \rangle$. It is clear that $S$ is reflexive, but not symmetric.' - - property_id: regular quotient object classifier + - property: regular quotient object classifier reason: 'The strategy is similar to the one for $\CRing$: Assume that $P \to R$ is a regular quotient object classifier. If $J$ denotes the kernel of $P \to R$, every ideal $I \subseteq A$ of any commutative $R$-algebra has the form $I = \langle \varphi(J) \rangle$ for a unique homomorphism $\varphi : P \to A$. If $\sigma : A \to A$ is an automorphism with $\sigma(I)=I$, then uniqueness gives us $\sigma \circ \varphi = \varphi$, which means that $\varphi(J)$ lies in $A^{\sigma}$, the fixed algebra of $\sigma$. But then $I$ is generated by elements in $A^{\sigma} \cap I$. If $K$ is a residue field of $R$, this fails for $A = K[X,Y]$, $I = \langle X,Y \rangle$, $\sigma(X)=Y$, $\sigma(Y)=X$. The fixed algebra is the subalgebra of symmetric polynomials, which is $K[X+Y,XY]$. So $\langle X,Y \rangle$ is generated by symmetric polynomials without constant term, which implies $\langle X,Y \rangle \subseteq \langle X+Y,XY \rangle$ in $K[X,Y]$. But reducing an equation like $X = a(X,Y) \cdot (X+Y) + b(X,Y) \cdot (XY)$ modulo $\langle X^2,Y^2,XY \rangle$ yields a contradiction.' - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: Let $K$ be a field over $R$. Consider the sequence of projections $\cdots \to K[X]/\langle X^2 \rangle \to K[X]/\langle X \rangle$ and the constant sequence $\cdots \to K[X] \to K[X]$. The surjective homomorphisms $K[X] \to K[X]/\langle X^n \rangle$ induce the inclusion $K[X] \hookrightarrow K[[X]]$ in the limit, where $K[[X]]$ is the algebra of formal power series. It is clearly not surjective, but this is not sufficient, we need to argue that it is not an epimorphism in $\CAlg(R)$, or equivalently, in $\CRing$. For a proof, see MSE/2391187. special_objects: diff --git a/databases/catdat/data/categories/CMon.yaml b/databases/catdat/data/categories/CMon.yaml index d08d869e..4f61787d 100644 --- a/databases/catdat/data/categories/CMon.yaml +++ b/databases/catdat/data/categories/CMon.yaml @@ -15,45 +15,45 @@ related_categories: - Mon satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\CMon \to \Set$ and $\Set$ is locally small. - - property_id: pointed + - property: pointed reason: The trivial monoid is a zero object. check_redundancy: false - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic of a commutative monoid. - - property_id: biproducts + - property: biproducts reason: This follows from the explicit construction of coproducts and products. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: The inclusion of additive monoids $\IN \hookrightarrow \IZ$ is a counterexample. - - property_id: Malcev + - property: Malcev reason: 'Consider the submonoid $\{(a,b) : a \leq b \}$ of $\IN^2$.' - - property_id: co-Malcev + - property: co-Malcev reason: 'See MO/509552: Consider the forgetful functor $U : \CMon \to \Set$ and the relation $R \subseteq U^2$ defined by $R(A) := \{(a,b) \in U(A)^2 : ab = a^2\}$. Both are representable: $U$ by the free monoid on a single generator and $R$ by the free commutative monoid on two generators $x,y$ subject to the relation $xy=x^2$. It is clear that $R$ is reflexive, but not symmetric.' - - property_id: cogenerator + - property: cogenerator reason: See MO/509232. - - property_id: coregular + - property: coregular reason: 'We can show this analogously to the case of commutative rings MSE/3746890. Consider the commutative monoid $\IN^2$ and its submonoid $U\coloneqq\{(m,n)\mid m\ge n\}$ with the inclusion $i\colon U\hookrightarrow\IN^2$. Then, the pushout of $i$ along itself is $\langle x,y,z : x+y=x+z \rangle$, and the equalizer of the cokernel pair of $i$ is $D\coloneqq\{(m,n)\mid m=0 \implies n=0 \}$. If the category $\CMon$ were coregular, the canonical inclusion $j\colon U\hookrightarrow D$ would have to be an epimorphism. However, it is not: let $I\coloneqq\{0,1\}$ be the two-element commutative monoid with $1+1=1$, and let $u,v\colon D \rightrightarrows I$ be the morphisms defined by $u^{-1}(0)=\{(0,0)\}$ and $v^{-1}(0)=\{(0,0),(1,2)\}$; then we have $u\circ j = v\circ j$.' - - property_id: regular subobject classifier + - property: regular subobject classifier reason: We can use exactly the same proof as for $\Mon$. - - property_id: regular quotient object classifier + - property: regular quotient object classifier reason: 'If $P \in \CMon$ is a regular quotient object classifier, this means that every surjective homomorphism of commutative monoids $A \to B$ is the cokernel of a unique homomorphism $P \to A$. But there are many surjective homomorphisms which are no cokernels at all: Consider the Boolean monoid $(\{0,1\},\vee)$ with $1 \vee 1 = 1$ and the surjective homomorphism $f : (\IN,+) \to (\{0,1\},\vee)$ defined by $f(0)=0$ and $f(n)=1$ for $n \geq 1$. It has trivial kernel, but is no isomorphism, so it cannot be a cokernel.' - - property_id: CSP + - property: CSP reason: First of all, epimorphisms in $\CMon$ are preserved and reflected by the forgetful functor to $\Mon$ (see below). Furthermore, if $M \to N$ is an epimorphism in $\Mon$ and $M$ is infinite, then $\card(N) \leq \card(M)$ (see MO/510431). This implies that in $\CMon$ the canonical homomorphism $\bigoplus_{n \geq 0} \IN \to \prod_{n \geq 0} \IN$ is not an epimorphism because its domain is countable and its codomain is uncountable. special_objects: diff --git a/databases/catdat/data/categories/CRing.yaml b/databases/catdat/data/categories/CRing.yaml index 5e8c1422..be5b145e 100644 --- a/databases/catdat/data/categories/CRing.yaml +++ b/databases/catdat/data/categories/CRing.yaml @@ -18,48 +18,48 @@ comments: - Regular monomorphisms are discussed in MSE/695685, but probably they cannot be classified. satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\CRing \to \Set$ and $\Set$ is locally small. - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic theory of a commutative ring. - - property_id: strict terminal object + - property: strict terminal object reason: 'If $f : 0 \to R$ is a homomorphism, then $R$ satisfies $1=f(1)=f(0)=0$, so that $R=0$.' check_redundancy: false - - property_id: Malcev + - property: Malcev reason: This follows in the same way as for $\Grp$, see also Example 2.2.5 in Malcev, protomodular, homological and semi-abelian categories. - - property_id: coextensive + - property: coextensive reason: '[Sketch] A ring homomorphism $f : A \times B \to R$ yields the idempotent element $e := f(1,0) \in R$, so that $R \cong eR \times (1-e)R$. Then $f$ decomposes into the ring homomorphisms $f_A : A \to eR$, $f_A(a) := f(a,0)$ and $f_B : B \to (1-e)R$, $f_B(b) := f(0,b)$.' unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: There is no homomorphism between $\IF_2$ and $\IF_3$. - - property_id: balanced + - property: balanced reason: The inclusion $\IZ \hookrightarrow \IQ$ is a counterexample. - - property_id: cogenerating set + - property: cogenerating set reason: 'We apply this lemma to the collection of fields: If $F$ is a field and $R$ is a non-trivial commutative ring, any ring homomorphism $F \to R$ is injective. For every infinite cardinal $\kappa$ the field of rational functions in $\kappa$ variables has cardinality $\geq \kappa$ and a non-trivial automorphism (swap two variables).' - - property_id: countably codistributive + - property: countably codistributive reason: 'The canonical homomorphism $\IQ \otimes \IZ^{\IN} \to (\IQ \otimes \IZ)^{\IN} = \IQ^{\IN}$ is not an isomorphism: its image consists of those sequences of rational numbers whose denominators can be bounded.' - - property_id: coregular + - property: coregular reason: See MSE/3745302. - - property_id: co-Malcev + - property: co-Malcev reason: 'See MO/509552: Consider the forgetful functor $U : \CRing \to \Set$ and the relation $R \subseteq U^2$ defined by $R(A) := \{(a,b) \in U(A)^2 : ab = a^2\}$. Both are representable: $U$ by $\IZ[X]$ and $R$ by $\IZ[X,Y] / \langle XY-X^2 \rangle$. It is clear that $R$ is reflexive, but not symmetric.' - - property_id: regular quotient object classifier + - property: regular quotient object classifier reason: 'Assume that $P \to \IZ$ is a regular quotient object classifier. If $J$ denotes its kernel, this means that every ideal $I \subseteq A$ of any commutative ring has the form $I = \langle \varphi(J) \rangle$ for a unique homomorphism $\varphi : P \to A$. If $\sigma : A \to A$ is an automorphism with $\sigma(I)=I$, then uniqueness gives us $\sigma \circ \varphi = \varphi$, which means that $\varphi(J)$ lies in $A^{\sigma}$, the fixed ring of $\sigma$. But then $I$ is generated by elements in the fixed ring. This fails for $A = \IZ[X]$, $I = \langle X \rangle$, $\sigma(X)=-X$. The fixed ring is $\IZ[X^2]$, and if $I$ was generated by elements $f \in \IZ[X^2] \cap I$, they would be multiples of $X^2$, but $X$ is not a multiple of $X^2$.' - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: 'For a prime $p$ consider the sequence of projections $\cdots \to \IZ/p^2 \to \IZ/p$ and the constant sequence $\cdots \to \IZ \to \IZ$. The surjective homomorphisms $\IZ \to \IZ/p^n$ induce the homomorphism $\IZ \to \IZ_p$ in the limit, where $\IZ_p$ is the ring of $p$-adic integers. It is not surjective since $\IZ_p$ is uncountable, but this is not sufficient (at least, for this category): We need to use SP/04W0 to conclude that it is no epimorphism in $\CRing$.' special_objects: diff --git a/databases/catdat/data/categories/Cat.yaml b/databases/catdat/data/categories/Cat.yaml index f04c7b04..bab8f2af 100644 --- a/databases/catdat/data/categories/Cat.yaml +++ b/databases/catdat/data/categories/Cat.yaml @@ -15,50 +15,50 @@ related_categories: - Set satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Cat \to \Set \times \Set$, $\C \mapsto (\Ob(\C),\Mor(\C))$, and $\Set \times \Set$ is locally small. - - property_id: cartesian closed + - property: cartesian closed reason: See p. 98 in Mac Lane. - - property_id: locally finitely presentable + - property: locally finitely presentable reason: See MO/84460. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Every non-empty category is weakly terminal (by using constant functors). - - property_id: generator + - property: generator reason: 'The interval category $\{0 \to 1\}$ is a generator: Assume that $F,G : \C \rightrightarrows \D$ are functors that agree when being precomposed with any functor from $\{0 \to 1\}$. This means that $F(f) = G(f)$ for all morphisms $f : X \to Y$ in $\C$. By comparing the domains and applying this to $f = \id_X$, we see that $F(X) = G(X)$ for all objects $X$. And we just saw that $F,G$ also agree on morphisms.' - - property_id: infinitary extensive + - property: infinitary extensive reason: '[Sketch] This is straight forward from the fact that $\Set$ is infinitary extensive: A functor $\C \to \coprod_i \D_i$ yields full subcategories $\C_i \subseteq \C$ (the preimages of $\D_i)$ with $\C = \coprod_i \C_i$.' unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: Since we know that $\Mon$ is not balanced, there is a monoid map $M \to N$ which is a monomorphism and an epimorphism which is not an isomorphism. Then $B(M) \to B(N)$ has the corresponding properties. - - property_id: cogenerating set + - property: cogenerating set reason: 'Assume that $S$ is a cogenerating set in $\Cat$. Then one checks that the set of monoids $\{\End(X) : X \in \C \in S\}$ is a cogenerating set in $\Mon$, which we know does not exist.' - - property_id: regular + - property: regular reason: See Example 3.14 at the nLab. - - property_id: coregular + - property: coregular reason: 'We already know that $\Mon$ is not coregular, in fact there is a regular monomorphism $M \to N$ of monoids and a morphism $M \to K$ such that $K \to K \sqcup_M N$ is not a monomorphism. The delooping functor $B : \Mon \to \Cat$ has a left adjoint (MSE/574745), hence it preserves regular monomorphisms. It also preserves pushouts (MSE/5130854), and it reflects monomorphisms since it is faithful. Therefore, $B(M) \to B(N)$ provides the desired counterexample of a non-stable regular monomorphism of categories.' - - property_id: Malcev + - property: Malcev reason: Use that $\Set$ is not Malcev and consider sets as discrete categories. - - property_id: co-Malcev + - property: co-Malcev reason: 'We can adapt the proof from $\Mon$ as follows: Consider the functor $U : \Cat \to \Set^+$ sending a category $\C$ to the (large) set $\{(x,u) : x \in \Ob(\C) ,\, u \in \End(x) \}$. It is represented by $B \IN$, the one-object category associated to the free monoid in one generator. Consider the relation $R \subseteq U^2$ consisting of those pairs $((x,u),(y,v))$ where $x = y$ and $uv = u^2$. This also representable, namely be the one-object category associated to the monoid with the presentation $\langle u,v : uv = u^2 \rangle$. Clearly, $R$ is reflexive, but not symmetric.' - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: We already know that $\Set$ does not have this property. Now apply the contrapositive of the dual of this lemma to the functor $\Set \to \Cat$ that maps a set to its discrete category. - - property_id: effective cocongruences + - property: effective cocongruences reason: >- The counterexample is similar to the one for $\Mon$: Let $X$ be the walking idempotent, and let $E$ be the delooping of the monoid with presentation $$\langle p, q \mid p^2=p,\, q^2=q,\, pq=q,\, qp=p \rangle.$$ diff --git a/databases/catdat/data/categories/CompHaus.yaml b/databases/catdat/data/categories/CompHaus.yaml index fcbde4b2..0ee0c8e6 100644 --- a/databases/catdat/data/categories/CompHaus.yaml +++ b/databases/catdat/data/categories/CompHaus.yaml @@ -14,39 +14,39 @@ related_categories: - Top satisfied_properties: - - property_id: locally small + - property: locally small reason: It is a full subcategory of $\Top$, which is locally small. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: This is already true for $\Top$. - - property_id: products + - property: products reason: By the Tychonoff product theorem, a product in $\Top$ of compact Hausdorff spaces is compact; it is also clearly Hausdorff. Since the forgetful functor from $\CompHaus$ to $\Top$ is fully faithful, this limit is reflected in $\CompHaus$ as well. check_redundancy: false - - property_id: equalizers + - property: equalizers reason: 'The equalizer in $\Top$ of two continuous functions $f, g : X \rightrightarrows Y$ between compact Hausdorff spaces is a closed subspace of $X$, and therefore it is also compact Hausdorff. Since the forgetful functor from $\CompHaus$ to $\Top$ is fully faithful, this limit is reflected in $\CompHaus$ as well.' check_redundancy: false - - property_id: cocomplete + - property: cocomplete reason: $\CompHaus$ is a reflective subcategory of $\Top$, with the reflector being the Stone-Čech compactification functor. See nLab for example. Therefore, as usual, we can form colimits in $\CompHaus$ by forming colimits in $\Top$ and then applying Stone-Čech compactification. check_redundancy: false - - property_id: generator + - property: generator reason: The one-point space is a generator because it represents the forgetful functor to $\Set$, which is faithful. - - property_id: cogenerator + - property: cogenerator reason: 'The unit interval $[0, 1]$ is a cogenerator: Suppose we have $f, g : X \rightrightarrows Y$ with $f \ne g$. Choose $x\in X$ such that $f(x) \ne g(x)$. Then by Urysohn''s lemma, there is a continuous function $h : Y \to [0, 1]$ such that $h(f(x)) = 0$ and $h(g(x)) = 1$. Therefore, $h\circ f \ne h\circ g$.' - - property_id: effective congruences + - property: effective congruences # TODO: rework this when Barr-exact is added reason: The forgetful functor from $\CompHaus$ to $\Set$ is monadic; see for example nLab. Therefore, by this result, $\CompHaus$ is Barr-exact, and in particular it has effective congruences. - - property_id: regular + - property: regular # TODO: rework this when Barr-exact is added reason: The forgetful functor from $\CompHaus$ to $\Set$ is monadic; see for example nLab. Therefore, by this result, $\CompHaus$ is Barr-exact and in particular is regular. - - property_id: coregular + - property: coregular reason: 'It suffices to show that pushouts preserve (regular) monomorphisms in $\CompHaus$. Thus, suppose we have a pushout square $$\begin{CD} @@ -56,31 +56,31 @@ satisfied_properties: \end{CD}$$ with $i : A \hookrightarrow B$ a monomorphism. Then for any pair of distinct elements $c, c'' \in C$, by Urysohn''s lemma there exists $\gamma : C \to [0, 1]$ with $\gamma(c) = 0$ and $\gamma(c'') = 1$. Also, by Tietze''s extension theorem, there exists $\beta : B \to [0, 1]$ such that $\beta \circ i = \gamma \circ f$. By the pushout property, there is a unique $\delta : D \to [0, 1]$ such that $\delta \circ g = \beta$ and $\delta \circ j = \gamma$. Since $\delta(j(c)) \ne \delta(j(c''))$, we conclude that $j(c) \ne j(c'')$. This shows that $j$ is injective, so it is a regular monomorphism.' - - property_id: extensive + - property: extensive reason: This follows as for $\Top$ or $\Haus$ since finite coproducts in $\CompHaus$ are formed as disjoint union spaces with the disjoint union topology. - - property_id: epi-regular + - property: epi-regular reason: |- First, any epimorphism $f : X \to Y$ is surjective: if not, its image would be a proper subset of $Y$, which is compact and hence closed. Then by Urysohn's lemma, there would be a non-zero continuous function $g : Y \to [0, 1]$ which is $0$ on the image; but then $g \circ f = 0 \circ f$, giving a contradiction. Now the identity morphism from $Y$, with the quotient topology of $f$, to $Y$ with its given topology is a bijective continuous function between compact Hausdorff spaces, so it is a homeomorphism. In other words, $f$ is a quotient map. Therefore, we see that if $g, h : E \rightrightarrows X$ is the kernel pair of $f$, and $U : \CompHaus \to \Top$ is the forgetful functor, then $U(f)$ is the coequalizer of $U(g)$ and $U(h)$. Since $U$ is fully faithful, that implies $f$ is the coequalizer of $g$ and $h$. - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: 'Suppose we have a cofiltered diagram of epimorphisms $(f_i : X_i \to Y_i)$, and $y = (y_i) \in \lim_i Y_i$. Then by lemma 1 here, the limit of $f_i^{-1}(\{ y_i \})$ is non-empty. If $x$ is in this limit, that implies that $(\lim_i f_i)(x) = y$.' - - property_id: locally copresentable + - property: locally copresentable reason: A proof can be found here. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: Malcev + - property: Malcev reason: This is clear since $\FinSet$ is not Malcev and can be interpreted as the subcategory of finite discrete spaces. - - property_id: regular subobject classifier + - property: regular subobject classifier reason: The proof is almost identical to the one for $\Haus$. - - property_id: natural numbers object + - property: natural numbers object reason: >- Let $I := [0, 1]$. If a natural numbers object $(N, z : 1 \to N, s : N \to N)$ existed, then we could iterate the initial conditions $I\to I\times I$, $x \mapsto (x, x)$ and the recursive step function $I\times I \to I \times I$, $(x, y) \mapsto (x, xy)$ to get a continuous function $N \times I \to I \times I$ such that $(s^n(z), x) \mapsto (x, x^n)$ for $x\in I$, $n \in \IN$. The sequence $(s^n(z)) \in N$ has a convergent subnet $(s^{n_\lambda}(z))_{\lambda \in \Lambda}$, say with limit $y$. Thus, for any $x\in I$ and $\lambda \in \Lambda$, we have $(s^{n_\lambda}(z), x) \mapsto (x, x^{n_\lambda})$. Taking limits, we see $(y, x) \mapsto (x, 0)$ if $x \ne 1$ or $(y, x) \mapsto (x, 1)$ if $x = 1$. In other words, $(y, x) \mapsto (x, \delta_{x, 1})$ for all $x\in I$. However, that contradicts the fact that the composition $$\begin{align*} @@ -89,10 +89,10 @@ unsatisfied_properties: \end{align*}$$ would have to be continuous. - - property_id: filtered-colimit-stable monomorphisms + - property: filtered-colimit-stable monomorphisms reason: 'The proof is similar to $\Haus$. For $n \geq 1$ let $X_n$ be the pushout of $[1/n, 1] \hookrightarrow [0, 1]$ with itself. That is, $X_n$ is the union of two unit intervals $[0, 1] \times \{ 1 \}$ and $[0, 1] \times \{ 2 \}$ where we identify $(x,1) \equiv (x,2)$ when $x \geq 1/n$. As in the construction for $\Haus$, we see that the colimit in $\Haus$ is $[0, 1]$ where all corresponding points of both unit intervals are identified. Since this is compact Hausdorff, it also provides the colimit in $\CompHaus$. Again, the injective continuous maps $\{1,2\} \to X_n$, $i \mapsto (0,i)$ (where $\{1,2\}$ is discrete) become the constant map $0 : \{1,2\} \to [0,1]$ in the colimit, which is not a monomorphism.' - - property_id: exact cofiltered limits + - property: exact cofiltered limits reason: |- Consider the $\IN$-codirected systems $X_n := [0, 1] \times [0, 1/n]$ with the maps $X_{n+1} \to X_n$ being inclusion maps, and $Y_n := [0, 1+1/n]$ with the maps $Y_{n+1} \to Y_n$ also being inclusion maps. We define $f_n : X_n \to Y_n$, $(x, y) \mapsto x$ and $g_n : X_n \to Y_n$, $(x, y) \mapsto x+y$. It is straightforward to check these give morphisms of $\IN$-codirected systems in $\CompHaus$. Now for each $n$, we claim the coequalizer of $f_n$ and $g_n$ is a singleton space. To see this, we prove the more general result that for $r, s > 0$ the coequalizer of $f, g : [0, r] \times [0, s] \rightrightarrows [0, r+s]$, $f(x,y) = x$, $g(x,y) = x+y$ is a singleton. We must show that for any $h : [0, r+s] \to T$ with $h\circ f = h\circ g$, then $h$ is constant. To this end, we show by induction on $n$ that whenever $x \in [0, r+s]$ and $x \le ns$, we have $h(x) = h(0)$. The base case $n=0$ is trivial. For the inductive step, if $x \le s$, then $f(0,x) = 0$ and $g(0,x) = x$, so $h(0) = h(x)$. Otherwise, we have $x-s \in [0,r]$ and $x-s \le (n-1)s$, so by inductive hypothesis $h(x-s) = h(0)$. Also, $f(x-s, s) = x-s$ and $g(x-s, s) = x$, so $h(x-s) = h(x)$, completing the induction. With this established, the desired result follows from the case $n := \lceil r/s \rceil + 1$. diff --git a/databases/catdat/data/categories/Delta.yaml b/databases/catdat/data/categories/Delta.yaml index 25517604..4e8df641 100644 --- a/databases/catdat/data/categories/Delta.yaml +++ b/databases/catdat/data/categories/Delta.yaml @@ -18,43 +18,43 @@ related_categories: - walking_coreflexive_pair satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: locally finite + - property: locally finite reason: There is a faithful functor $\Delta \to \FinSet$ and $\FinSet$ is locally finite. - - property_id: countable + - property: countable reason: This is obvious. - - property_id: terminal object + - property: terminal object reason: The ordered set $[0] = \{0\}$ is terminal. - - property_id: strongly connected + - property: strongly connected reason: For all $n,m$ there are morphisms $[n] \to [0] \to [m]$. - - property_id: generator + - property: generator reason: The ordered set $[0] = \{0\}$ is a generator. - - property_id: cogenerator + - property: cogenerator reason: The ordered set $[1] = \{0 < 1\}$ is a cogenerator, even for $\Pos$. - - property_id: skeletal + - property: skeletal reason: 'If $f : [n] \to [m]$ is an isomorphism, then $n + 1 = m + 1$ by comparing the cardinalities, hence $n = m$.' - - property_id: coequalizers + - property: coequalizers reason: Assume that $X \rightrightarrows Y$ are morphisms in $\FinOrd \setminus \{\varnothing\}$. Since $\FinOrd$ has coequalizers, we have a coequalizer $Y \to Q$. Since $Y$ is non-empty, $Q$ is non-empty as well, and clearly $Y \to Q$ is then also the coequalizer in $\FinOrd \setminus \{\varnothing\}$. - - property_id: core-thin + - property: core-thin reason: The category $\FinOrd \setminus \{\varnothing\}$ is core-thin because already $\FinOrd$ is core-thin. - - property_id: mono-regular + - property: mono-regular reason: The proof for $\FinOrd$ also works for $\FinSet \setminus \{\varnothing\}$. - - property_id: epi-regular + - property: epi-regular reason: The proof for $\FinOrd$ also works for $\FinSet \setminus \{\varnothing\}$. - - property_id: cosifted + - property: cosifted reason: >- Let $X,Y \in \Delta$. We may pick $x \in X$, $y \in Y$. Then there is a "point span" $X \xleftarrow{x} [0] \xrightarrow{y} Y$. Every span $X \xleftarrow{f} Z \xrightarrow{g} Y$ is connected to such a point span: Pick $z \in Z$. This defines a morphism of spans: $$\begin{CD} X @<{f(z)}<< [0] @>{g(z)}>> Y \\ @| @VV{z}V @| \\ X @<<{f}< Z @>>{g}> Y \end{CD}$$ @@ -63,22 +63,22 @@ satisfied_properties: This shows that the choice of $x \in X$ does not matter, and for $y \in Y$ the proof is the same. unsatisfied_properties: - - property_id: strict terminal object + - property: strict terminal object reason: This is trivial. - - property_id: cofiltered + - property: cofiltered reason: 'The two maps $d^0,d^1 : [0] \rightrightarrows [1]$ are not equalized by any morphism.' - - property_id: coreflexive equalizers + - property: coreflexive equalizers reason: 'The two maps $d^0,d^1 : [0] \rightrightarrows [1]$ have a common left inverse, the unique map $s^0 : [1] \to [0]$, but are not equalized by any morphism.' - - property_id: sequential colimits + - property: sequential colimits reason: We can just copy the proof for $\FinOrd$ to show that the sequence of inclusions $[0] \hookrightarrow [1] \hookrightarrow [2] \hookrightarrow \cdots$ has no colimit. - - property_id: sequential limits + - property: sequential limits reason: We can just copy the proof for $\FinOrd$ to show that the sequence of truncations $\cdots \twoheadrightarrow [2] \twoheadrightarrow [1] \twoheadrightarrow [0]$ has no limit. - - property_id: pushouts + - property: pushouts reason: Assume that the two inclusions $\{0 < 1\} \leftarrow \{0\} \rightarrow \{0 < 2\}$ have a pushout in $\FinOrd \setminus \{\varnothing\}$. This would be a universal non-empty finite ordered set $X$ with three elements $0,1,2$ satisfying $0 \leq 1$ and $0 \leq 2$. Assume w.l.o.g. $1 \leq 2$ (the case $2 \leq 1$ is similar). The universal property yields an order-preserving map $X \to \{a < b < c\}$ with $0 \mapsto a$, $1 \mapsto c$, $2 \mapsto b$. But then $c \leq b$, which is a contradiction. special_objects: {} diff --git a/databases/catdat/data/categories/FI.yaml b/databases/catdat/data/categories/FI.yaml index 4c29cba5..1397b75d 100644 --- a/databases/catdat/data/categories/FI.yaml +++ b/databases/catdat/data/categories/FI.yaml @@ -16,59 +16,59 @@ related_categories: - FinSet satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\FI \to \Set$ and $\Set$ is locally small. - - property_id: locally finite + - property: locally finite reason: There is a faithful functor $\FI \to \FinSet$ and $\FinSet$ is locally finite. - - property_id: left cancellative + - property: left cancellative reason: This is trivial. - - property_id: generator + - property: generator reason: The one-point set is a generator since it represents the forgetful functor $\FI \to \Set$. - - property_id: essentially countable + - property: essentially countable reason: Every finite set is isomorphic to some $\{1,\dotsc,n\}$ for some $n \in \IN$. - - property_id: equalizers + - property: equalizers reason: We construct equalizers just like in $\FinSet$ and observe that the universal property still holds. - - property_id: wide pullbacks + - property: wide pullbacks reason: 'We construct wide pullbacks just like in $\Set$, i.e., for a w.l.o.g. non-empty family of injective maps $f_i : X_i \to S$ we consider the subset $P \subseteq \prod_{i \in I} X_i$ of those tuples $x$ where $f_i(x_i) = f_j(x_j)$. Each projection $P \to X_i$ is injective, so in particular $P$ is finite, and $P \to X_i$ becomes a morphism in $\FI$. It is easy to check that the universal property still holds in $\FI$.' - - property_id: mono-regular + - property: mono-regular reason: 'If $f : X \to Y$ is an injective map of finite sets, it is the equalizer of the two injective maps $i_1,i_2 : Y \rightrightarrows Y \sqcup_X Y$, and $Y \sqcup_X Y$ is finite.' - - property_id: semi-strongly connected + - property: semi-strongly connected reason: If $X,Y$ are two finite sets, we have $\card(X) \leq \card(Y)$ or $\card(Y) \leq \card(X)$. In the first case there will be an injection $X \to Y$, in the second case there will be an injection $Y \to X$. - - property_id: locally cartesian closed + - property: locally cartesian closed reason: IF $X$ is a finite set, the slice category $\FI / X$ is equivalent to the poset of subsets of $X$. This is cartesian closed because $A \cap B \subseteq C$ holds if and only if $B \subseteq (X \setminus A) \cup C$, where $A,B,C \subseteq X$. unsatisfied_properties: - - property_id: small + - property: small reason: Even the collection of all singletons is not small. - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: core-thin + - property: core-thin reason: Its core is $\IB$, which we know is not thin. - - property_id: strongly connected + - property: strongly connected reason: There is no map from a non-empty set to the empty set. - - property_id: countable + - property: countable reason: This is trivial. - - property_id: cogenerator + - property: cogenerator reason: Let $Q$ be finite set. When $Q$ is empty, it is clearly no cogenerator. Otherwise, $Q + 1$ has at least two elements, so that there are two different morphisms $1 \rightrightarrows Q + 1$. But there is no morphism $Q + 1 \to Q$ at all. Hence, $Q$ is no cogenerator. - - property_id: binary powers + - property: binary powers reason: Assume that two finite sets $X,Y$ have a product $P$ in this category. Elements of $P$ are the same as maps $1 \to P$, and they are automatically injective. Therefore, $P \cong \Hom(1,P) \times \Hom(1,X) \times \Hom(1,Y) \cong X \times Y$, and the projections must agree as well. But they are usually not injective. In particular, the product $X \times X$ never exists when $X$ has $>1$ elements. - - property_id: sequential colimits + - property: sequential colimits reason: 'Let $X_n := \{1,\dotsc,n\}$. Assume the sequence of inclusion maps $X_n \hookrightarrow X_{n+1}$ has a colimit $(f_n : X_n \to X)$ in this category. But $f_n$ must be an injective map, so that $\card(X) \geq n$ for all $n$. Since $X$ is finite, this is a contradiction.' special_objects: diff --git a/databases/catdat/data/categories/FS.yaml b/databases/catdat/data/categories/FS.yaml index 2405de44..c4149c70 100644 --- a/databases/catdat/data/categories/FS.yaml +++ b/databases/catdat/data/categories/FS.yaml @@ -16,62 +16,62 @@ related_categories: - FinSet satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\FS \to \Set$ and $\Set$ is locally small. - - property_id: locally finite + - property: locally finite reason: There is a faithful functor $\FS \to \FinSet$ and $\FinSet$ is locally finite. - - property_id: essentially countable + - property: essentially countable reason: Every finite set is isomorphic to some $\{1,\dotsc,n\}$ for some $n \in \IN$. - - property_id: right cancellative + - property: right cancellative reason: This is trivial. - - property_id: cogenerator + - property: cogenerator reason: 'We prove that $\{0,1\}$ is a cogenerator: The surjective maps $X \to \{0,1\}$ correspond to the non-empty proper subsets of $X$. If $a,b \in X$ are elements that have the same image under each surjective map $X \to \{0,1\}$, it therefore means that they lie in the same non-empty proper subsets of $X$. This implies $a=b$: If $X = \{a\}$, this is trivial. Otherwise, use the subset $\{a\}$.' - - property_id: coequalizers + - property: coequalizers reason: We construct coequalizers as in $\FinSet$ (or $\Set$) and observe that the universal property still holds when we restrict to surjective maps. - - property_id: wide pushouts + - property: wide pushouts reason: 'We construct wide pushouts as in $\Set$ and observe that the universal property still holds when we restrict to surjective maps. If $f_i : S \to X_i$ are surjective maps and $P$ is their wide pushout, then each $X_i \to P$ is surjective, so that in particular $P$ is finite.' - - property_id: epi-regular + - property: epi-regular reason: 'If $f : X \to Y$ is a surjective map of finite sets, it is the coequalizer of the two projections $p_1, p_2 : X \times_Y X \rightrightarrows X$ in $\FinSet$, but also in $\FS$. Notice that $p_1,p_2$ are surjective. Even though $X \times_Y X$ is not a pullback in $\FS$, we can use this finite set here.' - - property_id: multi-terminal object + - property: multi-terminal object reason: The empty set and a singleton give a multi-terminal object. unsatisfied_properties: - - property_id: small + - property: small reason: Even the collection of all singletons is not small. - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: countable + - property: countable reason: This is trivial. - - property_id: core-thin + - property: core-thin reason: Its core is $\IB$, which we know is not thin. - - property_id: generator + - property: generator reason: Let $G$ be a finite set. There are at least two morphisms $G + 2 \rightrightarrows 2$, but there is no morphism $G \to G + 2$ at all. Hence, $G$ is not a generator. - - property_id: connected + - property: connected reason: 'If $f : \varnothing \to X$ is surjective, then $X = \varnothing$, and if $f : X \to \varnothing$ is any map, then also $X = \varnothing$. This shows that $\{ \varnothing \}$ is a connected component in this category. (The other connected component consists of all non-empty finite sets.)' - - property_id: sequential limits + - property: sequential limits reason: 'Let $X_n := \{1,\dotsc,n\}$. We define the truncation $p_n : X_{n+1} \to X_n$ by extending the identity of $X_n$ with $p_n(n+1) := n$. Assume the sequence of truncations $\cdots \to X_2 \to X_1$ has a limit $(f_n : X \to X_n)$ in this category. But $f_n$ is surjective, so that $\card(X) \geq n$ for all $n$. Since $X$ is finite, this is a contradiction.' - - property_id: pullbacks + - property: pullbacks reason: The connected component of non-empty sets has a terminal object, $1$, and it suffices to prove that it has no products. Let $X$ be a finite set with more than $1$ element. Assume that the product $P$ of $X$ with itself exists. The diagonal $X \to P$ is a split monomorphism, hence injective, but also surjective, i.e. an isomorphism. In other words, the two projections $P \rightrightarrows X$ are equal. The universal property of $P$ now implies that every two morphisms $Y \rightrightarrows X$ are equal, which is absurd. - - property_id: binary copowers + - property: binary copowers reason: Assume that the copower $X := 2+2$ exists. Since we have a surjective map $2 \to X$, the set $X$ has at most $2$ elements. The codiagonal $X \to 2$ shows that $X$ has at least $2$ elements. Thus, $X \cong 2$. For all finite sets $Y$ we get a bijection $\Hom(2,Y) \cong \Hom(2,Y)^2$, in particular the cardinalities are the same. For $Y=2$ this gives the contradiction $2 = 4$. - - property_id: locally cocartesian coclosed + - property: locally cocartesian coclosed reason: >- If $X$ is a finite set, the coslice category $X / \FS$ is thin and in fact equivalent to the lattice of equivalence relations on $X$. If $X$ has $\geq 3$ elements, it is not codistributive* and hence not cocartesian coclosed: For simplicity assume $X = \{a,b,c\}$. The bottom element $\bot$ corresponds to the partition $\{\{a\},\{b\},\{c\}\}$, the top element $\top$ to the partition $\{\{a,b,c\}\}$. Now consider the three equivalence relations $E_1,E_2,E_3$ corresponding to the three partitions $$\{\{a,b\},\{c\}\}, \, \{\{a,c\},\{b\}\}, \, \{\{b,c\},\{a\}\}.$$ @@ -81,7 +81,7 @@ unsatisfied_properties: $$(E_1 \vee E_2) \wedge (E_1 \vee E_3) = \top \wedge \top = \top.$$ *For thin categories, the properties codistributive and distributive are equivalent. - - property_id: multi-initial object + - property: multi-initial object reason: If a multi-initial object exists, then the connected component consisting of non-empty finite sets has an initial object $X$. Then, any non-empty finite set cannot have a cardinality strictly greater than $X$, which is a contradiction. special_objects: {} diff --git a/databases/catdat/data/categories/FinAb.yaml b/databases/catdat/data/categories/FinAb.yaml index defad2f8..4b868950 100644 --- a/databases/catdat/data/categories/FinAb.yaml +++ b/databases/catdat/data/categories/FinAb.yaml @@ -16,38 +16,38 @@ related_categories: - TorsAb satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\FinAb \to \Set$ and $\Set$ is locally small. - - property_id: locally finite + - property: locally finite reason: There is a faithful functor $\FinAb \to \FinSet$ and $\FinSet$ is locally finite. - - property_id: essentially countable + - property: essentially countable reason: The underlying set of a finite structure can be chosen to be a subset of $\IN$. - - property_id: abelian + - property: abelian reason: This follows from the fact for $\Ab$. - - property_id: self-dual + - property: self-dual reason: 'This is a simple special case of Pontryagin duality: The functor $\Hom(-,\IQ/\IZ)$ provides the equivalence.' - - property_id: ℵ₁-accessible + - property: ℵ₁-accessible reason: The proof works exactly as for $\FinSet$. unsatisfied_properties: - - property_id: small + - property: small reason: Even the collection of trivial groups is not small. - - property_id: skeletal + - property: skeletal reason: There are many trivial and hence isomorphic groups which are not equal. - - property_id: countable + - property: countable reason: This is trivial. - - property_id: split abelian + - property: split abelian reason: The sequence $0 \to \IZ/2 \to \IZ/4 \to \IZ/2 \to 0$ does not split. - - property_id: generator + - property: generator reason: If $A,B$ are finite abelian groups whose orders are coprime, then we know that $\Hom(A,B)$ is trivial. But a generator would admit a non-trivial homomorphism to any other non-trivial finite abelian group. special_objects: diff --git a/databases/catdat/data/categories/FinGrp.yaml b/databases/catdat/data/categories/FinGrp.yaml index faf132ea..5fd2c814 100644 --- a/databases/catdat/data/categories/FinGrp.yaml +++ b/databases/catdat/data/categories/FinGrp.yaml @@ -15,63 +15,63 @@ related_categories: - Grp_c satisfied_properties: - - property_id: locally small + - property: locally small reason: It is a full subcategory of $\Grp$, which is locally small. - - property_id: locally finite + - property: locally finite reason: There is a faithful functor $\FinGrp \to \FinSet$ and $\FinSet$ is locally finite. - - property_id: pointed + - property: pointed reason: The trivial group is a zero object. check_redundancy: false - - property_id: essentially countable + - property: essentially countable reason: The underlying set of a finite structure can be chosen to be a subset of $\IN$. - - property_id: coequalizers + - property: coequalizers reason: The quotient group of a finite group is still finite. - - property_id: mono-regular + - property: mono-regular reason: See Prop. 4.2 at the nLab. The proof also works for finite groups. - - property_id: conormal + - property: conormal reason: Since epimorphisms are surjective (see below), this is the first isomorphism theorem for finite groups. - - property_id: Malcev + - property: Malcev reason: A direct argument is possible, but this can also be derived from the observation that $\FinGrp$ is the category of group objects in $(\FinSet,\times)$ and Example 2.2.16 in Malcev, protomodular, homological and semi-abelian categories. - - property_id: effective congruences + - property: effective congruences reason: 'Suppose we have a congruence $f, g : E \rightrightarrows X$ in $\FinGrp$. Since the embedding $\FinGrp \hookrightarrow \Grp$ preserves finite limits, it is also a congruence in $\Grp$. We already know that $\Grp$ has effective congruences since it is algebraic. Using this result, we see that $E$ is the kernel pair of $X \to (X/E)_{\Grp}$ in $\Grp$. Also, the quotient $(X/E)_{\Grp}$ is finite; and the forgetful functor $\FinGrp \to \Grp$ is fully faithful and therefore reflects limits. Thus, we conclude that $E$ is the kernel pair of $X \to (X/E)_{\Grp}$ in $\FinGrp$ as well.' - - property_id: regular + - property: regular reason: The category is Malcev and hence finitely complete, and it has all coequalizers. The regular epimorphisms coincide with the surjective group homomorphisms (see below), hence are clearly stable under pullbacks. - - property_id: ℵ₁-accessible + - property: ℵ₁-accessible reason: The proof works exactly as for $\FinSet$. unsatisfied_properties: - - property_id: small + - property: small reason: Even the collection of trivial groups is not small. - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: countable + - property: countable reason: This is trivial. - - property_id: normal + - property: normal reason: Every non-normal subgroup of a finite group (such as $C_2 \hookrightarrow S_3$) provides a counterexample. - - property_id: binary copowers + - property: binary copowers reason: 'Assume that $C_2 \sqcup C_2$ exists. This is a finite group, say of order $N$, with two involutions $u,v$ such that for every finite group $G$ with two involutions $a,b$ there is a unique homomorphism $\varphi : C_2 \sqcup C_2 \to G$ with $\varphi(u)=a$ and $\varphi(v)=2$. In particular, when $G$ is generated by $a,b$, then $\ord(G) \leq N$. But then the dihedral group $G := D_N$ of order $2N$ yields a contradiction.' - - property_id: sequential colimits + - property: sequential colimits reason: This follows from this lemma. - - property_id: generator + - property: generator reason: If $A,B$ are finite groups whose orders are coprime, then we know that $\Hom(A,B)$ is trivial. But a generator would admit a non-trivial homomorphism to any other non-trivial finite group. - - property_id: cogenerator + - property: cogenerator reason: 'We apply this lemma to the collection of finite simple groups: Any non-trivial homomorphism from a finite simple group to a finite group must be injective, and for every $n \in \IN$ there is a finite simple group of size $\geq n$ (for example, the alternating group on $n+5$ elements).' special_objects: diff --git a/databases/catdat/data/categories/FinOrd.yaml b/databases/catdat/data/categories/FinOrd.yaml index 6cc7d2af..f8ef77b0 100644 --- a/databases/catdat/data/categories/FinOrd.yaml +++ b/databases/catdat/data/categories/FinOrd.yaml @@ -16,65 +16,65 @@ related_categories: - Pos satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\FinOrd \to \Set$ and $\Set$ is locally small. - - property_id: locally finite + - property: locally finite reason: There is a faithful functor $\FinOrd \to \FinSet$ and $\FinSet$ is locally finite. - - property_id: essentially countable + - property: essentially countable reason: Every finite ordered set is isomorphic to $\{0 < \cdots < n-1 \}$ for some $n \in \IN$. - - property_id: strict initial object + - property: strict initial object reason: The empty ordered set is initial and is clearly strict. - - property_id: terminal object + - property: terminal object reason: Take the singleton set with the unique ordering. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Every non-empty totally ordered set is weakly terminal (by using constant maps). - - property_id: generator + - property: generator reason: The one-point finite ordered set is a generator since it represents the forgetful functor $\FinOrd \to \Set$. - - property_id: cogenerator + - property: cogenerator reason: The ordered set $\{0 < 1\}$ is a cogenerator, even for $\Pos$. - - property_id: equalizers + - property: equalizers reason: Take the equalizer in $\FinSet$ and restrict the order. - - property_id: coequalizers + - property: coequalizers reason: It suffices to construct quotients by equivalence relations. Let $\sim$ be an equivalence relation on $X$, where $(X,\leq)$ is a finite ordered set. Since $X$ is finite, by induction we may assume that $\sim$ is generated by a single relation $(a,b)$. If $a=b$, there is nothing to prove. If $a < b$ and $X = \{0,1,\dotsc,n-1\}$ with the usual order, the quotient is $\{0,1,\dotsc,a,b+1,\dotsc,n-1\}$ with the usual order. - - property_id: mono-regular + - property: mono-regular reason: 'Let $i : A \to B$ be a monomorphism of finite ordered sets. If $A$ is empty, then $i$ is clearly regular, so assume it is not. The map $i$ is injective (see below), hence order-reflecting. Define maps $u,v : B \to A$ by $u(b) := \max \{a \in A : i(a) \leq b \}$ and $v(b) := \min \{a \in A : b \leq i(a) \}$. These are order-preserving and satisfy $u \circ i = v \circ i$, both sides are $\id_A$. Conversely, if $b \in B$ satisfies $u(b) = v(b) =: a$, then $i(a) \leq b$ and $b \leq i(a)$, hence $b = i(a)$. This shows that $i$ is the equalizer of $u,v$.' - - property_id: epi-regular + - property: epi-regular reason: 'Let $f : A \to B$ be an epimorphism of finite ordered sets. It is surjective (see below). Define $u,v : B \to A$ by $u(b) := \min(f^{-1}(b))$ and $v(b) := \max(f^{-1}(b))$. One can easily check that $u,v$ are order-preserving maps with $f \circ u = f \circ v$ (both sides are $\id_B$). Let $h : A \to T$ be an order-preserving map with $h \circ u = h \circ v$. Then $h(a)$ only depends on $b := f(a)$: We have $u(b) \leq a \leq v(b)$, hence $h(u(b)) \leq h(a) \leq h(v(b)) = h(u(b))$. Therefore, there is a unique map $\tilde{h} : B \to T$ with $\tilde{h}(f(a)) = h(a)$, and one easily checks that it is order-preserving. This shows that $f$ is the coequalizer of $u,v$.' - - property_id: core-thin + - property: core-thin reason: 'Let $f : \{1 < \cdots < n \} \to \{1 < \cdots < n \}$ be an automorphism. Then $f(i)$ is the smallest element not contained in $\{f(j) : j < i\}$. From this one can deduce $f(i)=i$ by induction.' unsatisfied_properties: - - property_id: small + - property: small reason: Even the collection of all singleton orders is not small. - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: one-way + - property: one-way reason: There are three different order-preserving maps $\{0 < 1\} \to \{0 < 1\}$. - - property_id: countable + - property: countable reason: This is trivial. - - property_id: strict terminal object + - property: strict terminal object reason: This is trivial. - - property_id: sequential limits + - property: sequential limits reason: Consider the (non-empty) ordered set $[n] := \{0 < \cdots < n\}$ for $n \in \IN$. The forgetful functor to $\Set$ is representable, hence preserves all limits. Thus, if the diagram of truncation maps $\cdots \twoheadrightarrow [2] \twoheadrightarrow [1] \twoheadrightarrow [0]$ has a limit in $\FinOrd$, its underlying set is isomorphic to the limit taken in $\Set$, which is $\IN \cup \{\infty\}$. But this is not a finite set. - - property_id: sequential colimits + - property: sequential colimits reason: 'Consider the (non-empty) ordered set $[n] := \{0 < \cdots < n\}$ for $n \in \IN$. Assume the sequence of inclusion maps $[0] \hookrightarrow [1] \hookrightarrow [2] \hookrightarrow \cdots$ has a colimit $(f_n : [n] \to X)$ in $\FinOrd$. Let $n_0 \geq 0$ be fixed. I claim that $f_{n_0}$ is injective, which will then yield a contradiction by taking $n_0 \geq \card(X)$. For $n \geq 0$ define $g_n : [n] \to [n_0]$ as follows. For $n \leq n_0$ it is the inclusion, and for $n \geq n_0$ it is the surjection which keeps all elements of $[n_0]$ and maps all other elements to $n_0$. Observe that $g_n$ preserves the order and $g_{n+1} |_{[n]} = g_n$. Hence, there is a unique order-preserving map $g : X \to [n_0]$ with $g \circ f_n = g_n$ for all $n$. For $n = n_0$ this shows $g \circ f_{n_0} = \id_{[n_0]}$, and $f_{n_0}$ is injective.' special_objects: diff --git a/databases/catdat/data/categories/FinSet.yaml b/databases/catdat/data/categories/FinSet.yaml index 7cccc389..820d795a 100644 --- a/databases/catdat/data/categories/FinSet.yaml +++ b/databases/catdat/data/categories/FinSet.yaml @@ -20,41 +20,41 @@ comments: - For the non-existence of sequential (co-)limits it is not sufficient to take a diagram of finite sets whose (co-)limit in $\Set$ is not contained in $\FinSet$. satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\FinSet \to \Set$ and $\Set$ is locally small. - - property_id: locally finite + - property: locally finite reason: This is trivial. - - property_id: essentially countable + - property: essentially countable reason: Every finite set is isomorphic to some $\{1,\dotsc,n\}$ for some $n \in \IN$. - - property_id: generator + - property: generator reason: The one-point set is a generator since it represents the forgetful functor $\FinSet \to \Set$. - - property_id: cogenerator + - property: cogenerator reason: The two-element set is a cogenerator. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Every non-empty finite set is weakly terminal (by using constant maps). - - property_id: elementary topos + - property: elementary topos reason: This follows easily from the fact that sets form an elementary topos. - - property_id: ℵ₁-accessible + - property: ℵ₁-accessible reason: The inclusion $\FinSet \hookrightarrow \Set$ is closed under ℵ₁-filtered colimits, that is, any ℵ₁-filtered colimit of finite sets is again finite. Since every finite set is ℵ₁-presentable in $\Set$, it is still ℵ₁-presentable in $\FinSet$. Therefore, $\FinSet$ is ℵ₁-accessible, where every object is ℵ₁-presentable. unsatisfied_properties: - - property_id: small + - property: small reason: Even the collection of all singletons is not a set. - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: countable + - property: countable reason: This is trivial. - - property_id: natural numbers object + - property: natural numbers object reason: >- If $(N,z,s)$ is a natural numbers object, then $$1 \xrightarrow{z} N \xleftarrow{s} N$$ diff --git a/databases/catdat/data/categories/Fld.yaml b/databases/catdat/data/categories/Fld.yaml index 510f6681..9d22dc65 100644 --- a/databases/catdat/data/categories/Fld.yaml +++ b/databases/catdat/data/categories/Fld.yaml @@ -16,56 +16,56 @@ comments: - Limits and colimits are discussed in MSE/359352. satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Fld \to \Set$ and $\Set$ is locally small. - - property_id: inhabited + - property: inhabited reason: This is trivial. - - property_id: left cancellative + - property: left cancellative reason: It is well-known that every field homomorphism is injective and hence a monomorphism. - - property_id: well-copowered + - property: well-copowered reason: Epimorphisms are the purely inseparable field extensions. If $K \to L$ is purely inseparable, then for all $x \in L$ there is some $n \in \IN$ with $x^n \in L$. An element of $K$ has at most $n$ $n$th-roots. So we can bound the size of $L$. - - property_id: multi-algebraic + - property: multi-algebraic reason: See Eg. 4.3(1) in [AR01]. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: locally finite + - property: locally finite reason: There are infinitely many homomorphisms $\IQ(X) \to \IQ(X)$, $X \mapsto X^k$. - - property_id: connected + - property: connected reason: A field of characteristic $0$ cannot be connected with a field of characteristic $p > 0$. in fact, the connected components of $\Fld$ are the subcategories $\Fld_p$ of fields of characteristic $p$, where $p$ is a prime or $0$. - - property_id: balanced + - property: balanced reason: Every non-trivial purely inseparable field extension, such as $\IF_p(X^p) \to \IF_p(X)$, provides a counterexample by the descriptions of special morphisms below. - - property_id: core-thin + - property: core-thin reason: If this category was core-thin, Galois theory would not exist. Specifically, the conjugation $\IC \to \IC$, $z \mapsto \overline{z}$ is a non-trivial automorphism. - - property_id: multi-terminal object + - property: multi-terminal object reason: Every field has a non-trivial extension, for instance, the rational function field over itself in one variable. Hence, a multi-terminal object never exists. - - property_id: pushouts + - property: pushouts reason: 'By MSE/359352, the pushout $E \sqcup_K F$ of two field homomorphisms $E \leftarrow K \rightarrow F$ exists if and only if the tensor product $E \otimes_K F$ has a "fieldification": this means that the nilradical is a prime ideal whose quotient ring is a field. This is quite rare: Consider $E = K(X)$, $F = K(Y)$. Then $E \otimes_K F$ is isomorphic to $K[X,Y] (K[X]-\{0\})^{-1} (K[Y]-\{0\})^{-1}$, which is an integral domain but not a field: for example, $X-Y$ has no inverse.' - - property_id: generator + - property: generator reason: Assume that $G$ is a generator, say of characteristic $p$. Then for all $q \neq p$ all homomorphisms between two fields of characteristic $q$ would be equal, which is absurd. - - property_id: cogenerating set + - property: cogenerating set reason: 'We apply this lemma to the collection of fields: Any homomorphism of fields is injective. For every infinite cardinal $\kappa$ the field of rational functions in $\kappa$ variables has cardinality $\geq \kappa$ and a non-trivial automorphism (swap two variables).' - - property_id: binary powers + - property: binary powers reason: 'Assume that the product $P := \IQ(\sqrt{2}) \times \IQ(\sqrt{2})$ exists. This field is isomorphic to a subfield of $\IQ(\sqrt{2})$, hence $P \cong \IQ$ or $P \cong \IQ(\sqrt{2})$. In the first case, the two projections $P \rightrightarrows \IQ(\sqrt{2})$ must be equal, which means that every two homomorphisms $K \rightrightarrows \IQ(\sqrt{2})$ are equal, which is absurd (take $K = \IQ(\sqrt{2})$ and its two automorphisms). In the second case, the projections induce for every field $K$ a bijection $\Hom(K,\IQ(\sqrt{2})) \cong \Hom(K,\IQ(\sqrt{2}))^2$, which however fails for $K = \IQ(\sqrt{2})$: the left hand side has $2$ elements, the right hand side has $4$ elements. A more general result about products in $\Fld$ can be found at MSE/359352.' - - property_id: locally cartesian closed + - property: locally cartesian closed reason: 'Assume that $K$ is a field such that $\Fld / K$ is cartesian closed. This slice category is equivalent to the poset of subfields of $K$. This poset is a lattice, and our assumption implies that it is distributive (see here). But this is quite rare: Consider $K = \IQ(\sqrt{2}, \sqrt{3})$. By Galois theory, the lattice of subfields is isomorphic to the diamond lattice $M_3$ which is not distributive. Specifically, $(\IQ(\sqrt{2}) \wedge \IQ(\sqrt{6})) \vee (\IQ(\sqrt{3}) \wedge \IQ(\sqrt{6})) = \IQ \vee \IQ = \IQ$, while $(\IQ(\sqrt{2}) \vee \IQ(\sqrt{3})) \wedge \IQ(\sqrt{6}) = \IQ(\sqrt{2},\sqrt{3}) \wedge \IQ(\sqrt{6}) = \IQ(\sqrt{6})$.' - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: >- Inside of $\IF_p(X)$ consider the descending sequence of subfields $$\IF_p(X) \supseteq \IF_p(X^p) \supseteq \IF_p(X^{p^2}) \supseteq \cdots,$$ diff --git a/databases/catdat/data/categories/FreeAb.yaml b/databases/catdat/data/categories/FreeAb.yaml index 4a77ebdb..5a0cd19c 100644 --- a/databases/catdat/data/categories/FreeAb.yaml +++ b/databases/catdat/data/categories/FreeAb.yaml @@ -14,53 +14,53 @@ related_categories: - TorsFreeAb satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\FreeAb \to \Set$ and $\Set$ is locally small. - - property_id: additive + - property: additive reason: The embedding $\FreeAb \hookrightarrow \Ab$ is closed under (finite) direct sums, and $\Ab$ is additive. - - property_id: coproducts + - property: coproducts reason: This is is because free abelian groups are closed under direct sums of abelian groups. - - property_id: generator + - property: generator reason: As for $\Ab$, the group $\IZ$ is a generator. - - property_id: cogenerator + - property: cogenerator reason: It is easy to check that $\IZ$ is a cogenerator for free abelian groups. - - property_id: well-powered + - property: well-powered reason: This is clear since the monomorphisms are injective. - - property_id: well-copowered + - property: well-copowered reason: See MSE/5025660. - - property_id: regular + - property: regular reason: |- This follows formally from the fact that $\Ab$ is regular and $\FreeAb$ is closed under subobjects and finite products: By Prop. 2.5 in the nLab it suffices to prove that there are pullback-stable (reg epi, mono)-factorizations. Every homomorphism $f : A \to B$ in $\FreeAb$ factors as $f = i \circ p : A \twoheadrightarrow C \hookrightarrow B$, where $C$ is a subgroup, hence free, and $A \to C$ is surjective. Clearly, surjective homomorphisms are pullback-stable. It remains to show that they coincide with the regular epimorphisms. (1) If $f : A \to B$ is surjective, it is the coequalizer of $A \times_B A \rightrightarrows A$ in $\Ab$. Since $A \times_B A$ is free abelian, $f$ is also an coequalizer in $\FreeAb$. (2) If $f : A \to B$ is a regular epimorphism in $\FreeAb$, consider the factorization $f = i \circ p$ as above. Since $f$ is an extremal epimorphism, $i$ must be an isomorphism, so that $f$ is surjective. unsatisfied_properties: - - property_id: balanced + - property: balanced reason: 'The homomorphism $2 : \IZ \to \IZ$ is a counterexample.' - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: countable powers + - property: countable powers reason: Assume that the power $P = \prod_{n \geq 0} \IZ$ exists in this category. The projections $P \to \IZ$ induce an isomorphism of abelian groups $\Hom(\IZ, P) \to \prod_n \Hom(\IZ, \IZ)$, hence $P \cong \IZ^{\IN}$. But it is well-known that $\IZ^{\IN}$ is not free abelian. - - property_id: sequential colimits + - property: sequential colimits reason: See MO/509715. - - property_id: effective cocongruences + - property: effective cocongruences reason: |- We will let $E$ be the abelian group with presentation $\langle a, b, c \mid a - b = 2c \rangle$, with two morphisms $\IZ \rightrightarrows E$ given by $1\mapsto a$, $1\mapsto b$. Note that $E$ is free with basis $\{ b, c \}$. Then $\Hom(E, G) \cong \{ (x, y, z) \in G^3 \mid x - y = 2z \}$. Observe that since $G$ is torsion-free, the projection onto the first two coordinates is injective; and $(x, y)$ is in the image precisely when $x \equiv y \pmod{2G}$, which gives an equivalence relation. Therefore, $E$ gives a cocongruence on $\IZ$. On the other hand, if $E$ were the cokernel pair of $h : H \to \IZ$, that would mean that for $x, y : \IZ \to G$, $x \equiv y \pmod{2G}$ if and only if $x \circ h = y \circ h$. In particular, from the case $G := \IZ$, $x := 2 \id$, $y := 0$, we would have $2h = 0$. That implies $h = 0$, but then that would give $\id_{\IZ} \equiv 0 \pmod{2}$, resulting in a contradiction. category_property_comments: - - property_id: accessible + - property: accessible comment: The question if this category is accessible is undecidable in ZFC. See MSE/720885. special_objects: diff --git a/databases/catdat/data/categories/Grp.yaml b/databases/catdat/data/categories/Grp.yaml index 02722a25..211446d6 100644 --- a/databases/catdat/data/categories/Grp.yaml +++ b/databases/catdat/data/categories/Grp.yaml @@ -17,58 +17,58 @@ related_categories: - SemiGrp satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Grp \to \Set$ and $\Set$ is locally small. - - property_id: pointed + - property: pointed reason: The trivial group is a zero object. check_redundancy: false - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic theory of a group. - - property_id: mono-regular + - property: mono-regular reason: See Prop. 4.2 at the nLab. - - property_id: conormal + - property: conormal reason: Since epimorphisms are surjective (see below), this is the first isomorphism theorem for groups. - - property_id: Malcev + - property: Malcev reason: See Example 2.2.4 in Malcev, protomodular, homological and semi-abelian categories. - - property_id: effective cocongruences + - property: effective cocongruences reason: A proof can be found here. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: normal + - property: normal reason: Every non-normal subgroup (such as $C_2 \hookrightarrow S_3$) provides a counterexample. - - property_id: cogenerator + - property: cogenerator reason: 'We apply this lemma to the collection of simple groups: Any non-trivial homomorphism from a simple group to a group must be injective, and for every infinite cardinal $\kappa$ there is a simple group of size $\geq \kappa$ (for example, the alternating group on $\kappa$ elements).' - - property_id: coregular + - property: coregular reason: This is because injective group homomorphisms are not stable under pushouts, see e.g. MSE/601463 or MSE/5088032. - - property_id: counital + - property: counital reason: The canonical morphism $F_2 = \IZ \sqcup \IZ \to \IZ \times \IZ$ is not a monomorphism since $F_2$ is not abelian. - - property_id: CIP + - property: CIP # TODO: remove code duplication with "counital" proof reason: The canonical morphism $F_2 = \IZ \sqcup \IZ \to \IZ \times \IZ$ is not a monomorphism since $F_2$ is not abelian. - - property_id: CSP + - property: CSP reason: The canonical homomorphism $\coprod_{n \geq 0} \IZ \to \prod_{n \geq 0} \IZ$ is not surjective because its domain is countable and its codomain is uncountable. Hence it is no epimorphism. - - property_id: regular quotient object classifier + - property: regular quotient object classifier reason: 'Assume that $\Grp$ has a (regular) quotient object classifier, i.e. a group $P$ such that every surjective homomorphism $G \to H$ is the cokernel of a unique homomorphism $\varphi : P \to G$. Equivalently, every normal subgroup $N \subseteq G$ is $\langle \langle \varphi(P) \rangle \rangle$ for a unique homomorphism $\varphi : P \to G$, where $\langle \langle - \rangle \rangle$ denotes the normal closure. If $c_g : G \to G$ denotes the conjugation with $g \in G$, then the images of $\varphi$ and $c_g \circ \varphi$ have the same normal closures, so the homomorphisms must be equal. In other words, $\varphi$ factors through the center $Z(G)$. But then every normal subgroup of $G$, in particular $G$ itself, would be contained in $Z(G)$, which is wrong for every non-abelian group $G$.' - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: We already know that $\Ab$ does not have this property. Now apply the contrapositive of the dual of this lemma to the forgetful functor $\Ab \to \Grp$ which indeed preserves epimorphisms. - - property_id: cocartesian cofiltered limits + - property: cocartesian cofiltered limits reason: >- For cofiltered diagrams of groups $(H_i)$ and a group $G$ the canonical homomorphism $$\textstyle \alpha : G \sqcup \lim_i H_i \to \lim_i (G \sqcup H_i)$$ diff --git a/databases/catdat/data/categories/Grp_c.yaml b/databases/catdat/data/categories/Grp_c.yaml index 66673a58..7c8fac0a 100644 --- a/databases/catdat/data/categories/Grp_c.yaml +++ b/databases/catdat/data/categories/Grp_c.yaml @@ -15,77 +15,77 @@ related_categories: - Set_c satisfied_properties: - - property_id: locally small + - property: locally small reason: There is an embedding $\Grp_\c \hookrightarrow \Grp$ and $\Grp$ is locally small. - - property_id: essentially small + - property: essentially small reason: Every countable group is isomorphic to a group whose underlying set is a subset of $\IN$. - - property_id: pointed + - property: pointed reason: The trivial group is countable and is a zero object. check_redundancy: false - - property_id: generator + - property: generator reason: The countable group $\IZ$ is a generator because it represents the forgetful functor $\Grp_\c \to \Set$. - - property_id: finite products + - property: finite products reason: This is because $\Grp$ has finite (in fact, all) products, and $\Grp_\c \hookrightarrow \Grp$ is closed under finite products. This is because a finite product of countable sets is again countable. check_redundancy: false - - property_id: equalizers + - property: equalizers reason: One can use the same construction as in $\Grp$ since a subgroup of a countable group is again countable. check_redundancy: false - - property_id: coequalizers + - property: coequalizers reason: One can use the same construction as in $\Grp$ since a quotient of a countable group is again countable. - - property_id: countable coproducts + - property: countable coproducts reason: This is because $\Grp$ has countable (in fact, all) coproducts, and $\Grp_\c \hookrightarrow \Grp$ is closed under countable coproducts. This is because a countable union of countable sets is again countable. - - property_id: mono-regular + - property: mono-regular reason: 'This can be deduced from the corresponding property of $\Grp$ as follows: Let $i : K \hookrightarrow G$ be a monomorphism in $\Grp_\c$, i.e. an injective homomorphism of countable groups. Since $\Grp$ is mono-regular, there is a group $H$ and two homomorphisms $f,g : G \rightrightarrows H$ with $i = \eq(f,g)$. Let $H'' \subseteq H$ be the subgroup generated by $\im(f) \cup \im(g)$. Since $G$ is countable, $H''$ is countable as well, and $f,g$ corestrict to homomorphisms $f'', g'' : G \rightrightarrows H''$. Hence, $i = \eq(f'',g'')$.' - - property_id: conormal + - property: conormal reason: 'If $f : G \to H$ is an epimorphism in $\Grp_\c$, i.e. a surjective homomorphism of countable groups, then $f$ is the cokernel of $K \hookrightarrow G$ in $\Grp$, where $K$ is the kernel of $f$. Since $K$ is countable, it is also the cokernel in $\Grp_\c$.' - - property_id: Malcev + - property: Malcev reason: We can use the same proof as for $\Grp$. - - property_id: regular + - property: regular reason: We already know that the category is finitely complete, and that it has all coequalizers. The regular epimorphisms coincide with the surjective group homomorphisms (see below), hence are clearly stable under pullbacks. - - property_id: effective congruences + - property: effective congruences reason: 'A congruence on a countable group $G$ has the form $\{(g,h) \in G^2 : g^{-1} h \in N \}$ for some normal subgroup $N \subseteq G$. It is the kernel pair of the projection $p : G \twoheadrightarrow G/N$ in $\Grp$, but also in $\Grp_\c$ since $G/N$ is countable.' - - property_id: effective cocongruences + - property: effective cocongruences reason: 'Let $G + G \twoheadrightarrow H$ be a cocongruence in $\Grp_\c$. Since $\Grp_\c \hookrightarrow \Grp$ is closed under finite colimits, this is the same as a cocongruence in $\Grp$ where $G,H \in \Grp$ happen to be countable groups. Since we already know that $\Grp$ has effective cocongruences, the cocongruence is the cokernel pair of some homomorphism of groups $K \to H$. If $K'' \subseteq H$ denotes the image of $K$, it is then also the cokernel pair of the inclusion $K'' \hookrightarrow H$, and $K''$ is countable.' unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: small + - property: small reason: Even the collection of all trivial groups is not a set. - - property_id: normal + - property: normal reason: Every non-normal subgroup of a countable group (such as $C_2 \hookrightarrow S_3$) provides a counterexample. - - property_id: counital + - property: counital reason: The canonical morphism $F_2 = \IZ \sqcup \IZ \to \IZ \times \IZ$ is not a monomorphism since $F_2$ is not abelian. - - property_id: countable powers + - property: countable powers reason: Since the forgetful functor $\Grp_\c \to \Set$ is representable, it preserves products. Therefore, if the power $\IZ^{\IN}$ exists in $\Grp_\c$, its underlying set must be the ordinary cartesian product, which however is uncountable. - - property_id: regular quotient object classifier + - property: regular quotient object classifier reason: We can copy the proof from $\Grp$. - - property_id: coregular + - property: coregular reason: Pushouts of injective homomorphisms between countable groups do not need to be injective, see MSE/5088032. - - property_id: cogenerator + - property: cogenerator reason: 'Assume that a cogenerator $Q$ exists in $\Grp_\c$. There are only countably many finitely generated subgroups of $Q$. But there are continuum many finitely generated simple groups; this follows from Corollary 1.5 in Finitely generated infinite simple groups of homeomorphisms of the real line by J. Hyde and Y. Lodha. Hence, there is a finitely generated (and hence countable) simple group $H$ which does not embed into $Q$. Since $H$ is simple, any homomorphism $H \to Q$ must be trivial then. But then $\id_H, 1 : H \rightrightarrows H$ are not separated by a homomorphism $H \to Q$.' - - property_id: ℵ₁-accessible + - property: ℵ₁-accessible reason: 'We can almost copy the proof from $\Set_\c$ to show that $\Grp_\c$ does not have $\aleph_1$-filtered colimits: Fix an uncountable set $X$, let $P_\c(X)$ be the poset of countable subsets of $X$, which is $\aleph_1$-filtered, and consider the functor $P_\c(X) \to \Grp_\c$ taking a subset $Y \subseteq X$ to the free group $F(Y)$. The colimit of this diagram in $\Grp$ is given by $F(X)$ itself, so if $G$ were a colimit in $\Grp_\c$, then $\Hom(G, C_2) \cong \Hom(F(X),C_2) \cong \{0,1\}^X$. But the former has cardinality at most $2^{\aleph_0}$ and the latter has cardinality $2^{\card(X)}$, so we have obtained a contradiction if we pick $X$ large enough (e.g. $\card(X)=2^{\aleph_0}$).' special_objects: diff --git a/databases/catdat/data/categories/Haus.yaml b/databases/catdat/data/categories/Haus.yaml index 931c4a85..04464bab 100644 --- a/databases/catdat/data/categories/Haus.yaml +++ b/databases/catdat/data/categories/Haus.yaml @@ -15,60 +15,60 @@ related_categories: - CompHaus satisfied_properties: - - property_id: locally small + - property: locally small reason: It is a full subcategory of $\Top$, which is locally small. - - property_id: generator + - property: generator reason: The one-point space is a generator since it represents the forgetful functor $\Haus \to \Set$. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Every non-empty Hausdorff space is weakly terminal (by using constant maps). - - property_id: equalizers + - property: equalizers reason: This follows from the corresponding fact for $\Top$ since subspaces of Hausdorff spaces are again Hausdorff. - - property_id: products + - property: products reason: This follows from the corresponding fact for $\Top$ since products of Hausdorff spaces are again Hausdorff. - - property_id: cocomplete + - property: cocomplete reason: This follows since $\Haus$ is a reflective subcategory of $\Top$, which is cocomplete. For the reflector, see e.g. the nLab. Explicitly, we construct the colimit of Hausdorff spaces by applying the reflector to the colimit of the underlying topological spaces. check_redundancy: false - - property_id: infinitary extensive + - property: infinitary extensive reason: This follows exactly as for $\Top$ since Hausdorff spaces are closed under taking subspaces and coproducts in $\Top$. - - property_id: well-powered + - property: well-powered reason: This is clear from the classification of monomorphisms as injective continuous maps. - - property_id: well-copowered + - property: well-copowered reason: 'Every epimorphism has dense image, so it suffices to prove that if a Hausdorff space $X$ has a dense subset $D \subseteq X$, we can bound the cardinality of $X$ in terms of the cardinality of $D$. For $x \in X$ let $\alpha(x)$ be the set of all $U \cap D$, where $U$ is an open neighborhood of $x$. This defines a map $\alpha : X \to P(P(D))$, and we claim that it is injective: this is because $\{x\}$ is the intersection of all $\overline{U} = \overline{U \cap D}$, where $U$ runs through the open neighborhoods of $x$.' - - property_id: co-Malcev + - property: co-Malcev reason: See MO/509548. - - property_id: effective cocongruences + - property: effective cocongruences reason: As the proof at MO/509548 shows, in fact any coreflexive corelation on $X$ in $\Haus$ is of the form $X +_S X$ for a closed subset $S$ of $X$. Such a cocongruence is clearly effective. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: The inclusion $\IQ \hookrightarrow \IR$ is a counterexample; it is an epimorphism since $\IQ$ is dense in $\IR$. - - property_id: Malcev + - property: Malcev reason: This is clear since $\Set$ is not Malcev and can be interpreted as the subcategory of discrete spaces (which are Hausdorff). - - property_id: regular subobject classifier + - property: regular subobject classifier reason: Assume that there is a regular subobject classifier $\Omega$. By the classification of regular monomorphisms, we would have an isomorphism between $\Hom(X,\Omega)$ and the set of closed subsets of $X$ for any Hausdorff space $X$. If we take $X = 1$ we see that $\Omega$ has two points. Since $\Omega$ is Hausdorff, $\Omega \cong 1 + 1$ must be discrete. But then $\Hom(X,\Omega)$ is isomorphic to the set of all clopen subsets of $X$, of which there are usually far fewer than closed subsets (consider $X = [0,1]$). - - property_id: cartesian filtered colimits + - property: cartesian filtered colimits reason: 'It is shown in MSE/1255678 that $\IQ \times - : \Top \to \Top$ does not preserve sequential colimits (so that it cannot be a left adjoint). The same example also works in $\Haus$: Surely $\IQ$ is Hausdorff, $X_n$ is Hausdorff, as is their colimit $X$, and the colimit (taken in $\Top$) of the $X_n \times \IQ$ admits a bijective continuous map to a Hausdorff space, therefore is also Hausdorff, meaning it is also the colimit taken in $\Haus$.' - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: 'Recall the counterexample for sets: The unique maps $\IN_{\geq n} \to 1$ are surjective, but their limit $0 = \bigcap_{n \geq 0} \IN_{\geq n} \to 1$ is not. This also works in $\Haus$ by using discrete topologies. We could also apply a variant of (the dual of) this lemma to the discrete topology functor $\Set \to \Haus$, which does not preserve all cofiltered limits, but does preserve intersections.' - - property_id: filtered-colimit-stable monomorphisms + - property: filtered-colimit-stable monomorphisms reason: |- The proof is similar to $\Met$. For $n \geq 1$ let $X_n$ be the pushout of $$(-\infty, -1/n] \cup [1/n, \infty) \hookrightarrow \IR$$ diff --git a/databases/catdat/data/categories/J2.yaml b/databases/catdat/data/categories/J2.yaml index 42af9f7a..1f897667 100644 --- a/databases/catdat/data/categories/J2.yaml +++ b/databases/catdat/data/categories/J2.yaml @@ -16,20 +16,20 @@ related_categories: - M-Set satisfied_properties: - - property_id: locally small + - property: locally small reason: This is trivial. - - property_id: finitary algebraic + - property: finitary algebraic reason: 'The structure of a Jónsson-Tarski algebra on a set $X$ is equivalent to one binary operation $\mu : X^2 \to X$ and two unary operations $\lambda, \rho : X \rightrightarrows X$ such that $\mu(\lambda(x),\rho(x)) = x$, $\lambda(\mu(x,y))=x$, and $\rho(\mu(x,y))=y$.' - - property_id: Grothendieck topos + - property: Grothendieck topos reason: See the nLab. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: >- There is a bijection $\alpha = (\lambda,\rho) : \IN \to \IN \times \IN$ such that $\lambda$ has a fixed point, but $\rho$ does not (see below). Then the isomorphism $\beta := (\rho,\lambda)$ has the opposite property. There cannot be any morphism $(\IN,\alpha) \to (\IN,\beta)$, as it would map the fixed point of $\lambda$ to a fixed point of $\rho$, and likewise there is no morphism $(\IN,\beta) \to (\IN,\alpha)$. diff --git a/databases/catdat/data/categories/LRS.yaml b/databases/catdat/data/categories/LRS.yaml index c975b542..034f3cbf 100644 --- a/databases/catdat/data/categories/LRS.yaml +++ b/databases/catdat/data/categories/LRS.yaml @@ -13,38 +13,38 @@ related_categories: - Sch satisfied_properties: - - property_id: locally small + - property: locally small reason: For two ringed spaces $(X,O_X), (Y,O_Y)$, the collection of morphisms $X \to Y$ of ringed spaces is the collection $\prod_{f \in \Hom(X,Y)} \Hom(O_Y,f_* O_X)$, which is a set since $\Hom(X,Y)$ is a set and each $\Hom(O_Y,f_* O_X)$ is a set. - - property_id: complete + - property: complete reason: See Localization of ringed spaces by W. Gillam. See also MSE/1033675. - - property_id: cocomplete + - property: cocomplete reason: See Demazure-Gabriel's "Groupes algébriques", I. §1. 1.6. Specifically, the forgetful functor to ringed spaces preserves colimits, and colimits of ringed spaces are built from colimits of topological spaces and limits of commutative rings, see MSE/1646202. - - property_id: well-powered + - property: well-powered reason: 'Let $f : X \to Y$ be a monomorphism of locally ringed spaces. I claim that $f$ is injective, from which the claim will follow. The diagonal $\Delta : X \to X \times_Y X$ is an isomorphism. By the explicit construction of fiber products, $X \times_Y X$ consists of triples $(x_1,x_2,\mathfrak{p})$ where $x_1,x_2 \in X$, $y := f(x_1) = f(x_2)$ and $\mathfrak{p}$ is a prime ideal in $k(x_1) \otimes_{k(y)} k(x_2)$. The map $\Delta$ is given by $\Delta(x) = \bigl(x,x,\ker(k(x) \otimes_{k(f(x))} k(x) \to k(x)\bigr)$, and it is bijective. This clearly implies that $f$ is injective (and that each tensor product $k(x) \otimes_{k(f(x))} k(x)$ has a unique prime ideal, namely the kernel mentioned).' - - property_id: well-copowered + - property: well-copowered reason: It is enough to prove that every epimorphism of locally ringed spaces is surjective. The forgetful functor $\LRS \to \RS$ has a right adjoint (see Localization of ringed spaces by W. Gillam), so it preserves epimorphisms. The forgetful functor $\RS \to \Top$ also has a right adjoint, namely $X \mapsto (X,\underline{\IZ})$, so it also preserves epimorphisms. - - property_id: infinitary extensive + - property: infinitary extensive reason: '[Sketch] Since $\Top$ is infinitary extensive, a morphism $f : Y \to \coprod_i X_i =: X$ yields a decomposition $Y = \coprod_i Y_i$ (as topological spaces) with continuous maps $f_i : Y_i \to X_i$. Endow the open subset $Y_i \subseteq Y$ with the restricted sheaf. Then each $f_i$ becomes a morphism of locally ringed spaces, and $Y = \coprod_i Y_i$ holds as locally ringed spaces.' unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: The canonical morphism $\Spec(\IZ/2 \times \IZ[1/2]) \longrightarrow \Spec(\IZ)$ is a mono- and an epimorphism, but no isomorphism. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: This is because already the full subcategory $\Sch$ of schemes is not semi-strongly connected. - - property_id: Malcev + - property: Malcev reason: This is because already $\Sch$ is not Malcev. - - property_id: co-Malcev + - property: co-Malcev reason: 'We can adjust the proof for $\Top$ (see MO/509548) as follows: Let $k$ be a field, $X$ be a singleton and $Y = \{u,v\}$ be the Sierpinski space where $\{u\}$ is open, but $\{v\}$ is not. Endow both with the sheaf of locally constant functions to $k$. Thus, $\O_X(X) = k$, $\O_Y(Y) = \O_Y(\{u\}) = k$. There is a canonical morphism $p : X + X \to Y$. It is a coreflexive corelation that is not cosymmetric.' special_objects: diff --git a/databases/catdat/data/categories/M-Set.yaml b/databases/catdat/data/categories/M-Set.yaml index a3fb57ce..5a918584 100644 --- a/databases/catdat/data/categories/M-Set.yaml +++ b/databases/catdat/data/categories/M-Set.yaml @@ -15,24 +15,24 @@ related_categories: - Set satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $M{-}\Set \to \Set$ and $\Set$ is locally small. - - property_id: Grothendieck topos + - property: Grothendieck topos reason: It is the category of sheaves on the opposite of the one-object category associated to $M$. - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic theory of an $M$-sets (having a unary operation for each $m \in M$). unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: trivial + - property: trivial reason: This is trivial. category_property_comments: - - property_id: semi-strongly connected + - property: semi-strongly connected comment: If this category is semi-strongly connected depends on the choice of $M$. For $M = 1$ it is, for $M = \IZ$ it is not. In general, if $G$ is a group, then $G{-}\Set$ is semi-strongly connected if and only if for all subgroups $H,K \subseteq G$, $H$ is subconjugated to $K$ or $K$ is subconjugated to $H$. If $G$ is abelian, this means that the poset of subgroups is linear, in which case $G$ is either isomorphic to $\IZ/p^n$ or to $\IZ/p^{\infty}$ for a prime $p$. See also MSE/5129804. special_objects: diff --git a/databases/catdat/data/categories/Man.yaml b/databases/catdat/data/categories/Man.yaml index fa6fecfe..088103b0 100644 --- a/databases/catdat/data/categories/Man.yaml +++ b/databases/catdat/data/categories/Man.yaml @@ -15,66 +15,66 @@ related_categories: - Top satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Man \to \Set$ and $\Set$ is locally small. - - property_id: finite products + - property: finite products reason: In short, this follows from the corresponding statement for topological spaces and $\IR^n \times \IR^m \cong \IR^{n+m}$. check_redundancy: false - - property_id: generator + - property: generator reason: The $0$-dimensional one-point manifold is a generator since it represents the forgetful functor $\Top \to \Set$. - - property_id: cogenerator + - property: cogenerator reason: 'The manifold $\IR$ is a cogenerator, since for every smooth manifold $M$ and points $p \neq q$ in $M$ there is a smooth function $f : M \to \IR$ with $f(p) = 1$ and $f(q) = 0$ (John Lee, Introduction to Smooth Manifolds, Prop. 2.25).' - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Every non-empty manifold is weakly terminal (by using constant maps). - - property_id: essentially small + - property: essentially small reason: 'This is a consequence of the Whitney embedding theorem. But there is also a more direct proof: Since a manifold is second-countable, it is Lindelöf (proof). In particular, there is a countable atlas. It is then completely determined by countable many open subsets of Euclidean spaces and the transition maps.' - - property_id: extensive + - property: extensive reason: '[Sketch] Since $\Top$ is infinitary extensive, a continuous map $f : M \to \coprod_i N_i$ corresponds to a decomposition $M = \coprod_i M_i$ (as topological spaces) with continuous maps $f_i : M_i \to N_i$. Endow the open subset $M_i \subseteq M$ with the smooth structure inherited from $M$. Now remark that $f$ is smooth iff each $f_i$ is smooth.' - - property_id: countably distributive + - property: countably distributive # TODO: maybe add "countably extensive" to make this more conceptual reason: To construct countable coproducts, take the usual disjoint union of spaces, which is clearly locally Euclidean and Hausdorff, and it is second countable since we are using only countable many spaces. (Without that condition, all coproducts would exist.) Now we need to check that the canonical smooth map $\coprod_i X \times Y_i \to X \times \coprod_i Y_i$ is a diffeomorphism (for countable families). It is a homeomorphism since $\Top$ is infinitary distributive. The inverse $X \times \coprod_i Y_i \to \coprod_i X \times Y_i$ is smooth since the domain is covered by the open subsets $X \times Y_i$ on which the map is clearly smooth. - - property_id: Cauchy complete + - property: Cauchy complete reason: See Theorem 2.1 at the nLab. - - property_id: coquotients of cocongruences + - property: coquotients of cocongruences reason: |- Let $p : X + X \twoheadrightarrow E$ be a cocongruence with coreflexivity morphism $r : E \to X$, so that $r \circ p : X + X \to X$ is the codiagonal. Since $p$ is an epimorphism, it has dense image (see below). We first claim that in fact $p$ also has closed image and therefore is surjective. Because $r \circ (p \circ i_1) : X \to X$ is the identity, the image of $p \circ i_1$ is the equalizer of $\id_E$ and $(p \circ i_1) \circ r$, hence closed. Likewise, the image of $p \circ i_2$ is closed. Thus, the image of $p$, which is the union of these images, is closed. Now, since the pushforward maps of tangent spaces compose to the identity, we see that $p$ must be a local immersion and $r$ must be a submersion. Also, since the fibers of $r$ have one or two points each, we see that the dimension of $E$ must locally be the same as the dimension of $X$. This implies that in fact $p$ and $r$ are local diffeomorphisms. Therefore, the cardinality of the fiber of $r$ is locally constant. Thus, if $U$ is the subset of $X$ where $r$ has fiber of a single point, with the subspace topology, then $U$ is a clopen submanifold of $X$ which serves as the equalizer of $p \circ i_1$ and $p \circ i_2$. - - property_id: effective cocongruences + - property: effective cocongruences reason: 'From the proof that $\Man$ has coquotients of cocongruences, we know that for any cocongruence $X \rightrightarrows E$, there is a clopen submanifold $U$ of $X$ such that the fibers of $r : E \twoheadrightarrow X$ have one point on $U$, and two points on $X \setminus U$. Therefore, $E$ is the cokernel pair of the inclusion map $U \hookrightarrow X$.' unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: small + - property: small reason: Even the collection of all singletons is not a set. - - property_id: balanced + - property: balanced reason: The irrational winding of a torus $\IR \to S^1 \times S^1$, $t \mapsto (e^{i t}, \, e^{i \alpha t})$, where $\alpha \in \IR \setminus \IQ$, provides a counterexample. It is injective and has a dense image. Hence, it is a mono- and an epimorphism. But it is not surjective, hence no isomorphism. - - property_id: countable powers + - property: countable powers reason: 'The power $\IR^{\IN}$ does not exist. More generally, let $(M_n)_{n \geq 0}$ be a sequence of smooth manifolds of positive dimension whose product $(\pi_n : P \to M_n)_{n \geq 0}$ exists. This product cone in $\Man$ yields a product cone in $\Set$ since the forgetful functor $\Man \to \Set$ is representable, hence preserves all limits. Choose points $x_n \in M_n$ with $T_{x_n}(M_n) \neq 0$. Choose the point $x \in P$ with $\pi_n(x) = x_n$. Consider the linear map $T_x(P) \to \prod_{n \geq 0} T_{x_n}(M_n)$ induced by the derivatives $d_x(\pi_n) : T_x(P) \to T_{x_n}(M_n)$. Since $T_x(P)$ is finite-dimensional and $\prod_{n \geq 0} T_{x_n}(M_n)$ is not, it cannot be surjective. But actually, it is: Choose tangent vectors $v_n \in T_{x_n}(M_n)$. Choose smooth curves $\gamma_n : \IR \to M_n$ with $\gamma_n(0)=x_n$ and ${\gamma_n}''(0) = v_n$. By the universal property there is a unique smooth curve $\gamma : \IR \to P$ with $\pi_n \gamma = \gamma_n$. In particular, $\gamma(0) = x$. The chain rule now implies that $\gamma''(0) \in T_x(P)$ is a preimage of $(v_n)$ – a contradiction.' - - property_id: pullbacks + - property: pullbacks reason: See MSE/5129579 or MSE/322485. - - property_id: sequential colimits + - property: sequential colimits reason: If $\Man$ had sequential colimits, then by this lemma there would be a manifold $M$ that admits a split epimorphism $M \to \IR^n$ for every $n$. But then $M$ will have an infinite-dimensional tangent space, which is a contradiction. - - property_id: ℵ₁-accessible + - property: ℵ₁-accessible reason: 'We already know that $\Set_\c$ does not have $\aleph_1$-filtered colimits. The functor $\pi_0: \Man \to \Set_\c$ is well-defined (because manifolds are second-countable), and it admits a fully faithful right adjoint (regarding a countable set as a discrete manifold). Therefore, $\Man$ does not have $\aleph_1$-filtered colimits.' - - property_id: quotients of congruences + - property: quotients of congruences reason: If $\Man$ had quotients of congruences, then by this lemma, it would have a pushout of $\IR \leftarrow \{ 0 \} \rightarrow \IR$. This contradicts MO/19916. special_objects: diff --git a/databases/catdat/data/categories/Meas.yaml b/databases/catdat/data/categories/Meas.yaml index 9eaccd6e..d2aea00f 100644 --- a/databases/catdat/data/categories/Meas.yaml +++ b/databases/catdat/data/categories/Meas.yaml @@ -16,56 +16,56 @@ comments: - The thread MSE/5024471 asks for the finitely presentable objects of this category. satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Meas \to \Set$ and $\Set$ is locally small. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Every non-empty measurable space is weakly terminal (by using constant maps). - - property_id: generator + - property: generator reason: The one-point measurable space (with the unique $\sigma$-algebra) is a generator since it represents the forgetful functor $\Meas \to \Set$. - - property_id: cogenerator + - property: cogenerator reason: Take the two-element set $2$ endowed with the trivial $\sigma$-algebra (where only $\varnothing$ and $2$ are measurable), and use that $2$ is a cogenerator for $\Set$. - - property_id: well-powered + - property: well-powered reason: This follows from the fact that monomorphisms are injective in this category. - - property_id: well-copowered + - property: well-copowered reason: This follows from the fact that epimorphisms are surjective in this category. - - property_id: complete + - property: complete reason: Take the limit of the underlying sets and take the smallest $\sigma$-algebra making all projections measurable. - - property_id: cocomplete + - property: cocomplete reason: Take the colimit of the underlying sets and take the largest $\sigma$-algebra making all inclusions measurable. That is, a set is measurable iff its preimage under each inclusion is measurable. - - property_id: infinitary extensive + - property: infinitary extensive reason: '[Sketch] Since $\Set$ is infinitary extensive, a map $f : Y \to \coprod_i X_i =: X$ corresponds to a decomposition $Y = \coprod_i Y_i$ (as sets) with maps $f_i : Y_i \to X_i$. Endow the measurable subset $Y_i \subseteq Y$ with the restricted $\sigma$-algebra. If $f$ is measurable, each $f_i$ is measurable, and $Y = \coprod_i Y_i$ holds as measurable spaces.' - - property_id: filtered-colimit-stable monomorphisms + - property: filtered-colimit-stable monomorphisms reason: This follows from this lemma applied to the forgetful functor to $\Set$. - - property_id: regular subobject classifier + - property: regular subobject classifier reason: The set $\{0,1\}$ with the trivial $\sigma$-algebra is a regular subobject classifier since measurable maps $X \to \{0,1\}$ correspond to subsets of $X$. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: Take a set $X$ with two different $\sigma$-algebras $\A \subset \B$ (for example, $\A = \{\varnothing,X\}$ and $\B = P(X)$ when $X$ has at least $2$ elements), then the identity map $(X,\B) \to (X,\A)$ provides a counterexample. - - property_id: Malcev + - property: Malcev reason: Use that $\Set$ is not Malcev and endow sets with the trivial $\sigma$-algebra. - - property_id: cartesian filtered colimits + - property: cartesian filtered colimits reason: See MSE/5027218. - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: We already know that $\Set$ does not have this property. Now apply the contrapositive of the dual of this lemma to the functor $\Set \to \Meas$ which equips a set with the trivial $\sigma$-algebra. - - property_id: effective cocongruences + - property: effective cocongruences reason: 'The proof is similar to the one for $\Top$: Use the trivial $\sigma$-algebra on a two-point set.' special_objects: diff --git a/databases/catdat/data/categories/Met.yaml b/databases/catdat/data/categories/Met.yaml index 3fbd3bf4..8e13a962 100644 --- a/databases/catdat/data/categories/Met.yaml +++ b/databases/catdat/data/categories/Met.yaml @@ -16,92 +16,92 @@ related_categories: - PMet satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Met \to \Set$ and $\Set$ is locally small. - - property_id: strict initial object + - property: strict initial object reason: The empty metric space is initial and clearly strict. - - property_id: generator + - property: generator reason: The one-point metric space is a generator since it represents the forgetful functor $\Met \to \Set$. - - property_id: cogenerator + - property: cogenerator reason: 'We claim that $\IR$ with the usual metric is a cogenerator. Let $a,b \in X$ be two points of a metric space such that $f(a)=f(b)$ for all non-expansive maps $f : X \to \IR$. This applies in particular to $f(x) := d(a,x)$ and shows that $0=d(a,a)=d(a,b)$, so that $a=b$.' - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Every non-empty metric space is weakly terminal (by using constant maps). - - property_id: equalizers + - property: equalizers reason: Just restrict the metric to the equalizer built from the sets. - - property_id: binary products + - property: binary products reason: The product of two metric spaces $(X,d)$, $(Y,d)$ is $(X \times Y,d)$ with $d((x_1,y_1),(x_2,x_2)) := \sup(d(x_1,x_2),d(y_1,y_2))$. check_redundancy: false - - property_id: terminal object + - property: terminal object reason: The one-point metric space is terminal. check_redundancy: false - - property_id: coequalizers + - property: coequalizers reason: This is because the category of pseudo-metric spaces $\PMet$ has coequalizers and $\Met \hookrightarrow \PMet$ has a left adjoint, mapping a pseudo-metric space $X$ to $X /{\sim}$ where $x \sim y \iff d(x,y)=0$. Concretely, we take the coequalizer in the category of pseudo-metric spaces and then identify points with distance zero. - - property_id: filtered colimits + - property: filtered colimits reason: This is because the category of pseudo-metric spaces $\PMet$ has filtered colimits and $\Met \hookrightarrow \PMet$ has a left adjoint, mapping a pseudo-metric space $X$ to $X /{\sim}$ where $x \sim y \iff d(x,y)=0$. Concretely, we take the filtered colimit in the category of pseudo-metric spaces and then identify points with distance zero. check_redundancy: false - - property_id: cartesian filtered colimits + - property: cartesian filtered colimits reason: >- We already saw that filtered colimits and finite products exist. The canonical map $\colim_i (X \times Y_i) \to X \times \colim_i Y_i$ is an isomorphism for filtered diagrams $(Y_i)$: It is surjective by the concrete description of filtered colimits. It is isometric because of the elementary observation $$\textstyle\inf_i \max(r, s_i) = \max(r, \inf_i s_i)$$ for $r, s_i \in \IR$, where $i \leq j \implies s_i \geq s_j$. - - property_id: well-powered + - property: well-powered reason: This follows since monomorphisms are injective. - - property_id: well-copowered + - property: well-copowered reason: 'If $f : X \to Y$ is an epimorphism, then $f(X)$ is dense in $Y$ (see below). Hence, there is an injective map $Y \to X^{\IN}$, which bounds the size of $Y$.' unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: essentially small + - property: essentially small reason: This is trivial. - - property_id: locally finite + - property: locally finite reason: This is obvious. - - property_id: strict terminal object + - property: strict terminal object reason: This is trivial. - - property_id: balanced + - property: balanced reason: The inclusion $\IQ \hookrightarrow \IR$ is a counterexample; it is an epimorphism since $\IQ$ is dense in $\IR$. Alternatively, consider the identity map $(X,2d) \to (X,d)$ for any non-trivial metric space $(X,d)$. - - property_id: Malcev + - property: Malcev reason: 'Consider the metric subspace $\{(a,b) \in \IR^2 : a \leq b\}$ of $\IR^2$.' - - property_id: countable powers + - property: countable powers reason: 'Assume that the power $P = \IR^{\IN}$ exists, where $\IR$ has the usual metric. Since the forgetful functor $\Met \to \Set$ is representable, it preserves limits, powers in particular. Thus, we may assume that $P$ is the set of sequences of numbers and that the projections $p_n : P \to \IR$ are given by $p_n(x) = x_n$. Now consider the sequences $x = (n)_n$ and $y = (0)_n$. Since each $p_n$ is non-expansive, we get $d(x,y) \geq d(p_n(x),p_n(y)) = d(n,0) = n$. But then $d(x,y) = \infty$, a contradiction.' - - property_id: binary copowers + - property: binary copowers reason: The coproduct of two non-empty metric spaces does not exist, see MSE/1778408. For example, the copower $\IR \sqcup \IR$ does not exist. We only get coproducts when allowing $\infty$ as a distance, as in $\Met_{\infty}$. - - property_id: cartesian closed + - property: cartesian closed reason: This is proven in MSE/5131457. - - property_id: filtered-colimit-stable monomorphisms + - property: filtered-colimit-stable monomorphisms reason: 'The following example is taken from Remark 2.7 in Approximate injectivity and smallness in metric-enriched categories by Adamek-Rosicky: For $n \geq 1$ let $X_n$ denote the metric space with underlying set $\{0,1\}$ in which $0,1$ have distance $1/n$. We have bijective non-expansive maps $X_n \to X_{n+1}$, $x \mapsto x$. The colimit of this sequence in $\PMet$ is $\{0,1\}$ where $0,1$ have distance $0$, so the colimit in $\Met$ collapses to $\{0\}$. Therefore, the colimit of the monomorphisms $X_0 \to X_n$, $x \mapsto x$ is the non-injective map $X_0 \to \{0\}$.' - - property_id: natural numbers object + - property: natural numbers object reason: >- If $(N,z,s)$ is a natural numbers object in $\Met$, then $$1 \xrightarrow{z} N \xleftarrow{s} N$$ is a coproduct cocone by Johnstone, Part A, Lemma 2.5.5. Since there is a map $1 \to N$, we have $N \neq \varnothing$. However, the coproduct of two non-empty metric spaces does not exist, see MSE/1778408. - - property_id: effective congruences + - property: effective congruences reason: 'Any kernel pair of $h : X \to Z$ in $\Met$ corresponds to a closed subset of $X\times X$. However, there are plenty of non-closed congruences, such as $\Delta \cup (\IQ \times \IQ) \subseteq \IR \times \IR$ with the subspace metric.' - - property_id: effective cocongruences + - property: effective cocongruences reason: >- We will define a cocongruence on the interval $(0,1) \subseteq \IR$ where $E := (-1, 0) \cup (0, 1) \subseteq \IR$, and the two maps $(0, 1) \rightrightarrows E$ are the inclusion map and $x \mapsto -x$. Then for any metric space $X$, the induced relation on non-expansive maps $(0, 1) \to X$ is that $f \sim g$ if and only if $$d(f(x), g(y)) \le x+y$$ diff --git a/databases/catdat/data/categories/Met_c.yaml b/databases/catdat/data/categories/Met_c.yaml index 0eb88a30..bb8c99c1 100644 --- a/databases/catdat/data/categories/Met_c.yaml +++ b/databases/catdat/data/categories/Met_c.yaml @@ -16,60 +16,60 @@ related_categories: - Top satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Met_c \to \Set$ and $\Set$ is locally small. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Every non-empty metric space is weakly terminal (by using constant maps). - - property_id: equalizers + - property: equalizers reason: Just restrict the metric to the equalizer built from the sets. - - property_id: countable products + - property: countable products reason: For finite products, we take the cartesian product with, say, the sup-metric. The product of countably many metric spaces $(X_i,d_i)_{i \geq 0}$ is given by the cartesian product $\prod_{i \geq 0} X_i$ with the metric $d(x,y) := \sum_{i \geq 0} d_i(x_i,y_i)/(1 + d_i(x_i,y_i))$. See Engelking's book General Topology. - - property_id: coproducts + - property: coproducts reason: See MSE/5004389. check_redundancy: false - - property_id: generator + - property: generator reason: The one-point metric space is a generator since it represents the forgetful functor $\Met_c \to \Set$. - - property_id: cogenerator + - property: cogenerator reason: The same proof as for $\Met$ shows that $\IR$ with the usual metric is a cogenerator. - - property_id: well-powered + - property: well-powered reason: This follows easily from the fact that monomorphisms are injective in this category. - - property_id: well-copowered + - property: well-copowered reason: 'If $f : X \to Y$ is an epimorphism, then $f(X)$ is dense in $Y$ (see below). Hence, there is an injective map $Y \to X^{\IN}$, which bounds the size of $Y$.' - - property_id: infinitary extensive + - property: infinitary extensive reason: This follows from the existence of coproducts and finite products, and from the fact that $\Top$ is infinitary extensive. - - property_id: effective cocongruences + - property: effective cocongruences reason: 'Suppose we have a cocongruence $f, g : X \rightrightarrows E$ in $\Met_\c$. Then the image in $\Haus$ is a coreflexive corelation (since epimorphisms in both categories are continuous maps with dense image). By MO/509548, that implies that image is of the form $X +_S X$ for a closed subset $S$ of $X$. Since $S$ is metrizable, and the functor $\Met_\c \to \Haus$ is fully faithful and therefore reflects colimits, we conclude that $E$ is effective in $\Met_\c$.' unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: The inclusion $\IQ \hookrightarrow \IR$ provides a counterexample. - - property_id: powers + - property: powers reason: See MSE/139168 for a proof that uncountable powers do not exist. - - property_id: Malcev + - property: Malcev reason: 'Consider the metric subspace $\{(a,b) \in \IR^2 : a \leq b\}$ of $\IR^2$.' - - property_id: regular subobject classifier + - property: regular subobject classifier reason: 'We recycle the proof from $\Haus$: Assume that there is a regular subobject classifier $\Omega$. By the classification of regular monomorphisms, we would have an isomorphism between $\Hom(X,\Omega)$ and the set of closed subsets of $X$ for any metric space $X$. If we take $X = 1$ we see that $\Omega$ has two points. Since $\Omega$ is Hausdorff, $\Omega \cong 1 + 1$ must be discrete. But then $\Hom(X,\Omega)$ is isomorphic to the set of all clopen subsets of $X$, of which there are usually far fewer than closed subsets (consider $X = [0,1]$).' - - property_id: sequential colimits + - property: sequential colimits reason: See MO/510316. - - property_id: quotients of congruences + - property: quotients of congruences reason: If $\Met_c$ had quotients of congruences, then by this lemma it would have sequential colimits of sequences of monomorphisms. This contradicts MO/510316. special_objects: diff --git a/databases/catdat/data/categories/Met_oo.yaml b/databases/catdat/data/categories/Met_oo.yaml index 11141988..f79db0b9 100644 --- a/databases/catdat/data/categories/Met_oo.yaml +++ b/databases/catdat/data/categories/Met_oo.yaml @@ -14,50 +14,50 @@ related_categories: - Met_c satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Met_{\infty} \to \Set$ and $\Set$ is locally small. - - property_id: generator + - property: generator reason: The singleton metric space $1$ is a generator, since morphisms $1 \to X$ correspond to the elements of $X$. - - property_id: cogenerator + - property: cogenerator reason: 'The proof is similar to $\Met$, a cogenerator is given by $\IR \cup \{\infty\}$ with the metric in which $d(a,\infty)=\infty$ for $a \in \IR$. Then one checks that the maps $d(a,-) : X \to \IR \cup \{\infty\}$ are non-expansive and finishes as for $\Met$.' - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Every non-empty metric space is weakly terminal (by using constant maps). - - property_id: locally ℵ₁-presentable + - property: locally ℵ₁-presentable reason: This is Example 4.5 in Metric abstract elementary classes as accessible categories. - - property_id: cartesian filtered colimits + - property: cartesian filtered colimits reason: We can use the same proof as for $\Met$ since the equation $\inf_i \max(r, s_i) = \max(r, \inf_i s_i)$ also holds for for $r, s_i \in \IR \cup \{\infty\}$. - - property_id: infinitary extensive + - property: infinitary extensive reason: '[Sketch] Since $\Set$ is infinitary extensive, a map $f : Y \to \coprod_i X_i =: X$ corresponds to a decomposition $Y = \coprod_i Y_i$ (as sets) with maps $f_i : Y_i \to X_i$. Endow $Y_i$ with the restricted metric. If $f$ is non-expansive, each $f_i$ is non-expansive, and for $x_i \in Y_i$ and $i \neq j$ we have $d_Y(x_i,x_j) \geq d_X(f(x_i),f(x_j)) = \infty$, so that $Y = \coprod_i Y_i$ holds as metric spaces.' unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: The inclusion $\IQ \hookrightarrow \IR$ provides a counterexample. Alternatively, consider the identity map $(X,2d) \to (X,d)$ for any non-trivial metric space $(X,d)$. - - property_id: Malcev + - property: Malcev reason: 'Consider the metric subspace $\{(a,b) \in \IR^2 : a \leq b\}$ of $\IR^2$.' - - property_id: co-Malcev + - property: co-Malcev reason: 'See MO/509552: Consider the forgetful functor $U : \Met_{\infty} \to \Set$ and the relation $R \subseteq U^2$ defined by $R(X) := \{(a,b) \in U(X)^2 : d(x,y) \leq 1 \}$. Both are representable: $U$ by the singleton metric space and $R$ by the metric space $\{ 0,1 \}$ where $d(0,1) := 1$. It is clear that $R$ is reflexive, but not transitive.' - - property_id: cartesian closed + - property: cartesian closed reason: This is proven in MSE/5131457. - - property_id: filtered-colimit-stable monomorphisms + - property: filtered-colimit-stable monomorphisms reason: We can copy the proof from $\Met$. - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: We already know that $\Set$ does not have this property. Now apply the contrapositive of the dual of this lemma to the functor $\Set \to \Met_{\infty}$ that equips a set with the discrete topology. - - property_id: effective cocongruences + - property: effective cocongruences reason: The same counterexample as for $\Met$ works here. The difference in this case is that a binary copower of two copies of $(0,1)$ does exist in $\Met_\infty$. However, this would assign a distance of $\infty$ between points in $(-1,0)$ and points in $(0,1)$, which does not agree with the chosen subspace metric on $(-1,0) \cup (0,1)$. special_objects: diff --git a/databases/catdat/data/categories/Mon.yaml b/databases/catdat/data/categories/Mon.yaml index ba01b087..cc00e780 100644 --- a/databases/catdat/data/categories/Mon.yaml +++ b/databases/catdat/data/categories/Mon.yaml @@ -16,58 +16,58 @@ related_categories: - SemiGrp satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Mon \to \Set$ and $\Set$ is locally small. - - property_id: pointed + - property: pointed reason: The trivial monoid is a zero object. check_redundancy: false - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic of a monoid. - - property_id: unital + - property: unital reason: If a submonoid of $X \times Y$ contains $X \times \{1\}$ and $\{1\} \times Y$, then for all $x \in X$ and $y \in Y$ it also contains $(x,1) \cdot (1,y) = (x,y)$. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: The inclusion of additive monoids $\IN \hookrightarrow \IZ$ is a counterexample. - - property_id: Malcev + - property: Malcev reason: 'Consider the submonoid $\{(a,b) : a \leq b \}$ of $\IN^2$.' - - property_id: cogenerator + - property: cogenerator reason: 'We apply this lemma to the collection of simple groups: Any non-trivial homomorphism $G \to M$ from a simple group $G$ to a monoid $M$ must be injective (as it corestricts to a homomorphism of groups $G \to M^{\times}$), and for every infinite cardinal $\kappa$ there is a simple group of size $\geq \kappa$ (for example, the alternating group on $\kappa$ elements).' - - property_id: counital + - property: counital reason: The canonical morphism $\IN \sqcup \IN \to \IN \times \IN$ is not a monomorphism since $\IN \sqcup \IN$ is not commutative. - - property_id: CIP + - property: CIP # TODO: remove code duplication with "counital" proof reason: The canonical morphism $\IN \sqcup \IN \to \IN \times \IN$ is not a monomorphism since $\IN \sqcup \IN$ is not commutative. - - property_id: CSP + - property: CSP reason: If $M \to N$ is an epimorphism in $\Mon$ and $M$ is infinite, then $\card(N) \leq \card(M)$ (see MO/510431). This implies that in $\Mon$ the canonical homomorphism $\coprod_{n \geq 0} \IN \to \prod_{n \geq 0} \IN$ is not an epimorphism because its domain is countable and its codomain is uncountable. - - property_id: coregular + - property: coregular reason: 'Consider the monoid $M := \langle x_0, x_1, s : x_0 s = x_1 s = 1 \rangle$. Notice that every element in $M$ has a unique expression as $s^k \cdot u$ with $k \in \IN$ and $u \in \langle x_0,x_1 \rangle_M$. Moreover, the canonical homomorphism $\iota : \langle x_0, x_1 \rangle \to M$ (from the free monoid) is injective. We will prove that it is a regular monomorphism, which however is not pushout-stable. Consider $N := \langle x_0, x_1, s_0, s_1 : x_i s_j = 1 \rangle$ and define $f_i : M \to N$ for $i=0,1$ by $f_i(x_j) = x_j$ and $f_i(s) = s_i$. Then $\iota$ is the equalizer of $f_0,f_1$. Now consider $g : \langle x_0,x_1 \rangle \to \langle y_0 \rangle$ defined by $g(x_0) = y_0$, $g(x_1) = 1$. The pushout of $\iota$ with $g$ is given by $\langle x_0, x_1, s, y_0 : x_0 s = x_1 s = 1 , \, x_0 = y_0, \, x_1 = 1 \rangle$, which simplifies to $\langle x_0, s : x_0 s = s = 1 \rangle$, which is trivial.' - - property_id: regular subobject classifier + - property: regular subobject classifier reason: 'Assume that $\Omega$ is a regular subobject classifier. Since the trivial monoid is a zero object, every regular submonoid $U \subseteq M$ of any monoid $M$ would have the form $\{m \in M : h(m) = 1 \}$ for some homomorphism $M \to \Omega$. Now take any monoid $M$ with zero that has two different homomorphisms with zero $f,g : M \rightrightarrows N$ (for example, let $M = N = \{0\} \cup \{x^n : n \geq 0\}$ be the free monoid with zero on one generator, $f(x) = 0$,and $g(x) = x$). Take their equalizer $U \subseteq M$, and choose a homomorphism $h : M \to \Omega$ with $U = \{m \in M : h(m) = 1\}$. Since $0 \in U$, we have $h(0)=1$. But then for all $m \in M$ we have $h(m) = h(m) h(0) = h(m 0) = h(0) = 1$, i.e. $U = M$, which yields the contradiction $f = g$.' - - property_id: regular quotient object classifier + - property: regular quotient object classifier reason: We can just copy the proof for $\CMon$. Alternatively, we may use this lemma (dualized). - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: We already know that $\Grp$ does not have this property. Now apply the contrapositive of the dual of this lemma to the forgetful functor $\Grp \to \Mon$. It preserves epimorphisms since it has a right adjoint, the unit group functor. - - property_id: cocartesian cofiltered limits + - property: cocartesian cofiltered limits reason: 'We know that $\Grp$ fails to satisfy this property. The same counterexample works here since the inclusion $\Grp \hookrightarrow \Mon$ preserves limits and colimits (it has a left and a right adjoint) and is conservative. A similar counterexample is given by the free monoids $N_n = \langle x_1,\dotsc,x_n \rangle$ and the Boolean monoid $M = \langle e : e^2=e \rangle$ with the maps $N_{n+1} \to N_n$, $x_{n+1} \mapsto 1$. Then the element $(x_1 e \cdots x_n e) \in \lim_n (M \sqcup N_n)$ does not come from $M \sqcup \lim_n N_n$ because its components have unbounded free product length.' - - property_id: effective cocongruences + - property: effective cocongruences reason: >- We adapt the counterexample from MO/510744 for $\Ring$. Namely, consider the monoids $$\begin{align*} X & := \langle p \mid p^2 = p \rangle \cong (\{ 0, 1 \}, \cdot),\\ E & := \langle p, q \mid p^2 = p,\, q^2 = q,\, pq = q,\, qp = p \rangle. \end{align*}$$ diff --git a/databases/catdat/data/categories/N.yaml b/databases/catdat/data/categories/N.yaml index d3434864..17ed0f07 100644 --- a/databases/catdat/data/categories/N.yaml +++ b/databases/catdat/data/categories/N.yaml @@ -16,34 +16,34 @@ related_categories: - Z_div satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: countable + - property: countable reason: This is trivial. - - property_id: thin + - property: thin reason: This is trivial. check_redundancy: false - - property_id: finitely cocomplete + - property: finitely cocomplete reason: Finitely many natural numbers have a supremum natural number. check_redundancy: false - - property_id: binary products + - property: binary products reason: Two natural numbers have a minimum. - - property_id: connected limits + - property: connected limits reason: Every non-empty set of natural numbers has a minimum. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: This is trivial. - - property_id: direct + - property: direct reason: This is because the natural numbers with respect to $<$ are well-founded. unsatisfied_properties: - - property_id: countable coproducts + - property: countable coproducts reason: The numbers $0,1,2,\dotsc$ have no supremum, i.e. no coproduct. special_objects: diff --git a/databases/catdat/data/categories/N_oo.yaml b/databases/catdat/data/categories/N_oo.yaml index 78ed12f2..9f153895 100644 --- a/databases/catdat/data/categories/N_oo.yaml +++ b/databases/catdat/data/categories/N_oo.yaml @@ -15,36 +15,36 @@ related_categories: - On satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: countable + - property: countable reason: This is trivial. - - property_id: coproducts + - property: coproducts reason: Take the supremum. check_redundancy: false - - property_id: semi-strongly connected + - property: semi-strongly connected reason: This is trivial. - - property_id: direct + - property: direct reason: This is because the natural numbers with $\infty$ with respect to $<$ are well-founded. - - property_id: locally strongly finitely presentable + - property: locally strongly finitely presentable reason: We already saw that coproducts, and therefore colimits exist. Every natural number is strongly finitely presentable, and $\infty$ is the colimit of all $n < \infty$. unsatisfied_properties: - - property_id: essentially finite + - property: essentially finite reason: This is trivial. - - property_id: self-dual + - property: self-dual reason: This is because every upper set is infinite, but every lower set is finite. - - property_id: inverse + - property: inverse reason: Consider the strictly increasing sequence $0 < 1 < 2 < \cdots$. - - property_id: finitary algebraic + - property: finitary algebraic reason: This follows from this lemma. special_objects: diff --git a/databases/catdat/data/categories/On.yaml b/databases/catdat/data/categories/On.yaml index c07ec375..b16a0323 100644 --- a/databases/catdat/data/categories/On.yaml +++ b/databases/catdat/data/categories/On.yaml @@ -14,39 +14,39 @@ related_categories: - N satisfied_properties: - - property_id: thin + - property: thin reason: This is trivial. check_redundancy: false - - property_id: locally small + - property: locally small reason: This is trivial. - - property_id: cocomplete + - property: cocomplete reason: Every set of ordinal numbers has a supremum. - - property_id: binary products + - property: binary products reason: For ordinal numbers $\alpha,\beta$ their minimum exists, it is $\alpha$ when $\alpha \leq \beta$ and $\beta$ otherwise. - - property_id: connected limits + - property: connected limits reason: Every non-empty set of ordinal numbers has a minimum. - - property_id: well-powered + - property: well-powered reason: This is because for every ordinal $\alpha$ the collection of ordinals $\leq \alpha$ is a set (namely, $\alpha + 1$). - - property_id: semi-strongly connected + - property: semi-strongly connected reason: It is well-known that for ordinals $\alpha,\beta$ we have $\alpha \leq \beta$ or $\beta \leq \alpha$. - - property_id: direct + - property: direct reason: This is because the ordinal numbers with respect to $<$ are well-founded (by definition). unsatisfied_properties: - - property_id: terminal object + - property: terminal object reason: There is no largest ordinal $\alpha$ since $\alpha + 1$ will always be larger. - - property_id: well-copowered + - property: well-copowered reason: The "quotients" of $0$ are all ordinals. - - property_id: inverse + - property: inverse reason: Consider the strictly increasing sequence $0 < 1 < 2 < \cdots$. special_objects: diff --git a/databases/catdat/data/categories/PMet.yaml b/databases/catdat/data/categories/PMet.yaml index f409cd3a..5237d792 100644 --- a/databases/catdat/data/categories/PMet.yaml +++ b/databases/catdat/data/categories/PMet.yaml @@ -13,84 +13,84 @@ related_categories: - Met satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\PMet \to \Set$ and $\Set$ is locally small. - - property_id: generator + - property: generator reason: The one-point (pseudo-)metric space is a generator since it represents the forgetful functor $\PMet \to \Set$. - - property_id: cogenerator + - property: cogenerator reason: The set $\{0,1\}$ equipped with the pseudo-metric $d(0,1)=0$ is a cogenerator since every map into is automatically non-expansive and since $\{0,1\}$ is a cogenerator in $\Set$. - - property_id: strict initial object + - property: strict initial object reason: The empty (pseudo-)metric space is initial and clearly strict. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Every non-empty pseudo-metric space is weakly terminal (by using constant maps). - - property_id: well-powered + - property: well-powered reason: This follows since monomorphisms are injective (see below). - - property_id: well-copowered + - property: well-copowered reason: This follows since epimorphisms are surjective (see below). - - property_id: equalizers + - property: equalizers reason: Just restrict the pseudo-metric to the equalizer built from the sets. check_redundancy: false - - property_id: binary products + - property: binary products reason: The product of two pseudo-metric spaces $(X,d)$, $(Y,d)$ is $(X \times Y,d)$ with $d((x_1,y_1),(x_2,x_2)) := \sup(d(x_1,x_2),d(y_1,y_2))$. check_redundancy: false - - property_id: terminal object + - property: terminal object reason: The one-point (pseudo-)metric space is terminal. check_redundancy: false - - property_id: coequalizers + - property: coequalizers reason: See MO/123739. - - property_id: filtered colimits + - property: filtered colimits reason: 'Given a filtered diagram $(X_i)$ of pseudo-metric spaces, take the filtered colimit $X$ of the underlying sets with the following pseudo-metric: If $x,y \in X$, let $d(x,y)$ be infimum of all $d(x_i,y_i)$, where $x_i,y_i \in X_i$ are some preimages of $x,y$ in some $X_i$. The definition ensures that each $X_i \to X$ is non-expansive, and the universal property is easy to check.' check_redundancy: false - - property_id: exact filtered colimits + - property: exact filtered colimits reason: 'We already saw that finite limits and filtered colimits exist. Now let $\I$ be a finite category and $\J$ be a small filtered category, w.l.o.g. a directed poset. Let $X : \I \times \J \to \PMet$ be a diagram. We need to show that the canonical map $\colim_{j \in \J} \lim_{i \in \I} X(i,j) \to \lim_{i \in \I} \colim_{j \in \J} X(i,j)$ is an isomorphism. It is bijective since the forgetful functor to $\Set$ preserves finite limits and filtered colimits and since $\Set$ has exact filtered colimits. That the map is isometric can easily be reduced to the following lemma: If $d_{i,j} \in \IR_{\geq 0}$ are numbers for $i \in \I$, $j \in \J$ with $j \leq k \implies d_{i,k} \leq d_{i,j}$, then $\inf_j \sup_i d_{i,j} = \sup_i \inf_j d_{i,j}$. This can be proven directly. Alternatively, use that the thin category $(\IR_{\geq 0} \cup \{\infty\},\leq)$ is isomorphic to $([0,1],\leq)$, and we already know that it has exact filtered colimits.' unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: essentially small + - property: essentially small reason: This is trivial. - - property_id: locally finite + - property: locally finite reason: This is obvious. - - property_id: strict terminal object + - property: strict terminal object reason: This is trivial. - - property_id: balanced + - property: balanced reason: 'Let $d : \IR \times \IR \to \IR_{\geq 0}$ be the usual Euclidean metric on $\IR$ and $0 : \IR \times \IR \to \IR_{\geq 0}$ be the zero pseudo-metric. Then the identity map $(\IR,d) \to (\IR,0)$ provides a counterexample.' - - property_id: Malcev + - property: Malcev reason: Take any counterexample in $\Set$ and equip it with the zero pseudo-metric. - - property_id: cartesian closed + - property: cartesian closed reason: This is proven in MSE/5131457. - - property_id: countable powers + - property: countable powers reason: 'Assume that the power $P = \IR^{\IN}$ exists, where $\IR$ has the usual (pseudo-)metric. Since the forgetful functor $\PMet \to \Set$ is representable, it preserves limits, powers in particular. Thus, we may assume that $P$ is the set of sequences of numbers and that the projections $p_n : P \to \IR$ are given by $p_n(x) = x_n$. Now consider the sequences $x = (n)_n$ and $y = (0)_n$. Since each $p_n$ is non-expansive, we get $d(x,y) \geq d(p_n(x),p_n(y)) = d(n,0) = n$. But then $d(x,y) = \infty$, a contradiction.' - - property_id: binary copowers + - property: binary copowers reason: The coproduct of two non-empty pseudo-metric spaces does not exist, see MSE/1778408 (the proof also works for pseudo-metric spaces). For example, the copower $\IR \sqcup \IR$ does not exist. We only get coproducts when allowing $\infty$ as a distance. - - property_id: natural numbers object + - property: natural numbers object reason: >- If $(N,z,s)$ is a natural numbers object in $\PMet$, then $$1 \xrightarrow{z} N \xleftarrow{s} N$$ is a coproduct cocone by Johnstone, Part A, Lemma 2.5.5. Since there is a map $1 \to N$, we have $N \neq \varnothing$. However, the coproduct of two non-empty pseudo-metric spaces does not exist, see MSE/1778408. - - property_id: effective cocongruences + - property: effective cocongruences reason: 'The proof is similar to the one for $\Top$: Equip a two-point set with the zero metric; this pseudo-metric space represents the functor taking a pseudo-metric space to the pairs of points with $d(x,y) = 0$. In this case, once you conclude $Z = \varnothing$, the map $h : Z \to 1$ does not have any cokernel pair, since that would have to be a coproduct $1+1$, which does not exist.' special_objects: diff --git a/databases/catdat/data/categories/Pos.yaml b/databases/catdat/data/categories/Pos.yaml index 402d17a5..6bd3a976 100644 --- a/databases/catdat/data/categories/Pos.yaml +++ b/databases/catdat/data/categories/Pos.yaml @@ -14,53 +14,53 @@ related_categories: - Prost satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Pos \to \Set$ and $\Set$ is locally small. - - property_id: locally finitely presentable + - property: locally finitely presentable reason: See Adamek-Rosicky, Example 1.10. - - property_id: cartesian closed + - property: cartesian closed reason: For posets $P,Q$ we endow $\Hom(P,Q)$ with the partial order in which $f \leq g$ holds iff $f(p) \leq g(p)$ for all $p \in P$. The universal evaluation map is $\Hom(P,Q) \times P \to Q$, $(f,p) \mapsto f(p)$, it is order-preserving, and it satisfies the universal property. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Every non-empty poset is weakly terminal (by using constant maps). - - property_id: generator + - property: generator reason: The singleton poset $1$ is a generator, since morphisms $1 \to P$ correspond to the elements of $P$. - - property_id: cogenerator + - property: cogenerator reason: 'We prove that the poset $\{0 < 1\}$ is a cogenerator: Let $P$ be a poset and $a,b \in P$ be two elements such that $f(a) = f(b)$ for all order-preserving maps $f : P \to \{0 < 1 \}$. This means that $a$ and $b$ lie in the same upper sets. In particular, $b$ lies in the upper set generated by $a$, meaning $a \leq b$, and similarly we deduce $b \leq a$. Thus, $a = b$.' - - property_id: infinitary extensive + - property: infinitary extensive reason: '[Sketch] Since $\Set$ is infinitary extensive, a map $f : P \to \coprod_i Q_i$ corresponds to a decomposition $P = \coprod_i P_i$ (as sets) with maps $f_i : P_i \to Q_i$. Endow $P_i$ with the induced order. If $f$ is order-preserving, the elements in different $P_i$ cannot be comparable (since their $f$-images are not comparable), so that $P = \coprod_i P_i$ as posets, and each $f_i$ is order-preserving.' - - property_id: coregular + - property: coregular reason: See MSE/5130295. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: The inclusion $\{0,1\} \to \{0 < 1\}$ provides a counterexample (where in the domain there is no relation between $0$ and $1$). - - property_id: regular + - property: regular reason: See Example 3.14 at the nLab. - - property_id: Malcev + - property: Malcev reason: 'Consider the subposet $\{(a,b) : a \leq b \}$ of $\IN^2$.' - - property_id: co-Malcev + - property: co-Malcev reason: 'See MO/509552: Consider the forgetful functor $U : \Pos \to \Set$ and the relation $R \subseteq U^2$ defined by $R(A) := \{(a,b) \in U(A)^2 : a \leq b\}$. Both are representable: $U$ by the singleton poset and $R$ by $\{0 \leq 1 \}$. It is clear that $R$ is reflexive, but not symmetric.' - - property_id: regular subobject classifier + - property: regular subobject classifier reason: Assume that there is a regular subobject classifier $\Omega$, so that (by the classification of regular monomorphisms) $\Hom(P,\Omega)$ is isomorphic to the set of subsets of $P$. By taking $P = \{0\}$ we see that $\Omega$ has $2$ elements, so that either $\Omega \cong \{0,1\}$ (discrete) or $\Omega \cong \{0 < 1\}$. By taking $P = \{0 < 1\}$ we see that $\Omega$ has four pairs $(x,y)$ with $x \leq y$. But $\{0,1\}$ has only two and $\{0 < 1\}$ has only three such pairs. - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: Pick any poset $X$ which has a decreasing sequence of non-empty sets $X = X_0 \supseteq X_1 \supseteq \cdots$ with empty intersection, such as $X_n = \IN_{\geq n}$ with the natural ordering. The unique map $X_n \to 1$ is surjective, but their limit $\varnothing \to 1$ is not surjective. - - property_id: effective cocongruences + - property: effective cocongruences reason: |- Let $X$ be $\IR$ with the standard (total) order, and let $E$ be the poset with underlying set $\IR \times \{ 0, 1 \}$ and partial order such that $(x, m) \le (y, n)$ if and only if $x < y$ or $(x, m) = (y, n)$. The two maps $\IR \rightrightarrows E$ will be $x \mapsto (x, 0)$ and $x \mapsto (x, 1)$ respectively. For any partial order $(\IP, \le)$, the induced equivalence relation on the set of order-preserving functions $\IR \to \IP$ is that $f \sim g$ if and only if $f(x) \le g(y)$ and $g(x) \le f(y)$ whenever $x < y$. This relation is clearly reflexive and symmetric; for transitivity, if $f \sim g$ and $g \sim h$, then whenever $x < y$, we have $f(x) \le g(\frac{x+y}{2}) \le h(y)$ and similarly $h(x) \le g(\frac{x+y}{2}) \le f(y)$, showing that $f \sim h$. On the other hand, if this cocongruence on $\IR$ were effective, then by the dual of this result, $E$ would be the cokernel pair of the equalizer of the two maps $\IR \rightrightarrows E$. However, that equalizer is the empty poset, so $E$ would have to be the coproduct poset $\IR + \IR$, giving a contradiction. diff --git a/databases/catdat/data/categories/Prost.yaml b/databases/catdat/data/categories/Prost.yaml index b5ad23fc..efb9412f 100644 --- a/databases/catdat/data/categories/Prost.yaml +++ b/databases/catdat/data/categories/Prost.yaml @@ -13,53 +13,53 @@ related_categories: - Pos satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Pos \to \Set$ and $\Set$ is locally small. - - property_id: locally finitely presentable + - property: locally finitely presentable reason: The same proof as for $\Pos$ works, cf. Adamek-Rosicky, Example 1.10. - - property_id: cartesian closed + - property: cartesian closed reason: For prosets $P,Q$ we endow $\Hom(P,Q)$ with the preorder in which $f \leq g$ holds iff $f(p) \leq g(p)$ for all $p \in P$. The universal evaluation map is $\Hom(P,Q) \times P \to Q$, $(f,p) \mapsto f(p)$, it is order-preserving, and it satisfies the universal property. - - property_id: generator + - property: generator reason: The singleton proset $1$ is a generator, since morphisms $1 \to P$ correspond to the elements of $P$. - - property_id: cogenerator + - property: cogenerator reason: Endow the set $\{ 0,1 \}$ with the preorder $0 \leq 1$, $1 \leq 0$ (which is not a partial order). Then every map $P \to \{0,1\}$ is order-preserving. Now the claim follows since the set $\{ 0,1 \}$ is a cogenerator in $\Set$. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Every non-empty proset is weakly terminal (by using constant maps). - - property_id: infinitary extensive + - property: infinitary extensive reason: '[Sketch] Since $\Set$ is infinitary extensive, a map $f : P \to \coprod_i Q_i$ corresponds to a decomposition $P = \coprod_i P_i$ (as sets) with maps $f_i : P_i \to Q_i$. Endow $P_i$ with the induced order. If $f$ is order-preserving, the elements in different $P_i$ cannot be comparable (since their $f$-images are not comparable), so that $P = \coprod_i P_i$ as prosets, and each $f_i$ is order-preserving.' - - property_id: coregular + - property: coregular reason: See MSE/5130295. - - property_id: regular subobject classifier + - property: regular subobject classifier reason: The set $\{0,1\}$ with the chaotic preorder $(0 \leq 1$, $1 \leq 0)$ is a regular subobject classifier since order-preserving maps $P \to \{0,1\}$ correspond to subsets of $P$. unsatisfied_properties: - - property_id: regular + - property: regular reason: See Example 3.14 at the nLab. - - property_id: balanced + - property: balanced reason: The inclusion $\{0,1\} \to \{0 < 1\}$ provides a counterexample (where in the domain there is no relation between $0$ and $1$). - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: Malcev + - property: Malcev reason: 'Consider the subproset $\{(a,b) : a \leq b \}$ of $\IN^2$.' - - property_id: co-Malcev + - property: co-Malcev reason: 'See MO/509552: Consider the forgetful functor $U : \Prost \to \Set$ and the relation $R \subseteq U^2$ defined by $R(A) := \{(a,b) \in U(A)^2 : a \leq b\}$. Both are representable: $U$ by the singleton preordered set and $R$ by $\{0 \leq 1 \}$. It is clear that $R$ is reflexive, but not symmetric.' - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: We know that $\Set$ does not have this property. Now use the contrapositive of the dual of this lemma applied to the functor $\Set \to \Prost$ that equips a set with the chaotic preorder. - - property_id: effective cocongruences + - property: effective cocongruences reason: 'Consider the proset $E := \{ a, b \}$ with the chaotic preorder. This represents the functor which sends a proset to the pairs of elements $x,y$ with $x \le y$ and $y \le x$. Therefore, it defines a cocongruence $1 \rightrightarrows E$, where the maps are the two possible functions. However, this cannot be effective: for any map $h : Z \to 1$ which equalizes the two functions, $Z$ must be empty. But that means the cokernel pair of $h$ is the two-element proset with the trivial preorder.' special_objects: diff --git a/databases/catdat/data/categories/R-Mod.yaml b/databases/catdat/data/categories/R-Mod.yaml index 8a69c37a..d0de0abb 100644 --- a/databases/catdat/data/categories/R-Mod.yaml +++ b/databases/catdat/data/categories/R-Mod.yaml @@ -17,23 +17,23 @@ related_categories: - Vect satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $R{-}\Mod \to \Set$ and $\Set$ is locally small. - - property_id: abelian + - property: abelian reason: This is standard, see Mac Lane, Ch. VIII. - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic theory of an $R$-module (given by the algebraic theory of an abelian group and for each $r \in R$ a unary operation). unsatisfied_properties: - - property_id: split abelian + - property: split abelian reason: By assumption, $R$ is not semisimple, so that there is a non-projective $R$-module, which yields a non-split sequence. - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: CSP + - property: CSP reason: The canonical homomorphism $\bigoplus_{n \geq 0} R \to \prod_{n \geq 0} R$ is not surjective, hence no epimorphism. special_objects: diff --git a/databases/catdat/data/categories/R-Mod_div.yaml b/databases/catdat/data/categories/R-Mod_div.yaml index 4854507b..d95fa991 100644 --- a/databases/catdat/data/categories/R-Mod_div.yaml +++ b/databases/catdat/data/categories/R-Mod_div.yaml @@ -14,20 +14,20 @@ related_categories: - Vect satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $R{-}\Mod \to \Set$ and $\Set$ is locally small. - - property_id: split abelian + - property: split abelian reason: It is a standard fact that the category of $R$-modules is abelian for any ring $R$, see Mac Lane, Ch. VIII. If $R$ is a division ring, then by linear algebra every $R$-module has a basis, hence is projective, so that every short exact sequence splits. - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic theory of an $R$-module (given by the algebraic theory of an abelian group and for each $r \in R$ a unary operation). unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: CSP + - property: CSP reason: The canonical homomorphism $\bigoplus_{n \geq 0} R \to \prod_{n \geq 0} R$ is not surjective, hence no epimorphism. special_objects: diff --git a/databases/catdat/data/categories/Rel.yaml b/databases/catdat/data/categories/Rel.yaml index cc610ddf..f5dab65f 100644 --- a/databases/catdat/data/categories/Rel.yaml +++ b/databases/catdat/data/categories/Rel.yaml @@ -13,52 +13,52 @@ related_categories: - Set satisfied_properties: - - property_id: locally small + - property: locally small reason: The set of morphisms $X \to Y$ is the set $P(X \times Y)$. - - property_id: self-dual + - property: self-dual reason: 'There is an isomorphism $\Rel \to \Rel^{\op}$ that is the identity on objects (sets) and maps a relation $R \subseteq X \times Y$ to the opposite relation $R^{\op} \subseteq Y \times X$ defined by $R^{\op} := \{(y,x) : (x,y) \in R \}$.' - - property_id: pointed + - property: pointed reason: The empty set is clearly both initial and terminal. The zero morphisms are the empty relations. check_redundancy: false - - property_id: generator + - property: generator reason: One checks that the the one-point set is a generator. - - property_id: coproducts + - property: coproducts reason: It is an easy exercise to deduce this from the corresponding fact for sets and that sets form a distributive category. check_redundancy: false - - property_id: biproducts + - property: biproducts reason: This is a consequence of the description of coproducts and products, both are disjoint unions (even for infinite families). - - property_id: CIP + - property: CIP reason: The canonical morphism from the coproduct to the product is the identity. - - property_id: well-powered + - property: well-powered reason: 'Any relation $R : A \to B$ induces an injective map $P(A) \to P(B)$, so that in particular there is an injective map $A \to P(B)$.' - - property_id: balanced + - property: balanced reason: See MSE/5030393. - - property_id: kernels + - property: kernels reason: 'Let $R : A \to B$ be a relation. Define $K = \bigl\{a \in A : \neg \exists b \in B ((a,b) \in R) \bigr\}$. We have an inclusion map $I : K \to A$, which can also be regarded as relation, and $R \circ I = \empty$ is the empty relation, i.e. the zero morphism. It is easy to check the universal property.' - - property_id: quotients of congruences + - property: quotients of congruences reason: A proof can be found here. - - property_id: effective congruences + - property: effective congruences reason: A proof can be found here. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: Cauchy complete + - property: Cauchy complete reason: See MSE/1931577. - - property_id: normal + - property: normal reason: The construction of equalizers in $\Rel$ shows that they are injective functions, but MSE/350716 shows that monomorphisms in $\Rel$ don't have to be functions. special_objects: diff --git a/databases/catdat/data/categories/Ring.yaml b/databases/catdat/data/categories/Ring.yaml index 56bab09e..08ee049a 100644 --- a/databases/catdat/data/categories/Ring.yaml +++ b/databases/catdat/data/categories/Ring.yaml @@ -18,56 +18,56 @@ comments: - It is likely that the epimorphisms can be described as for $\CRing$. satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Ring \to \Set$ and $\Set$ is locally small. - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic theory of a ring. - - property_id: strict terminal object + - property: strict terminal object reason: 'If $f : 0 \to R$ is a homomorphism, then $R$ satisfies $1=f(1)=f(0)=0$, so that $R=0$.' - - property_id: Malcev + - property: Malcev reason: This follows in the same way as for $\Grp$, see also Example 2.2.5 in Malcev, protomodular, homological and semi-abelian categories. - - property_id: disjoint finite products + - property: disjoint finite products reason: 'To show that $A \sqcup_{A \times B} B$ is trivial, let $R$ be a ring which admits homomorphisms $f : A \to R$, $g : B \to R$ with $f(p_1(a,b))=g(p_2(a,b))$ for all $(a,b) \in A \times B$, i.e. $f(a)=g(b)$. Applying this to $a=0$, $b=1$ yields $1=0$ in $R$.' unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: The inclusion $\IZ \hookrightarrow \IQ$ is a counterexample. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: This is because already the full subcategory $\CRing$ does not have this property. - - property_id: cogenerating set + - property: cogenerating set reason: 'We apply this lemma to the collection of fields: If $F$ is a field and $R$ is a non-trivial ring, any ring homomorphism $F \to R$ is injective. For every infinite cardinal $\kappa$ the field of rational functions in $\kappa$ variables has cardinality $\geq \kappa$ and a non-trivial automorphism (swap two variables).' - - property_id: codistributive + - property: codistributive reason: 'If $\sqcup$ denotes the coproduct of rings (see MSE/625874 for their description) and $R$ is a ring, the canonical morphism $R \sqcup \IZ^2 \to (R \sqcup \IZ)^2 = R^2$ is usually no isomorphism. For example, for $R = \IZ[X]$ the coproduct on the LHS is not commutative, it has the ring presentation $\langle X,E : E^2=E \rangle$.' - - property_id: co-Malcev + - property: co-Malcev reason: 'See MO/509552: Consider the forgetful functor $U : \Ring \to \Set$ and the relation $R \subseteq U^2$ defined by $R(A) := \{(a,b) \in U(A)^2 : ab = a^2\}$. Both are representable: $U$ by $\IZ[X]$ and $S$ by $\IZ \langle X,Y \rangle / \langle XY-X^2 \rangle$. It is clear that $R$ is reflexive, but not symmetric.' - - property_id: coregular + - property: coregular reason: 'Let $B := M_2(\IQ)$ and $A := \IQ^2$. Then $A \to B$, $(x,y) \mapsto \diag(x,y)$ is a regular monomorphism: A direct calculation shows that a matrix is diagonal iff it commutes with $M := \bigl(\begin{smallmatrix} 1 & 0 \\ 0 & 2 \end{smallmatrix}\bigr)$, so that $A \to B$ is the equalizer of the identity $B \to B$ and the conjugation $B \to B$, $X \mapsto M X M^{-1}$. Consider the homomorphism $A \to K$, $(a,b) \mapsto a$. We claim that $K \to K \sqcup_A B$ is not a monomorphism, because in fact, the pushout $K \sqcup_A B$ is zero: Since $A \to K$ is surjective with kernel $0 \times K$, the pushout is $B/\langle 0 \times K \rangle$, which is $0$ because $B$ is simple (proof) or via a direct calculation with elementary matrices.' - - property_id: regular quotient object classifier + - property: regular quotient object classifier reason: We may copy the proof for $\CRing$ (since the proof there did not use that $P$ is commutative). Alternatively, any regular quotient object classifier in $\Ring$ would produce one in $\CRing$ by this lemma (dualized). - - property_id: cocartesian cofiltered limits + - property: cocartesian cofiltered limits reason: >- Consider the ring $A = \IZ[X]$ and the sequence of rings $B_n = \IZ[Y]/(Y^{n+1})$ with projections $B_{n+1} \to B_n$, whose limit is $\IZ[[Y]]$. Every element in the coproduct of rings $\IZ[X] \sqcup \IZ[[Y]]$ has a finite "free product" length. Now consider the elements $$w_n = (1 + XY) (1+XY^2) \cdots (1+X Y^n) \in A \sqcup B_n.$$ Because of $w_n \equiv w_{n-1} \bmod Y^n$ these form an element $w \in \lim_n (A \sqcup B_n)$. Expanding $w_n$, the longest term is $XY XY^2 \cdots X Y^n$ of "free product" length $2n$, which is unbounded. - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: We know that $\CRing$ does not have this property. Now use the contrapositive of the dual of this lemma applied to the forgetful functor $\CRing \to \Ring$. It preserves epimorphisms by MSE/5133488. - - property_id: effective cocongruences + - property: effective cocongruences reason: See MO/510744. special_objects: diff --git a/databases/catdat/data/categories/Rng.yaml b/databases/catdat/data/categories/Rng.yaml index 3a033967..75ea489d 100644 --- a/databases/catdat/data/categories/Rng.yaml +++ b/databases/catdat/data/categories/Rng.yaml @@ -17,63 +17,63 @@ comments: - It is likely that the epimorphisms can be described as for $\CRing$. satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Rng \to \Set$ and $\Set$ is locally small. - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic theory of a rng. - - property_id: pointed + - property: pointed reason: The zero ring is a zero object. - - property_id: Malcev + - property: Malcev reason: This follows in the same way as for $\Grp$, see also Example 2.2.5 in Malcev, protomodular, homological and semi-abelian categories. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: The inclusion $\IZ \hookrightarrow \IQ$ is a counterexample. (The proof can be reduced to the unital case.) - - property_id: cogenerator + - property: cogenerator reason: 'We apply this lemma to the collection of fields: Any non-zero rng homomorphism from a field to a rng must be injective, and for every infinite cardinal $\kappa$ the field of rational functions in $\kappa$ variables has cardinality $\geq \kappa$.' - - property_id: counital + - property: counital reason: >- If $\IZ\langle X_1,\dotsc,X_n \rangle_0$ denotes the free rng on $n$ generators (non-commutative polynomials without constant term), then the canonical homomorphism $$\IZ\langle X,Y \rangle_0 = \IZ\langle X \rangle_0 \sqcup \IZ\langle Y \rangle_0 \to \IZ\langle X \rangle_0 \times \IZ\langle Y \rangle_0$$ is not a monomorphism since $\IZ\langle X,Y \rangle_0$ is not commutative. - - property_id: CIP + - property: CIP # TODO: remove code duplication with "counital" proof reason: >- If $\IZ\langle X_1,\dotsc,X_n \rangle_0$ denotes the free rng on $n$ generators (non-commutative polynomials without constant term), then the canonical homomorphism $$\IZ\langle X,Y \rangle_0 = \IZ\langle X \rangle_0 \sqcup \IZ\langle Y \rangle_0 \to \IZ\langle X \rangle_0 \times \IZ\langle Y \rangle_0$$ is not a monomorphism since $\IZ\langle X,Y \rangle_0$ is not commutative. - - property_id: CSP + - property: CSP reason: Assume that $\coprod_n \IZ \to \prod_n \IZ$ is an epimorphism in $\Rng$. Then $((\coprod_n \IZ)^+)^{\ab} \to \prod_n \IZ$ would be an epimorphism in $\CRing$, where $(-)^+$ denotes the unitalization and $(-)^{\ab}$ the abelianization. But if $R \to S$ is an epimorphism of commutative rings, then $\card(S) \leq \card(R)$ by SP/04W0. Since $((\coprod_n \IZ)^+)^{\ab}$ is countable and $\prod_n \IZ$ is not, we get a contradiction. - - property_id: regular subobject classifier + - property: regular subobject classifier reason: 'Assume that $\Rng$ has a subobject classifier $\Omega$. Since $0$ is a zero object, every regular subobject $R \subseteq S$ would be the kernel of some homomorphism $S \to \Omega$. In particular, it would be an ideal. Now take any pair of homomorphisms $f,g : S \rightrightarrows T$ in $\Ring$. Their equalizer $R \subseteq S$ is also the equalizer in $\Rng$, and it contains $1 \in S$. If it was an ideal, then $R = S$, and hence $f = g$, which is absurd.' - - property_id: coregular + - property: coregular reason: 'We can copy the proof for $\Ring$. In short, the inclusion of diagonal matrices $\IQ^2 \hookrightarrow M_2(\IQ)$ is a regular monomorphism, but becomes zero after taking the pushout with $p_1 : \IQ^2 \twoheadrightarrow \IQ$ because $M_2(\IQ)$ is simple.' - - property_id: regular quotient object classifier + - property: regular quotient object classifier reason: 'Assume that $\Rng$ has a regular quotient object classifier $P$. Consider the functor $N : \Ab \to \Rng$ that equips an abelian group with zero multiplication. It is fully faithful and has a left adjoint mapping a rng $R$ to the abelian group $R/R^2$. If $R$ is a rng with zero multiplication and $R \to S$ is a surjective homomorphism, then $S$ has zero multiplication. Therefore, the assumptions of this lemma (dualized) apply and we conclude that $P/P^2$ is a regular quotient object classifier of $\Ab$. But we already know that $\Ab$ has no such object (in fact, the only additive categories with such an object are trivial by MSE/4086192).' - - property_id: cocartesian cofiltered limits + - property: cocartesian cofiltered limits reason: >- Consider the ring $A = \IZ[X]$ and the sequence of rings $B_n = \IZ[Y]/(Y^{n+1})$ with projections $B_{n+1} \to B_n$, whose limit is $\IZ[[Y]]$ (both in $\Ring$ and $\Rng$). Every element in the coproduct of rngs $\IZ[X] \sqcup \IZ[[Y]]$ has a finite "free product" length. Now consider the elements $$w_n = (1 + XY) (1+XY^2) \cdots (1+X Y^n) - 1 \in A \sqcup B_n.$$ Because of $w_n \equiv w_{n-1} \bmod Y^n$ these form an element $w \in \lim_n (A \sqcup B_n)$. Expanding $w_n$, the longest term is $XY XY^2 \cdots X Y^n$ of "free product" length $2n$, which is unbounded. - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: 'We know that $\Ring$ does not have this property. Now use the contrapositive of the dual of this lemma applied to the forgetful functor $\Ring \to \Rng$. We only need to verify that it preserves epimorphisms: Let $f : R \to S$ be an epimorphism in $\Ring$ and let $g,h : S \rightrightarrows T$ be two homomorphisms of rngs with $gf = hf$. The element $e = g(1) = h(1) \in T$ is idempotent, and $g,h$ become homomorphisms of rings $S \rightrightarrows eTe$. Hence, $g=h$.' - - property_id: effective cocongruences + - property: effective cocongruences reason: >- The counterexample is similar to the one at MO/510744 for $\Ring$: in this case, $$X := \langle p \mid p^2 = p \rangle_{\Rng} \cong \IZ$$ diff --git a/databases/catdat/data/categories/Sch.yaml b/databases/catdat/data/categories/Sch.yaml index fb8404c6..535776be 100644 --- a/databases/catdat/data/categories/Sch.yaml +++ b/databases/catdat/data/categories/Sch.yaml @@ -19,41 +19,41 @@ comments: - Epimorphisms are discussed at MO/56564. Probably they cannot be classified. satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Sch \to \LRS$ and $\LRS$ is locally small. - - property_id: terminal object + - property: terminal object reason: The scheme $\Spec(\IZ)$ is terminal. - - property_id: pullbacks + - property: pullbacks reason: This is the well-known construction of the fiber product of schemes, see e.g. EGA I, Chap. I, Thm. 3.2.1. - - property_id: well-powered + - property: well-powered reason: See MO/160681. - - property_id: infinitary extensive + - property: infinitary extensive reason: One uses the same proof as for locally ringed spaces, using that open subspaces of schemes are also schemes. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: The canonical morphism $\Spec(\IZ/2 \times \IZ[1/2]) \longrightarrow \Spec(\IZ)$ is a mono- and an epimorphism, but no isomorphism. - - property_id: countable powers + - property: countable powers reason: While all diagrams of affine schemes have a limit in the category of schemes, one can show that an infinite product of quasi-compact non-empty schemes only exists when almost all of them are affine, see MO/65506. Thus, for example the power $(\IP^1)^{\IN}$ does not exist in $\Sch$. - - property_id: Malcev + - property: Malcev reason: Consider the subscheme $V(x-y) \cup V(y) \subseteq \IA^2$. It contains the diagonal but it is not symmetric. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: This is because already the full subcategory of affine schemes is not semi-strongly connected, because $\CRing$ is not semi-strongly connected. Specifically, there is no morphism between $\Spec(\IF_2)$ and $\Spec(\IF_3)$. - - property_id: generating set + - property: generating set reason: If $S$ is a generating set of schemes, then the set of affine open subsets of the schemes in $S$ would also be a generating set. This is then also a generating set in the category of affine schemes, corresponding to a cogenerating set in $\CRing$, which we know does not exist. - - property_id: quotients of congruences + - property: quotients of congruences reason: If $\Sch$ had quotients of congruences, then by this lemma it would also have pushouts of monomorphisms, contradicting the fact that the span $\IA^1 \leftarrow \Spec(k(t)) \rightarrow \IA^1$ has no pushout - see MO/9961. special_objects: diff --git a/databases/catdat/data/categories/SemiGrp.yaml b/databases/catdat/data/categories/SemiGrp.yaml index a22fd044..6acc26e6 100644 --- a/databases/catdat/data/categories/SemiGrp.yaml +++ b/databases/catdat/data/categories/SemiGrp.yaml @@ -14,19 +14,19 @@ related_categories: - Mon satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\SemiGrp \to \Set$ and $\Set$ is locally small. - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic theory of a semigroup. - - property_id: strict initial object + - property: strict initial object reason: This is because the initial object is the empty semigroup, and a non-empty set has no map to an empty set. - - property_id: disjoint coproducts + - property: disjoint coproducts reason: This follows easily from the concrete description of coproducts as (a variant of) free products. - - property_id: cocartesian cofiltered limits + - property: cocartesian cofiltered limits reason: >- We need to prove that for two cofiltered diagram of semigroups $(B_i)$, $(C_i)$ the canonical map $$\textstyle \alpha : \lim_i B_i + \lim_i C_i \to \lim_i (B_i + C_i)$$ @@ -35,26 +35,26 @@ satisfied_properties: with any positive product length, starting and ending either with $B_i$ or $C_i$. Moreover, $\lim_i$ commutes with arbitrary coproducts in $\Set$, and of course also with products. This shows that $\alpha$ is bijective. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: |- The inclusion of additive semigroups $\IN \hookrightarrow \IZ$ is a counterexample. Indeed, if $f,g : \IZ \to G$ are semigroup homomorphisms which agree on $\IN$, the element $e := f(0) = g(0)$ provides a neutral element for $eGe$, which therefore becomes a monoid, and $f,g$ corestrict to monoid homomorphisms $f,g : \IZ \to eGe$ which agree on $\IN$. And we already know that $f = g$ in that case (inverses are uniquely determined). Another example can be found in MO/510431. - - property_id: Malcev + - property: Malcev reason: 'Consider the subsemigroup $\{(a,b) : a \leq b \}$ of $\IN^2$ under addition.' - - property_id: co-Malcev + - property: co-Malcev reason: 'See MO/509552: Consider the forgetful functor $U : \SemiGrp \to \Set$ and the relation $R \subseteq U^2$ defined by $R(A) := \{(a,b) \in U(A)^2 : ab = a^2\}$. Both are representable: $U$ by the free semigroup on a single generator and $R$ by the free semigroup on two generators $x,y$ subject to the relation $xy=x^2$. It is clear that $R$ is reflexive, but not symmetric.' - - property_id: semi-strongly connected + - property: semi-strongly connected reason: |- Let us first remark that every non-empty finite semigroup $A$ has an idempotent element $e$, and then $B \to A$, $x \mapsto e$ does define a semigroup homomorphism for any $B$. Therefore, counterexamples need to be infinite and also without idempotent elements. Let $A$ be the set of positive rational numbers of the form $m/2^n$ (with $m > 0$, $n \geq 0$), and let $B$ be the set of positive rational numbers of the form $m/3^n$ (with $m > 0$, $n \geq 0$). Both are semigroups under addition. The element $1 \in A$ is $2^\infty$-divisible, meaning that for every $n \geq 0$ there is some $a \in A$ with $1 = 2^n \cdot a$. But $B$ has no $2^\infty$-divisible element. Hence, there is no semigroup homomorphism $A \to B$. Likewise, there is no semigroup homomorphism $B \to A$. - - property_id: cogenerating set + - property: cogenerating set # TODO: find a variant of the lemma missing_cogenerating_sets # (or missing_cogenerator) which handles this. reason: >- @@ -62,16 +62,16 @@ unsatisfied_properties: $$N := \{g \in G : f(g) = f(1)\}$$ is a normal subgroup of $G$. It is proper, and hence trivial. But then $f$ is injective, which is a contradiction. - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: We already know that $\Set$ does not have this property (by this result). Now apply the contrapositive of the dual of this lemma to the functor $\Set \to \SemiGrp$ that equips a set with the multiplication $a \cdot b := a$. - - property_id: effective cocongruences + - property: effective cocongruences reason: >- The proof is similar to $\Mon$, i.e. we adapt the counterexample from MO/510744. Namely, consider the semigroups $$\begin{align*} X & := \langle p \mid p^2 = p \rangle,\\ E & := \langle p, q \mid p^2 = p,\, q^2 = q,\, pq = q,\, qp = p \rangle, \end{align*}$$ whose underlying sets are $\{p\}$ and $\{p,q\}$, respectively. Then $X$ represents the functor sending a semigroup $A$ to its idempotents, and $E$ represents the relation on idempotents $a, b$ of $A$ that $ab = b$, $ba = a$. It is easy to check that this defines an equivalence relation (see MO/510744 for details). Since $p \ne q$ in $E$, the equalizer of the two maps $X \rightrightarrows E$ is the empty semigroup. Therefore, if $E$ were effective, it would be isomorphic to the coproduct $X \sqcup X$, whose underlying set consists of non-empty words in $p,q$ with $p,q$ strictly alternating. In particular, in this coproduct, $pq \ne q$. - - property_id: natural numbers object + - property: natural numbers object reason: >- Assume that a natural numbers object exists. Then by this result, for every semigroup $A$ the natural homomorphism $$\textstyle\alpha : \coprod_{n \geq 0} A \to A \times \coprod_{n \geq 0} 1$$ @@ -83,7 +83,7 @@ unsatisfied_properties: $$\alpha(y_0 x_0) = \alpha(x_0 y_0),$$ showing that $\alpha$ is not injective. - - property_id: coregular + - property: coregular reason: >- We will find a regular monomorphism $\iota : F \to M$ of semigroups and a homomorphism $F \to K$ such that $K \to K \sqcup_F M$ is not injective. It is similar to our example for $\Mon$. Consider these semigroups defined by generators and relations: $$\begin{align*} @@ -96,7 +96,7 @@ unsatisfied_properties: $$K \sqcup_F M \cong \langle x,c,d,s : x s = c,\, x s = d \rangle$$ shows that $c,d \in K$ have the same image in the pushout. - - property_id: regular subobject classifier + - property: regular subobject classifier reason: 'Assume that a regular subobject classifier $\Omega$ exists in $\SemiGrp$. The universal regular monomorphism $\top : 1 \to \Omega$ corresponds to an idempotent element $e \in \Omega$. It follows that $e \Omega e$ is a monoid with neutral element $e$. We claim that it is a regular subobject classifier in $\Mon$, which we know does not exist. Indeed, let $\iota : A \to B$ be a regular monomorphism of monoids. Since the forgetful functor $\Mon \to \SemiGrp$ preserves limits, we can also see $\iota$ as a regular monomorphism of semigroups. Hence, there is a unique homomorphism of semigroups $f : B \to \Omega$ with $\iota(A) = \{b \in B : f(b) = e\}$. Since $1 \in \iota(A)$, we have $f(1) = e$. Then $f$ corresponds to a homomorphism of monoids $f : B \to e \Omega e$ with kernel $\iota$, which proves our claim.' special_objects: diff --git a/databases/catdat/data/categories/Set.yaml b/databases/catdat/data/categories/Set.yaml index 75d719e5..8d967a19 100644 --- a/databases/catdat/data/categories/Set.yaml +++ b/databases/catdat/data/categories/Set.yaml @@ -5,7 +5,7 @@ objects: sets morphisms: maps description: The category of sets plays a fundamental role in category theory. Due to the Yoneda embedding, many results about general categories can be reduced to the category of sets. It is also usually the first example of a category that one encounters. nlab_link: https://ncatlab.org/nlab/show/Set -dual_category_id: Set_op +dual_category: Set_op tags: - set theory @@ -18,23 +18,23 @@ related_categories: - SetxSet satisfied_properties: - - property_id: locally small + - property: locally small reason: The collection of maps between two sets $X,Y$ is a subset of $X \times Y$ and therefore a set. - - property_id: finitary algebraic + - property: finitary algebraic reason: Use the empty algebraic theory. - - property_id: Grothendieck topos + - property: Grothendieck topos reason: It is equivalent to the category of sheaves on a one-point space. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Every non-empty set is weakly terminal (by using constant maps). unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: trivial + - property: trivial reason: This is trivial. special_objects: diff --git a/databases/catdat/data/categories/Set_c.yaml b/databases/catdat/data/categories/Set_c.yaml index 73416df7..20fa28b6 100644 --- a/databases/catdat/data/categories/Set_c.yaml +++ b/databases/catdat/data/categories/Set_c.yaml @@ -15,61 +15,61 @@ related_categories: - Grp_c satisfied_properties: - - property_id: locally small + - property: locally small reason: The collection of maps between two (countable) sets $X,Y$ is a subset of $X \times Y$ and therefore a set. - - property_id: essentially small + - property: essentially small reason: Every countable set is isomorphic to a subset of $\IN$. - - property_id: finitely complete + - property: finitely complete reason: The embedding $\Set_\c \hookrightarrow \Set$ is closed under finite products and equalizers, hence under finite limits. check_redundancy: false - - property_id: finitely cocomplete + - property: finitely cocomplete reason: The embedding $\Set_\c \hookrightarrow \Set$ is closed under finite coproducts and coequalizers, hence under finite colimits. check_redundancy: false - - property_id: subobject classifier + - property: subobject classifier reason: This is because $\{0,1\}$ is a subobject classifier in $\Set$, which is countable, and the monomorphisms coincide. - - property_id: epi-regular + - property: epi-regular reason: If $X \to Y$ is an epimorphism in $\Set_\c$, i.e. a surjective map, it is coequalizer of the two maps $X \times_Y X \rightrightarrows X$ in $\Set$ and hence also in $\Set_\c$. - - property_id: generator + - property: generator reason: The one-point set is clearly a generator. - - property_id: cogenerator + - property: cogenerator reason: The two-point set is a cogenerator in $\Set$, hence also in $\Set_\c$. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: This is because the larger category $\Set$ has this property. - - property_id: extensive + - property: extensive reason: The same proof as for $\Set$ applies. Actually, the category is "countably extensive". - - property_id: countably distributive + - property: countably distributive reason: By elementary set theory, a countable (disjoint) union of countable sets is again countable. Hence, countable coproducts exist in $\Set_\c$, and we already saw that finite products exist. The distributivity morphism is an isomorphism since this is the case in $\Set$ and the forgetful functor $\Set_\c \to \Set$ preserves finite products and countable coproducts. - - property_id: effective congruences + - property: effective congruences reason: 'Let $f, g : E \rightrightarrows X$ be a congruence in $\Set_\c$. Then using $1$ as a test object, we see that this induces an equivalence relation on $X$. We already know that $\Set$ has effective congruences (as does every topos). Using this result, we see that $E$ is the kernel pair of $X \to (X/E)_{\Set}$ in $\Set$. Also, the quotient $(X/E)_{\Set}$ is countable; and the forgetful functor $\Set_\c \to \Set$ is fully faithful and therefore reflects limits. Thus, we conclude that $E$ is the kernel pair of $X \to (X/E)_{\Set}$ in $\Set_\c$ as well.' - - property_id: regular + - property: regular reason: From the other properties we know that the category is finitely complete and that it has coequalizers. The regular epimorphisms are stable under pullback since this holds in $\Set$ and both regular epimorphisms (they are surjective maps) and pullbacks coincide. - - property_id: coregular + - property: coregular reason: From the other properties we know that the category is finitely cocomplete and that it has equalizers. The regular monomorphisms are stable under pushout since this holds in $\Set$ and both regular monomorphisms (they are injective maps) and pushouts coincide. unsatisfied_properties: - - property_id: small + - property: small reason: Even the collection of all singletons is not a set. - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: countable powers + - property: countable powers reason: Since the forgetful functor $\Set_\c \to \Set$ is representable, it preserves (countable) products. Therefore, if the power $\{0,1\}^{\IN}$ exists in $\Set_\c$, it must be the ordinary cartesian product, which however is uncountable. - - property_id: ℵ₁-accessible + - property: ℵ₁-accessible reason: 'In fact, $\Set_\c$ does not have $\aleph_1$-filtered colimits: Fix an uncountable set $X$, let $P_\c(X)$ be the poset of countable subsets of $X$, which is $\aleph_1$-filtered, and consider the functor $P_\c(X) \to \Set_\c$ taking a subset $Y \subseteq X$ to $Y$. The colimit of this diagram in $\Set$ is given by $X$ itself, so if $X_c$ were a colimit in $\Set_\c$, then $\Hom(X_c, \{0,1\}) \cong \Hom(X, \{0,1\})$. But the former has cardinality at most $2^{\aleph_0}$ and the latter has cardinality $2^{\card(X)}$, so we have obtained a contradiction if we pick $X$ large enough (e.g. $\card(X)=2^{\aleph_0}$).' special_objects: diff --git a/databases/catdat/data/categories/Set_f.yaml b/databases/catdat/data/categories/Set_f.yaml index b9964b0d..604ba432 100644 --- a/databases/catdat/data/categories/Set_f.yaml +++ b/databases/catdat/data/categories/Set_f.yaml @@ -14,68 +14,68 @@ related_categories: - Set satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Set_\f \to \Set$ and $\Set$ is locally small. - - property_id: generator + - property: generator reason: The singleton set (which is not terminal) is a generator as it represents the forgetful functor $\Set_\f \to \Set$. - - property_id: cogenerator + - property: cogenerator reason: 'The set $\{0,1\}$ is a cogenerator in $\Set_\f$: Assume that $f,g : X \rightrightarrows Y$ are two finite-to-one maps such that $h \circ f = h \circ g$ for all finite-to-one maps $h : Y \to \{0,1\}$. This exactly means $f^*(A)=g^*(A)$ for all finite subsets $A \subseteq Y$. Applying this to $A = \{f(x)\}$ for $x \in X$ we get $x \in f^*(\{f(x)\}) = g^*(\{f(x)\})$, so that $g(x) = f(x)$.' - - property_id: semi-strongly connected + - property: semi-strongly connected reason: From set theory it is known that for all sets $X,Y$ there is an injective map $X \to Y$ or an injective map $Y \to X$, and injective maps are finite-to-one. - - property_id: extensive + - property: extensive reason: 'We first show that finite coproducts exist. The empty set is clearly initial. The disjoint union $X+Y$ of two sets $X,Y$ with the inclusion maps $X \rightarrow X+Y \leftarrow Y$ is a coproduct: The inclusions are injective, hence finite-to-one. If $f : X \to T$, $g : Y \to T$ are finite-to-one maps, the induced map $(f;g) : X + Y \to T$ is finite-to-one since the fiber of $t \in T$ is $f^*(\{t\}) + g^*(\{t\})$, which is finite. Hence, finite coproducts exist. A map $A \to X + Y$ yields a decomposition $A = A_X + A_Y$ with maps $A_X \to X$, $A_Y \to Y$ (since $\Set$ is extensive). Here, $A \to X + Y$ is finite-to-one iff $A_X \to X$ and $A_Y \to Y$ are finite-to-one.' - - property_id: equalizers + - property: equalizers reason: 'Equalizers can be constructed as in $\Set$ because of the following trivial observation: if $f : X \to Y$ is a finite-to-one map and $E \subseteq Y$ is a subset with $f(X) \subseteq E$, then the induced map $f^E : X \to E$ is also finite-to-one.' - - property_id: epi-regular + - property: epi-regular reason: 'If $f : X \to Y$ is an epimorphism in $\Set_\f$, i.e. a surjective finite-to-one map, it is a coequalizer of the two maps $p_1, p_2 : X \times_Y Y \rightrightarrows Y$ in $\Set$. These maps are finite-to-one since $p_i^*(\{y\}) \cong f^*(\{y\})$ for $i=1,2$, and their coequalizer is also $f$ in $\Set_\f$: It suffices to observe that if $h : Y \to T$ is a map such that $h \circ f$ is finite-to-one, then $h$ is finite-to-one as well. In fact, surjectivity of $f$ implies $h^*(\{t\}) = f_*((h \circ f)^*(\{t\}))$ for $t \in T$.' - - property_id: well-copowered + - property: well-copowered reason: This is clear since the epimorphisms are surjective. - - property_id: quotients of congruences + - property: quotients of congruences reason: 'A congruence on a set $X$ in $\Set_\f$ is the same as an equivalence relation $R$ on $X$ whose equivalence classes are finite. In that case, the usual quotient map $p : X \to X/R$ is finite-to-one. Moreover, if $h : X/R \to Y$ is a map such that $h \circ p : X \to Y$ is finite-to-one, then $h$ is finite-to-one as well because $h^*(\{y\}) \subseteq p^*((h \circ p)^*(\{y\}))$ for all $y \in Y$. Therefore, $p$ is also the quotient in $\Set_\f$.' - - property_id: effective congruences + - property: effective congruences reason: 'Let $f, g : E \rightrightarrows X$ be a congruence in $\Set_\f$. From the proof on quotients of congruences in $\Set_\f$, we have a quotient map $p : X \to X/E$ in $\Set_\f$, and $E$ is the kernel pair of $p$ in $\Set$. It remains to see that $E$ is also the kernel pair of $p$ in $\Set_f$. Thus, suppose we have $x_1, x_2 : T \rightrightarrows X$ with $p \circ x_1 = p \circ x_2$. Then there is a unique $e : T \to E$ in $\Set$ with $x_1 = f\circ e$ and $x_2 = g\circ e$. Since $f\circ e$ is finite-to-one, we must have $e$ is finite-to-one as well.' - - property_id: effective cocongruences + - property: effective cocongruences reason: 'Suppose we have a cocongruence $f, g : X \rightrightarrows E$ in $\Set_f$. Then it is a coreflexive corelation in $\Set$. Since $\Set$ is co-Malcev and has effective cocongruences, that implies $E$ is the cokernel pair of some function $h : Z \to X$ in $\Set$. By the dual of this result, if $\inc_Y : Y \hookrightarrow X$ is the equalizer of $f$ and $g$, then $E$ is also the cokernel pair of $\inc_Y$ in $\Set$. It remains to see that $E$ is the cokernel pair of $\inc_Y$ in $\Set_\f$ as well. Thus, suppose $a, b : X \rightrightarrows T$ are such that $a |_Y = b |_Y$. Then there is a unique $c : E\to T$ in $\Set$ with $a = c\circ f$ and $b = c\circ g$. Since $(f;g) : X + X \to E$ is surjective and $c \circ (f;g) = (a;b)$ is finite-to-one, we see $c$ is finite-to-one as well.' - - property_id: locally cartesian closed + - property: locally cartesian closed reason: If $X$ is a set, the equivalence $\Set/X \simeq \Set^X$, $f \mapsto (f^*(\{x\}))_{x \in X}$ restricts to an equivalence $\Set_\f / X \simeq \FinSet^X$. This category is cartesian closed since $\FinSet$ is cartesian closed and products of cartesian closed categories are cartesian closed. - - property_id: ℵ₁-accessible + - property: ℵ₁-accessible reason: 'We first show that $\aleph_1$-directed colimits exist and are preserved by the forgetful functor to $\Set$. Let $(X_i)_{i \in I}$ be a diagram in $\Set_\f$ indexed by a $\aleph_1$-directed poset $I$. Let $(u_i : X_i \to X)$ be the colimit in $\Set$. Each map $u_i$ is finite-to-one: Otherwise, some fiber $u_i^*(\{x\}) \subseteq X_i$ contains infinitely many elements $a_1,a_2,\dotsc$. For every $n \geq 1$ we find $i_n \geq i$ such that $a_1,a_n$ have the same image in $X_{i_n}$. Let $i_\infty \in I$ be an upper bound of all $i_n$. Then all $a_n$ have the same image in $X_{i_\infty}$. This is a contradiction since $X_i \to X_{i_\infty}$ is finite-to-one. Now if $f : X \to Y$ is a map such that each $f \circ u_i$ is finite-to-one, then $f$ is finite-to-one as well: If a fiber $f^*(\{y\})$ contains infinitely many distinct elements $x_1,x_2,\dotsc$, there is some index $i$ such that all have a preimage in $X_i$, but then $(f \circ u_i)^*(\{y\})$ would be infinite. This concludes the proof that $\aleph_1$-directed colimits exist. In $\Set$, every set $X$ is the $\aleph_1$-directed colimit of its countable subsets. This remains true in $\Set_\f$ because a map $X \to Y$ is finite-to-one as long as each restriction to a countable subset of $X$ is finite-to-one. Moreover, every countable set is $\aleph_1$-presentable in $\Set$, but also in $\Set_\f$.' unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: essentially small + - property: essentially small reason: This is obvious. - - property_id: locally finite + - property: locally finite reason: If $X$ is an infinite set, there are infinitely many maps $1 \to X$. They are injective and hence finite-to-one. - - property_id: strongly connected + - property: strongly connected reason: Already $\Set$ is not strongly connected. - - property_id: binary powers + - property: binary powers reason: 'More generally, if $X,Y$ are two non-empty sets such that $X \times Y$ exists in $\Set_\f$, then both $X$ and $Y$ must be finite. In fact, the forgetful functor to $\Set$ is representable, so it must preserve products. This means we can assume $X \times Y$ is the usual cartesian product with the usual projections. Since $p_1 : X \times Y \to X$ is finite-to-one and $X$ is non-empty, $Y$ is finite. By symmetric, also $X$ is finite. (Conversely, if $X$ and $Y$ are finite, or one of them is empty, then indeed $X \times Y$ exists.)' - - property_id: countable copowers + - property: countable copowers reason: 'Assume that the copower $X := \IN \otimes 1$ exists, where $1$ is the singleton set. This is a set with a map $i : \IN \to X$ (not necessarily finite-to-one) such that for every other such map $j : \IN \to Y$ there is a unique finite-to-one map $f : X \to Y$ with $f \circ i = j$. Applying this to $j : \IN \to 1$, we see that $X$ is finite. Applying the universal property to maps $j : \IN \to \{0,1\}$, we see that for every subset $E \subseteq \IN$ there is a unique finite subset $F \subseteq X$ with $i^*(F) = E$. But finiteness of $F$ is automatic, so $i^* : P(X) \to P(\IN)$ is bijective. But then $P(\IN)$ is finite, which is absurd.' - - property_id: filtered + - property: filtered reason: 'Consider the maps $f,g : \IN \rightrightarrows \IN$ defined by $f(x)=x$ and $g(x)=2x$. They are injective, hence finite-to-one. If a map $h : \IN \to X$ coequalizes them, we have $h(x)=h(2x)$, in particular $h(1)=h(2^n)$ for all $n \in \IN$. Thus, $h$ is not finite-to-one.' - - property_id: sequential limits + - property: sequential limits reason: Consider the set $[n] := \{0,\dotsc,n\}$ for $n \in \IN$. The forgetful functor to $\Set$ is representable (by the singleton set), hence preserves all limits. Thus, if the diagram of truncation maps $\cdots \twoheadrightarrow [2] \twoheadrightarrow [1] \twoheadrightarrow [0]$ has a limit in $\Set_\f$, its underlying set is isomorphic to the limit taken in $\Set$, which is $\IN \cup \{\infty\}$. But there is no finite-to-one map $\IN \cup \{\infty\} \to [0]$. special_objects: diff --git a/databases/catdat/data/categories/Set_op.yaml b/databases/catdat/data/categories/Set_op.yaml index ab8c4082..c6980355 100644 --- a/databases/catdat/data/categories/Set_op.yaml +++ b/databases/catdat/data/categories/Set_op.yaml @@ -5,7 +5,7 @@ objects: sets morphisms: 'A morphism $f : X \to Y$ is a map of sets $Y \to X$.' description: By definition, this category is the dual (or opposite) of the category $\Set$. nlab_link: https://ncatlab.org/nlab/show/opposite+category -dual_category_id: Set +dual_category: Set tags: - set theory diff --git a/databases/catdat/data/categories/Set_pointed.yaml b/databases/catdat/data/categories/Set_pointed.yaml index 7bae64f1..47075655 100644 --- a/databases/catdat/data/categories/Set_pointed.yaml +++ b/databases/catdat/data/categories/Set_pointed.yaml @@ -14,59 +14,59 @@ related_categories: - Top* satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Set_* \to \Set$ and $\Set$ is locally small. - - property_id: pointed + - property: pointed reason: The singleton set is a zero object. check_redundancy: false - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic theory with just one constant. - - property_id: subobject classifier + - property: subobject classifier reason: The pointed set $(\{0,1\},1)$ is a subobject classifier. - - property_id: cogenerator + - property: cogenerator reason: The pointed set $(\{0,1\},1)$ is a cogenerator. - - property_id: epi-regular + - property: epi-regular reason: Every epimorphism is surjective for this category, and in an algebraic category every surjective homomorphism is a regular epimorphism. - - property_id: coregular + - property: coregular reason: From the other properties we know that (co-)limits exist and that monomorphisms coincide with injective pointed maps. So it suffices to prove that these maps are stable under pushouts. This follows from the corresponding fact for $\Set$ and the observation that the forgetful functor $\Set_* \to \Set$ preserves pushouts. - - property_id: co-Malcev + - property: co-Malcev reason: Malcev categories are closed under slice categories by Prop. 2.2.14 in Malcev, protomodular, homological and semi-abelian categories. It follows that co-Malcev categories are closed under coslice categories, and $\Set_*$ is a coslice category of $\Set$, which is co-Malcev since every elementary topos is co-Malcev. - - property_id: cocartesian cofiltered limits + - property: cocartesian cofiltered limits reason: |- Let $X$ be a pointed set and $(Y_i)$ be a filtered diagram of pointed sets. Base points will be denoted by $0$. The canonical map $X \vee \lim_i Y_i \to \lim_i (X \vee Y_i)$ is injective since the wedge sum naturally embeds into the product and the natural map $X \vee \prod_i Y_i \to \prod_i (X \times Y_i)$ is injective. Now let $z = (z_i) \in \lim_i (X \vee Y_i)$. Case 1: There is some index $i$ with $z_i \in X \setminus \{0\}$. We claim $z_j \in X$ for any index $j$ and $z_j = z_i$ in $X$, so that $z$ has a preimage in $X$. To see this, choose an index $k \geq i,j$. Since $X \vee Y_i \to X \vee Y_k$ maps $z_i \mapsto z_k$ and is the identity on $X$, we see that $z_k \in X$ and $z_k = z_i$ in $X$. Since $X \vee Y_j \to X \vee Y_k$ maps $z_j \mapsto z_k$, we see that $z_j \notin Y_j$, since otherwise $z_k \in Y_k \cap X = \{0\}$. Hence, $z_j \in X \setminus \{0\}$, and then $z_j = z_k = z_i$. Case 2: We have $z_i \in Y_i$ for all $i$. Then clearly $(z_i) \in \lim_i Y_i$ is a preimage. - - property_id: CIP + - property: CIP reason: The coproduct (wedge sum) of a family of pointed sets $(X_i)_{i \in I}$ can be realized as the subset of $\prod_{i \in I} X_i$ consisting of those tuples $x$ such that $x_i = 0$ for all but (at most) one index. - - property_id: effective cocongruences + - property: effective cocongruences # TODO: rework this when Barr-exact is added reason: We have that $\Set_*^{\op}$ is a slice category of $\Set^{\op}$, which in turn is monadic over $\Set$. Therefore, by combining results from Borceux and Bourn Appendix A and nLab, $\Set_*^{\op}$ is Barr-exact, and in particular it has effective congruences. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: unital + - property: unital reason: 'The joint image of $X \to X \times Y \leftarrow Y$ is just $\{(x,0) : x \in X\} \cup \{(0,y) : y \in Y\}$ (where $0$ denotes the base point), which is clearly a proper subset of $X \times Y$ when both $X,Y$ are non-trivial.' - - property_id: CSP + - property: CSP # TODO: remove duplication with unital proof reason: 'The image of $X \vee Y$ in $X \times Y$ is just $\{(x,0) : x \in X\} \cup \{(0,y) : y \in Y\}$ (where $0$ denotes the base point), which is clearly a proper subset when both $X,Y$ are non-trivial.' - - property_id: conormal + - property: conormal reason: 'Every cokernel is "injective away from the base point". Formally, if $p : A \to B$ is a cokernel in $\Set_*$, it has the property that $p(x)=p(y) \neq 0$ implies $x=y$ (where $0$ denotes the base point). Clearly this is not satisfied for every surjective pointed map, consider $(\IN,0) \to (\{0,1\},0)$ defined by $0 \mapsto 0$ and $x \mapsto 1$ for $x > 0$.' - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: We already know that $\Set$ does not have this property. Now apply the contrapositive of the dual of this lemma to the functor $\Set \to \Set_*$ that freely adds a base point. special_objects: diff --git a/databases/catdat/data/categories/Setne.yaml b/databases/catdat/data/categories/Setne.yaml index 046e9674..dfc8e122 100644 --- a/databases/catdat/data/categories/Setne.yaml +++ b/databases/catdat/data/categories/Setne.yaml @@ -13,65 +13,65 @@ related_categories: - Set satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Setne \to \Set$ and $\Set$ is locally small. - - property_id: generator + - property: generator reason: The one-point set is clearly a generator. - - property_id: cogenerator + - property: cogenerator reason: The two-point set is a cogenerator, this follows as for $\Set$. - - property_id: products + - property: products reason: Take the product of non-empty sets inside of $\Set$ and observe that it is non-empty by the axiom of choice. - - property_id: cartesian closed + - property: cartesian closed reason: This follows as for $\Set$, since for non-empty sets $X,Y$ there is at least one function $X \to Y$. - - property_id: binary coproducts + - property: binary coproducts reason: The disjoint union of two non-empty sets is non-empty. - - property_id: mono-regular + - property: mono-regular reason: This follows as for $\Set$. - - property_id: epi-regular + - property: epi-regular reason: This follows as for $\Set$. - - property_id: strongly connected + - property: strongly connected reason: Use constant maps. - - property_id: finitely accessible + - property: finitely accessible reason: Since the inclusion $\Setne \hookrightarrow \Set$ is closed under non-empty colimits, it is also closed under filtered colimits. Therefore, non-empty finite sets are still finitely presentable in $\Setne$, and every non-empty set is written as a filtered colimit of them. - - property_id: generalized variety + - property: generalized variety reason: Since the inclusion $\Setne \hookrightarrow \Set$ is closed under non-empty colimits, it is also closed under sifted colimits. Therefore, non-empty finite sets are still strongly finitely presentable in $\Setne$, and every non-empty set is written as a sifted colimit of them. - - property_id: natural numbers object + - property: natural numbers object reason: Any natural numbers object in $\Set$, such as $(\IN,0,n \mapsto n+1)$, is clearly also one in $\Setne$. - - property_id: filtered-colimit-stable monomorphisms + - property: filtered-colimit-stable monomorphisms reason: This follows from this lemma applied to the forgetful functor to $\Set$. - - property_id: multi-complete + - property: multi-complete reason: Let $D$ be a diagram in $\Setne$, and let $L$ be a limit of $D$ in $\Set$. If $L$ is non-empty, it gives a limit in $\Setne$ as well. If $L$ is the empty set, there is no cone over $D$ in $\Setne$; hence the empty set of cones gives a multi-limit of $D$ in $\Setne$. - - property_id: effective congruences + - property: effective congruences reason: 'If a congruence $E \rightrightarrows X$ is the kernel pair of $h : X \to Z$, with both $E$ and $X$ non-empty, then certainly $Z$ must also be non-empty.' unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: cofiltered + - property: cofiltered reason: The two maps $\{0\} \rightrightarrows \{0,1\}$ are equalized by no map $X \to \{0\}$ in this category. - - property_id: sequential limits + - property: sequential limits reason: Assume that the sequence of inclusions $\cdots \to \IN_{\geq 2} \to \IN_{\geq 1} \to \IN_{\geq 0} = \IN$ as a limit $X$, consisting of maps $X \to \IN_{\geq n}$. Since $X$ is non-empty, there is a map $1 \to X$. This corresponds to a family of compatible maps $ 1 \to \IN_{\geq n}$, i.e. to compatible elements in $\IN_{\geq n}$. But the set $\bigcap_{n \geq 0} \IN_{\geq n}$ is empty. - - property_id: coquotients of cocongruences + - property: coquotients of cocongruences reason: The two maps $\{0\} \rightrightarrows \{0,1\}$ form a cocongruence on $\{0\}$ — namely the cofull cocongruence on $\{0\}$ — but they do not have an equalizer. - - property_id: effective cocongruences + - property: effective cocongruences reason: The two maps $\{0\} \rightrightarrows \{0,1\}$ form a cocongruence on $\{0\}$ — namely the cofull cocongruence on $\{0\}$ — but there is no map $Z \to \{0\}$ making the required commutative diagram, much less a cocartesian square. special_objects: diff --git a/databases/catdat/data/categories/SetxSet.yaml b/databases/catdat/data/categories/SetxSet.yaml index fedbd2e2..c65a2261 100644 --- a/databases/catdat/data/categories/SetxSet.yaml +++ b/databases/catdat/data/categories/SetxSet.yaml @@ -14,23 +14,23 @@ related_categories: - Sh(X) satisfied_properties: - - property_id: locally small + - property: locally small reason: This is obvious. - - property_id: Grothendieck topos + - property: Grothendieck topos reason: It is equivalent to the category of sheaves on a discrete space with two points. - - property_id: locally strongly finitely presentable + - property: locally strongly finitely presentable reason: Take the two-sorted algebraic theory with no operations and no equations. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: When $X$ is non-empty, there is no morphism between $(\varnothing,X)$ and $(X,\varnothing)$. - - property_id: generator + - property: generator reason: Assume that $(A,B)$ is a generator. Of course, $A$ and $B$ cannot be both empty. Assume w.l.o.g. that $A$ is non-empty. Then there is no morphism $(A,B) \to (0,1)$, but there are two different morphisms $(0,1) \rightrightarrows (0,2)$. special_objects: diff --git a/databases/catdat/data/categories/Sh(X).yaml b/databases/catdat/data/categories/Sh(X).yaml index 3cdf469d..9df7e6ca 100644 --- a/databases/catdat/data/categories/Sh(X).yaml +++ b/databases/catdat/data/categories/Sh(X).yaml @@ -19,17 +19,17 @@ comments: - It is likely that neither of the currently remaining unknown properties (finitary algebraic, locally ℵ₁-presentable, exact filtered colimits, etc.) are satisfied for a generic space $X$, but we need to make this precise by adding additional requirements to $X$. Maybe we need to create separate entries for specific spaces $X$. satisfied_properties: - - property_id: locally small + - property: locally small reason: This is easy. - - property_id: Grothendieck topos + - property: Grothendieck topos reason: This holds by definition of a Grothendieck topos. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: Consider constant sheaves for isomorphic but non-equal sets. - - property_id: trivial + - property: trivial reason: This is because $X$ is assumed to be non-empty. special_objects: diff --git a/databases/catdat/data/categories/Sh(X,Ab).yaml b/databases/catdat/data/categories/Sh(X,Ab).yaml index 665e94bd..937451c1 100644 --- a/databases/catdat/data/categories/Sh(X,Ab).yaml +++ b/databases/catdat/data/categories/Sh(X,Ab).yaml @@ -18,17 +18,17 @@ comments: - It is likely that neither of the currently remaining unknown properties (finitary algebraic, locally ℵ₁-presentable, CSP, etc.) are satisfied for a generic space $X$, but we need to make this precise by adding additional requirements to $X$. Maybe we need to create separate entries for specific spaces $X$. satisfied_properties: - - property_id: locally small + - property: locally small reason: This is easy. - - property_id: Grothendieck abelian + - property: Grothendieck abelian reason: This is standard, see for example Theorem 18.1.6. in Kashiwara-Schapira. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: Consider constant sheaves for isomorphic but non-equal abelian groups. - - property_id: split abelian + - property: split abelian reason: 'Choose a point $x \in X$. The functor $x_* : \Ab \to \Sh(X,\Ab)$ (skyscraper sheaf) is exact, and its left adjoint $x^* : \Sh(X,\Ab) \to \Ab$ (stalk) satisfies $x^* x_* \cong \id_{\Ab}$. Now, since $\Ab$ is not split abelian, there is a short exact sequence of abelian groups $0 \to A \to B \to C \to 0$ that does not split. Then $0 \to x_* A \to x_* B \to x_* C \to 0$ is also exact, but it does not split: Otherwise it would also be split after applying $x^*$, which however gives the original sequence in $\Ab$.' special_objects: diff --git a/databases/catdat/data/categories/Sp.yaml b/databases/catdat/data/categories/Sp.yaml index c01342b2..317929af 100644 --- a/databases/catdat/data/categories/Sp.yaml +++ b/databases/catdat/data/categories/Sp.yaml @@ -14,38 +14,38 @@ related_categories: - FinSet satisfied_properties: - - property_id: essentially small + - property: essentially small reason: This holds because $\FinSet$ and $\IB$ are essentially small. - - property_id: elementary topos + - property: elementary topos reason: The category is equivalent to $\prod_{n \geq 0} \Sigma_n{-}\FinSet$ (where $\Sigma_n$ denotes the symmetric group of order $n$), and each $\Sigma_n{-}\FinSet$ is an elementary topos since is $\FinSet$ an elementary topos and $\Sigma_n$ is a finite group, cf. Johnstone, Part B, Corollary 2.3.18. - - property_id: cogenerator + - property: cogenerator reason: 'This follows from $\Sp \simeq \prod_{n \geq 0} \Sigma_n{-}\FinSet$, this lemma, and the fact that if $G$ is a (finite) group, the power set $P(G)$ with the evident $G$-action is a weakly terminal cogenerator in $G{-}\Set$ (resp. $G{-}\FinSet$). For the proof, notice that $\varnothing,G \in P(G)$ are fixed points, yielding two $G$-maps $1 \rightrightarrows P(G)$. In particular, $P(G)$ is weakly terminal. If $X$ is a $G$-set with distinct points $x,y$, we construct a $G$-map $f : X \to P(G)$ that separates $x,y$: First, $X$ is a coproduct of orbits. If $x,y$ lie in different orbits, let $f|_{Gx}$ be constant $\varnothing$, $f|_{Gy}$ be constant $G$, and, say, $f$ be constant $\varnothing$ on all other orbits. If $x,y$ lie in the same orbit, say $y = g_0 x$, define $f|_{Gx} : Gx \to P(G)$ by $f(x) = G_x$ (stabilizer), which is well-defined, and choose $f$ to be $\varnothing$ on all other orbits. Then $f(y) = g_0 G_x \neq G_x = f(x)$.' unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: locally finite + - property: locally finite reason: If $1$ denotes the terminal species, there are infinitely many morphisms $1 \to 1 \sqcup 1$ since they correspond to functions $\IN \to \{1,2\}$. - - property_id: locally small + - property: locally small reason: 'Disclaimer: This result and its proof are not relevant for category theory and are also depending on the concrete model of set theory. That this category is locally essentially small is only what matters. Now, consider the terminal species $F=G=1$. Then $\Hom(F,G)$ has just a single element, namely the natural transformation $\alpha$ that sends every finite set $X$ to the unique map $\alpha_X : 1 \to 1$. Formally, $\alpha$ is a map, modelled as a set of ordered pairs $(X,\id_1)$, where $X$ is a finite set. Hence, $\alpha$ is not a set (since finite sets do not form a set), and therefore $\Hom(F,G) = \{\alpha\}$ is also not a set.' - - property_id: essentially countable + - property: essentially countable reason: Any function $f\colon\IN \to \IN$ can be regarded as a combinatorial species with trivial actions, and distinct functions yield non-isomorphic species. - - property_id: countable powers + - property: countable powers reason: If $\Sp \simeq \FinSet \times \prod_{n > 0} \Sigma_n{-}\FinSet$ has countable powers, then $\FinSet$ has countable powers as well by this lemma, which we already know is false. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: 'For $n \geq 0$ let $E_n$ be the combinatorial species of $n$-sets: $E_n(A) = \{A\}$ when $A$ has cardinality $n$, otherwise $E_n(A) = \varnothing$. Then there is no morphism between $E_n$ and $E_m$ unless $n = m$.' - - property_id: generator + - property: generator reason: 'Assume that a generator $G$ exists. For $n \geq 0$ let $F_n$ be the combinatorial species of sets of cardinality $\neq n$: $F_n(A) = \varnothing$ when $A$ has cardinality $n$, otherwise $F_n(A) = \{A\}$. There are two different morphisms $F_n \rightrightarrows F_n \sqcup F_n$. Hence, there must be at least one morphism $G \to F_n$. If $A$ has cardinality $n$, this implies $G(A) = \varnothing$. Since this holds for all $n$, $G$ is the initial object. But this is clearly no generator (it would mean that the category is thin).' - - property_id: natural numbers object + - property: natural numbers object reason: >- If $(N,z,s)$ is a natural numbers object, then $$1 \xrightarrow{z} N \xleftarrow{s} N$$ diff --git a/databases/catdat/data/categories/Top.yaml b/databases/catdat/data/categories/Top.yaml index fa7cfd47..87bdc1f7 100644 --- a/databases/catdat/data/categories/Top.yaml +++ b/databases/catdat/data/categories/Top.yaml @@ -15,72 +15,72 @@ related_categories: - Top* satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Top \to \Set$ and $\Set$ is locally small. - - property_id: complete + - property: complete reason: Take the limit of the underlying sets and endow it with the coarsest topology making all projections continuous. - - property_id: cocomplete + - property: cocomplete reason: Take the colimit of the underlying sets and endow it with the finest topology making all inclusions continuous. check_redundancy: false - - property_id: well-powered + - property: well-powered reason: This is clear from the classification of monomorphisms as injective continuous maps. - - property_id: well-copowered + - property: well-copowered reason: This is clear from the classification of epimorphisms as surjective continuous maps. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Every non-empty space is weakly terminal (by using constant maps). - - property_id: generator + - property: generator reason: The one-point space is a generator since it represents the forgetful functor $\Top \to \Set$. - - property_id: cogenerator + - property: cogenerator reason: It is easily checked that the indiscrete two-point space is a cogenerator. - - property_id: infinitary extensive + - property: infinitary extensive reason: '[Sketch] Since $\Set$ is infinitary extensive, a map $f : Y \to \coprod_i X_i$ corresponds to a decomposition $Y = \coprod_i Y_i$ (as sets) with maps $f_i : Y_i \to X_i$. Endow $Y_i$ with the subspace topology. If $f$ is continuous, each $Y_i = f^{-1}(X_i)$ is open in $Y$, so that $Y = \coprod_i Y_i$ holds as topological spaces, and each $f_i$ is continuous.' - - property_id: regular subobject classifier + - property: regular subobject classifier reason: The indiscrete two-point space $\{0,1\}$ is a regular subobject classifier since continuous maps $X \to \{0,1\}$ correspond to subsets of $X$. - - property_id: coregular + - property: coregular reason: The category has all limits and colimits, and the regular monomorphisms are the subspace inclusions. Thus, it suffices to prove that subspace inclusions are stable under pushouts. For a proof see e.g. Lemma 3.6 at the nLab. - - property_id: filtered-colimit-stable monomorphisms + - property: filtered-colimit-stable monomorphisms reason: This follows from this lemma applied to the forgetful functor to $\Set$. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: If $X$ is a set, consider the discrete space $X_d$ on $X$ and the indiscrete space $X_i$ on $X$. The identity map $X \to X$ lifts to a continuous map $X_d \to X_i$, which is bijective and therefore both a mono- and an epimorphism, but it is not an isomorphism unless $X$ has at most one element. - - property_id: cartesian filtered colimits + - property: cartesian filtered colimits reason: 'The functor $\IQ \times - : \Top \to \Top$ does not preserve colimits, see MSE/2969372.' - - property_id: regular + - property: regular reason: See Example 3.14 at the nLab. - - property_id: locally presentable + - property: locally presentable reason: In fact, it does not have any small dense subcategory by MSE/4097315. For a related result, see MO/288648. - - property_id: Malcev + - property: Malcev reason: This is clear since $\Set$ is not Malcev and can be interpreted as the subcategory of discrete spaces. - - property_id: co-Malcev + - property: co-Malcev reason: 'See MO/509548. We can also phrase the proof as follows: Consider the forgetful functor $U : \Top \to \Set$ and the relation $R \subseteq U^2$ defined by $R(X) := \{(x,y) \in U(X)^2 : x \in \overline{\{y\}} \}$. Both are representable: $U$ by the singleton and $R$ by the Sierpinski space. It is clear that $R$ is reflexive, but not symmetric.' - - property_id: coaccessible + - property: coaccessible reason: Assume $\Top$ is coaccessible. Let $p\colon S \to I$ be the identity map from the Sierpinski space to the two-element indiscrete space. Then, a topological space is discrete if and only if it is projective to the morphism $p$. This implies that the full subcategory spanned by all discrete spaces, which is equivalent to $\Set$, is coaccessible by Prop. 4.7 in Adamek-Rosicky. However, since $\Set$ is not coaccessible, this is a contradiction. - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: We already know that $\Set$ does not have this property. Now apply the contrapositive of the dual of this lemma to the functor $\Set \to \Top$ which equips a set with the indiscrete topology. - - property_id: effective cocongruences + - property: effective cocongruences reason: 'Consider the indiscrete topological space $I$ on two points. This represents the functor which takes a topological space $X$ to the pairs of indistinguishable points of $X$. Therefore, we get a cocongruence $1 \rightrightarrows I$, where the maps are the two possible functions. However, this cannot be effective: if we have $h : Z\to 1$ which equalizes the two maps, then $Z$ must be empty. But that means the cokernel pair of $h$ is the discrete space on two points.' special_objects: diff --git a/databases/catdat/data/categories/Top_pointed.yaml b/databases/catdat/data/categories/Top_pointed.yaml index 5108c523..e4f769b1 100644 --- a/databases/catdat/data/categories/Top_pointed.yaml +++ b/databases/catdat/data/categories/Top_pointed.yaml @@ -14,97 +14,97 @@ related_categories: - Top satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Top_* \to \Set_*$ and $\Set_*$ is locally small. - - property_id: pointed + - property: pointed reason: The singleton space $\{0\}$ with base point $0$ is a zero object. check_redundancy: false - - property_id: complete + - property: complete reason: This follows from $\Top_* \cong 1 / \Top$ and the fact that $\Top$ is complete. Concretely, the limit of pointed spaces $(X_i,x_i)$ is the limit of the underlying spaces $X_i$ equipped with the base point that projects down to each $x_i$. check_redundancy: false - - property_id: coequalizers + - property: coequalizers reason: This follows immediately from the fact that $\Top$ has coequalizers. check_redundancy: false - - property_id: coproducts + - property: coproducts reason: This follows from $\Top_* \cong 1 / \Top$ and the fact that $\Top$ has wide pushouts. check_redundancy: false - - property_id: well-powered + - property: well-powered reason: This is clear from the classification of monomorphisms as injective pointed continuous maps. - - property_id: well-copowered + - property: well-copowered reason: This is clear from the classification of epimorphisms as surjective pointed continuous maps. - - property_id: generator + - property: generator reason: The discrete space $\{0,1\}$ with base point $0$ is a generator since it represents the forgetful functor $\Top_* \to \Set$. - - property_id: cogenerator + - property: cogenerator reason: It is easily checked that the indiscrete two-point space $\{0,1\}$ with base point $1$ is a cogenerator. - - property_id: regular subobject classifier + - property: regular subobject classifier reason: The indiscrete two-point space $\{0,1\}$ with base point $1$ is a regular subobject classifier since pointed continuous maps $X \to \{0,1\}$ correspond to pointed subsets of $X$ (by taking the fiber of $1$ as usual). - - property_id: counital + - property: counital reason: Since embeddings are regular monomorphisms in this category (see below) and hence strong monomorphisms, it suffices to prove that the canonical morphism $X \vee Y \hookrightarrow X \times Y$ is an embedding. For a proof, see MSE/4055988. - - property_id: CIP + - property: CIP reason: This follows since $\Set_*$ has this property and the forgetful functor preserves products and coproducts. - - property_id: cocartesian cofiltered limits + - property: cocartesian cofiltered limits reason: >- We continue the proof for $\Set_*$ by showing that the natural bijective map $$\textstyle \alpha : X \vee \lim_i Y_i \to \lim_i (X \vee Y_i)$$ is open. It suffices to consider open sets of two types: (1) If $U \subseteq X$ is open, the $\alpha$-image of $U \vee \lim_i Y_i$ is $p_{i_0}^{-1}(U \vee Y_{i_0})$ for any chosen index $i_0$, hence open. (2) If $i$ is an index and $V_i \subseteq Y_i$ is open, then the $\alpha$-image of $X \vee (p_i^{-1}(V_i) \cap \lim_i Y_i)$ is $p_i^{-1}(X \vee V_i)$, hence open. - - property_id: filtered-colimit-stable monomorphisms + - property: filtered-colimit-stable monomorphisms reason: This follows from this lemma applied to the forgetful functor to $\Set$. - - property_id: coregular + - property: coregular reason: Regular monomorphisms coincide with the embeddings (see below). Since $\Top$ is coregular, they are stable under pushouts, and pushouts in $\Top_*$ are the same. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: If $X$ is a set with a base point $x_0$, consider the discrete space $X_d$ on $X$ and the indiscrete space $X_i$ on $X$. The identity map $X \to X$ lifts to a continuous map $X_d \to X_i$ preserving $x_0$, which is bijective and therefore both a mono- and an epimorphism, but it is not an isomorphism unless $X = \{x_0\}$. - - property_id: regular + - property: regular reason: See Example 3.14 at the nLab. The proof also works for pointed spaces (resp. posets) by using the base points $a$ and $0$. - - property_id: locally presentable + - property: locally presentable reason: In fact, it does not have any small dense subcategory by MSE/4097315. The proof easily adapts to pointed spaces. - - property_id: cartesian filtered colimits + - property: cartesian filtered colimits reason: 'The functor $\IQ \times - : \Top_* \to \Top_*$ does not preserve colimits, see MSE/2969372. The counterexample also works for pointed spaces.' - - property_id: co-Malcev + - property: co-Malcev reason: 'We can adjust the proof for $\Top$ as follows: Consider the forgetful functor $U : \Top_* \to \Set$ and the relation $R \subseteq U^2$ defined by $R(X) := \{(x,y) \in U(X)^2 : x \in \overline{\{y\}} \}$. Both are representable: $U$ by the discrete space $\{0,1\}$ with base point $0$ and $R$ by the Sierpinski space with an isolated base point added. It is clear that $R$ is reflexive, but not symmetric.' - - property_id: unital + - property: unital reason: 'The joint image of $X \to X \times Y \leftarrow Y$ is just $\{(x,0) : x \in X\} \cup \{(0,y) : y \in Y\}$ (where $0$ denotes the base point), which is clearly a proper subset of $X \times Y$ when both $X,Y$ are non-trivial.' - - property_id: CSP + - property: CSP # TODO: remove duplication with unital proof reason: 'The image of $X \vee Y$ in $X \times Y$ is just $\{(x,0) : x \in X\} \cup \{(0,y) : y \in Y\}$ (where $0$ denotes the base point), which is clearly a proper subset when both $X,Y$ are non-trivial.' - - property_id: regular quotient object classifier + - property: regular quotient object classifier reason: We can recycle the proof for $\Set_*$ using discrete topological spaces. - - property_id: coaccessible + - property: coaccessible reason: 'We can adjust the proof for $\Top$ as follows: Assume $\Top_*$ is coaccessible. Let $S_0=\{x,*\}$ be the pointed topological space such that $\{*\}$ is the only non-trivial open set, and let $S_1=\{x,*\}$ be the pointed space such that $\{x\}$ is the only non-trivial open set. Let $p_i\colon S_i \to \{x,*\}$ be the identity function to the two-element indiscrete pointed space. Then, a pointed topological space is discrete if and only if it is projective to the morphisms $p_0$ and $p_1$. This implies that the full subcategory spanned by all discrete pointed spaces, which is equivalent to $\Set_*$, is coaccessible by Prop. 4.7 in Adamek-Rosicky. However, since $\Set_*$ is not coaccessible, this is a contradiction.' - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: We already know that $\Set_*$ does not have this property. Now apply the contrapositive of the dual of this lemma to the functor $\Set_* \to \Top_*$ that equips a pointed set with the indiscrete topology. - - property_id: effective congruences + - property: effective congruences reason: Suppose that $\Top_*$ had effective congruences. Then by this result, $\Top$ would also have effective congruences, which we know is not the case. - - property_id: effective cocongruences + - property: effective cocongruences reason: 'This counterexample is adapted from the counterexample for $\Top$. Consider the pointed topological space $I := \{ *, a, b \}$ with topology $\{ \varnothing, \{ * \}, \{ a, b \}, \{ *, a, b \} \}$. This represents the functor which sends a pointed topological space $X$ to the pairs of indistinguishable points of $X$. Therefore, we get a cocongruence $\{ *, a \} \rightrightarrows I$ on the discrete space $\{ *, a \}$, where the maps are $*\mapsto *, a\mapsto a$ and $*\mapsto *, a\mapsto b$ respectively. However, this cannot be effective: if we have $h : Z \to \{ *, a \}$ which equalizes the cocongruence, then $h$ must be the constant function with value $*$. But that means the cokernel pair of $h$ is the discrete space on $\{ *, a, b \}$.' special_objects: diff --git a/databases/catdat/data/categories/TorsAb.yaml b/databases/catdat/data/categories/TorsAb.yaml index 94bfee59..bf9737bd 100644 --- a/databases/catdat/data/categories/TorsAb.yaml +++ b/databases/catdat/data/categories/TorsAb.yaml @@ -15,40 +15,40 @@ related_categories: - TorsFreeAb satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\TorsAb \to \Ab$ and $\Ab$ is locally small. - - property_id: cocomplete + - property: cocomplete reason: The embedding $\TorsAb \hookrightarrow \Ab$ is closed under colimits and $\Ab$ is cocomplete. check_redundancy: false - - property_id: complete + - property: complete reason: The embedding $\TorsAb \hookrightarrow \Ab$ has a right adjoint, sending an abelian group $A$ to its torsion subgroup $T(A)$. Since $\Ab$ is complete, $\TorsAb$ is complete as well. The limit of a diagram of torsion abelian groups is the torsion subgroup of the limit of the underlying abelian groups. Notice that the torsion subgroup is not required in the case of equalizers, since a subgroup of a torsion abelian group is already torsion. Also, a finite product of torsion abelian groups is already torsion. check_redundancy: false - - property_id: preadditive + - property: preadditive reason: It is a full subcategory of the preadditive category $\Ab$. - - property_id: normal + - property: normal reason: 'If $f : A \to B$ is a monomorphism, it is injective (see below). In $\Ab$ it is then the kernel of $B \to B/f(A)$. Since $B/f(A)$ is torsion, it is also the kernel in $\TorsAb$.' - - property_id: conormal + - property: conormal reason: 'If $f : A \to B$ is an epimorphism, it is surjective (see below). In $\Ab$ it is then the cokernel of its kernel $K \hookrightarrow A$. Since $K$ is torsion, it is also the cokernel in $\TorsAb$.' - - property_id: finitely accessible + - property: finitely accessible reason: We already know that (filtered) colimits exist and are preserved by the forgetful functor to $\Ab$. Every torsion abelian group is the filtered colimit of its finitely generated subgroups (which are finite). These are finitely presentable in $\Ab$, hence also in $\TorsAb$. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: split abelian + - property: split abelian reason: The sequence $0 \to \IZ/2 \to \IZ/4 \to \IZ/2 \to 0$ does not split. - - property_id: CSP + - property: CSP reason: The canonical homomorphism $\bigoplus_{n \geq 0} \IZ/2 \to T(\prod_{n \geq 0} \IZ/2)$ is not surjective, since the torsion element $(1,1,\dotsc)$ does not lie in the image. Hence, it is no epimorphism. - - property_id: locally strongly finitely presentable + - property: locally strongly finitely presentable reason: >- Assume that it is locally strongly finitely presentable, and therefore the category of algebras of a multi-sorted finitary algebraic theory. For each sort $s$ let $F_s$ be the free algebra on one generator of sort $s$. One of them must be non-trivial, since otherwise the category would be thin. So assume $F_s \neq 0$. It is finitely presentable, hence a finite abelian group, and regular projective. Any direct summand will have the same properties. So we find that some $\IZ/p^k$ (with $k \geq 1$ and $p$ prime) is regular projective. However, the exact sequence in $\TorsAb$ $$0 \to \IZ/p \to \IZ/p^{k+1} \to \IZ/p^k \to 0$$ diff --git a/databases/catdat/data/categories/TorsFreeAb.yaml b/databases/catdat/data/categories/TorsFreeAb.yaml index 1d3981b8..8f616ba0 100644 --- a/databases/catdat/data/categories/TorsFreeAb.yaml +++ b/databases/catdat/data/categories/TorsFreeAb.yaml @@ -15,43 +15,43 @@ related_categories: - TorsAb satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\TorsFreeAb \to \Ab$ and $\Ab$ is locally small. - - property_id: complete + - property: complete reason: The embedding $\TorsFreeAb \hookrightarrow \Ab$ is closed under limits and $\Ab$ is complete. check_redundancy: false - - property_id: cocomplete + - property: cocomplete reason: 'The embedding $\TorsFreeAb \hookrightarrow \Ab$ has a left adjoint, sending an abelian group $A$ to its torsion-free reflection $A/T(A)$, where $T(A)$ is the torsion subgroup of $A$. Since $\Ab$ is cocomplete, $\TorsFreeAb$ is cocomplete as well. The colimit of a diagram of torsion-free abelian groups is the torsion-free reflection of the colimit of the underlying abelian groups. Notice that the reflection is not required in the case of coproducts: the direct sum of torsion-free abelian groups is again torsion-free. It is also not required for filtered colimits.' check_redundancy: false - - property_id: finitely accessible + - property: finitely accessible reason: We already saw that filtered colimits exist and are preserved by the forgetful functor to $\Ab$. Every torsion-free abelian group is the filtered colimit of its finitely generated subgroups, which are in fact free. Finitely generated free abelian groups are finitely presentable in $\Ab$ and therefore also in $\TorsFreeAb$. - - property_id: preadditive + - property: preadditive reason: It is a full subcategory of the preadditive category $\Ab$. - - property_id: cogenerator + - property: cogenerator reason: The additive group $\IQ$ is a cogenerator since every torsion-free abelian group $A$ embeds into $A \otimes \IQ$, which is a vector space over $\IQ$, and by linear algebra $K$ is a cogenerator in the category of vector spaces over $K$. - - property_id: regular + - property: regular reason: The regular epimorphisms are exactly the surjective homomorphisms (see below), and these are clearly stable under pullbacks. - - property_id: coregular + - property: coregular reason: >- It suffices to prove that regular monomorphisms (which are classified below) are stable under pushouts. Let $i : A \to B$ be a regular monomorphism in $\TorsFreeAb$, i.e. $i$ is injective and its $\Ab$-cokernel $B/i(A)$ is torsion-free, and let $f : B \to C$ be any morphism in $\TorsFreeAb$. Their $\Ab$-pushout is $$P = (B \times C)/\{(i(a),-f(a)): a \in A\}.$$ It is torsion-free: If $n \in \IZ \setminus \{0\}$ and $n (b,c) = (i(a),-f(a))$, there is some $a' \in A$ with $b = i(a')$ since $B/i(A)$ is torsion-free. It follows $n a' = a$, and then $c = -f(a')$ since $C$ is torsion-free. Thus, $(b,c) = (i(a'),-f(a'))$, which proves our claim. Therefore, $P$ is also the pushout in $\TorsFreeAb$. The homomorphism $j : C \to P$, $j(c) = [0,c]$ is injective (since $\Ab$ is coregular, but a direct proof is also easy), and by the universal property of $P$ its $\Ab$-cokernel is isomorphic to the $\Ab$-cokernel of $i$, which is torsion-free. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: balanced + - property: balanced reason: 'It can be checked directly that $2 : \IZ \to \IZ$ is both a monomorphism and an epimorphism, but no isomorphism. This also follows from the general classification of mono- and epimorphisms below.' - - property_id: CSP + - property: CSP reason: The canonical homomorphism $\bigoplus_{n \geq 0} \IZ \to \prod_{n \geq 0} \IZ$ is injective, but not an epimorphism, since the quotient $\prod_{n \geq 0} \IZ / \bigoplus_{n \geq 0} \IZ$ is not torsion. In fact, it is torsion-free and non-zero. special_objects: diff --git a/databases/catdat/data/categories/Vect.yaml b/databases/catdat/data/categories/Vect.yaml index 438c19dc..05431984 100644 --- a/databases/catdat/data/categories/Vect.yaml +++ b/databases/catdat/data/categories/Vect.yaml @@ -14,20 +14,20 @@ related_categories: - R-Mod_div satisfied_properties: - - property_id: locally small + - property: locally small reason: There is a forgetful functor $\Vect \to \Set$ and $\Set$ is locally small. - - property_id: split abelian + - property: split abelian reason: That $\Vect$ is abelian is a standard fact, see Mac Lane, Ch. VIII. Furthermore, it is a fact from linear algebra that every subspace has a complement, which is why every short exact sequence splits. - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic theory of a vector space. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: CSP + - property: CSP reason: The canonical homomorphism $\bigoplus_{n \geq 0} K \to \prod_{n \geq 0} K$ is not surjective, hence no epimorphism. special_objects: diff --git a/databases/catdat/data/categories/Z.yaml b/databases/catdat/data/categories/Z.yaml index 25f1f027..7a4f42c0 100644 --- a/databases/catdat/data/categories/Z.yaml +++ b/databases/catdat/data/categories/Z.yaml @@ -15,63 +15,63 @@ related_categories: - Set satisfied_properties: - - property_id: complete + - property: complete reason: This follows immediately from the fact for $\Set$. - - property_id: cocomplete + - property: cocomplete reason: This follows immediately from the fact for $\Set$. check_redundancy: false - - property_id: infinitary extensive + - property: infinitary extensive reason: This follows immediately from the fact for $\Set$. - - property_id: exact filtered colimits + - property: exact filtered colimits reason: This follows immediately from the fact for $\Set$. - - property_id: mono-regular + - property: mono-regular reason: This follows immediately from the fact for $\Set$. check_redundancy: false - - property_id: epi-regular + - property: epi-regular reason: This follows immediately from the fact for $\Set$. - - property_id: regular + - property: regular reason: This follows immediately from the fact for $\Set$. - - property_id: coregular + - property: coregular reason: This follows immediately from the fact for $\Set$. - - property_id: co-Malcev + - property: co-Malcev reason: This follows immediately from the fact for $\Set$. check_redundancy: false - - property_id: effective congruences + - property: effective congruences reason: 'If we have a congruence $E \rightrightarrows X$ in $[\CRing, \Set]$, then evaluating at any commutative ring gives a congruence in $\Set$. Defining $Y$ pointwise to be the quotient of this congruence, we get a morphism of functors $h : X \to Y$, and by this result applied pointwise, the kernel pair of $h$ is $E$.' - - property_id: effective cocongruences + - property: effective cocongruences reason: 'If we have a cocongruence $X\rightrightarrows E$ in $[\CRing, \Set]$, then evaluating at any commutative gives a cocongruence in $\Set$. Defining $Y$ pointwise to be the equalizer of the pair, we get a morphism of functors $h : Y \to X$, and by the dual of this result applied pointwise, the cokernel pair of $h$ is $E$.' check_redundancy: false unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: Malcev + - property: Malcev reason: Any counterexample for $\Set$ (i.e., any non-symmetric reflexive relation) yields one for this category by taking constant functors. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: This is because already the full subcategory of representable functors is not semi-strongly connected, because $\CRing$ is not semi-strongly connected. Specifically, there is no morphism between $\Hom(\IF_2,-)$ and $\Hom(\IF_3,-)$. - - property_id: locally essentially small + - property: locally essentially small reason: See MO/390611 for example. - - property_id: cartesian closed + - property: cartesian closed reason: 'There are functors $F,G : \CRing \rightrightarrows \Set$ such that $\Hom(F,G)$ is not essentially small, see MO/390611 for example. Now if the exponential $[F,G] : \CRing \to \Set$ exists, we get $[F,G](\IZ) \cong \Hom(\Hom(\IZ,-),[F,G])$ by Yoneda, which simplifies to $\Hom(1,[F,G]) \cong \Hom(1 \times F,G) \cong \Hom(F,G)$, a contradiction.' - - property_id: well-powered + - property: well-powered reason: 'Consider the functor $F$ from MO/390611 for example. The collection of subobjects of $F$ is not isomorphic to a set: for each infinite cardinal $\kappa$, simply cut off the construction of $F$ at $\kappa$. This yields a different subobject for each $\kappa$.' - - property_id: cofiltered-limit-stable epimorphisms + - property: cofiltered-limit-stable epimorphisms reason: We already know that $\Set$ does not have this property. Now apply the contrapositive of the dual of this lemma to the functor $\Set \to [\CRing, \Set]$ that maps a set to its constant functor. special_objects: diff --git a/databases/catdat/data/categories/Z_div.yaml b/databases/catdat/data/categories/Z_div.yaml index ebddd5fc..12712066 100644 --- a/databases/catdat/data/categories/Z_div.yaml +++ b/databases/catdat/data/categories/Z_div.yaml @@ -14,33 +14,33 @@ related_categories: - N satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: countable + - property: countable reason: This is trivial. - - property_id: thin + - property: thin reason: This is trivial. check_redundancy: false - - property_id: distributive + - property: distributive reason: 'We need to prove $\lcm_i \gcd(a, b_i) \cong \gcd(a, \lcm_i b_i)$ for finite families. If $x$ denotes the LHS and $y$ denotes the RHS, the relation $x \mid y$ is formal. If $v_p(-) : \IZ \to \IN_{\infty}$ denotes the multiplicity of a prime $p$, then $v_p(x)$ equals $\max_i \min(v_p(a),v_p(b_i))$, and $v_p(y)$ equals $\min(v_p(a), \max_i v_p(b_i))$. Since our family is finite, there is some $i_0$ with $\max_i v_p(b_i) = v_p(b_{i_0})$. Then $v_p(x) \geq \min(v_p(a),v_p(b_{i_0})) = v_p(y)$. This proves $y \mid x$.' - - property_id: locally ℵ₁-presentable + - property: locally ℵ₁-presentable reason: Every $\aleph_1$-directed diagram is eventually constant. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: The integers $+1$ and $-1$ are isomorphic, but not equal. - - property_id: self-dual + - property: self-dual reason: The only integer with infinitely many divisors (up to isomorphism) is $0$. But many integers have infinitely many multiples (up to isomorphism). - - property_id: countably distributive + - property: countably distributive reason: We have $2 \times \coprod_n 3^n = \gcd(2,\lcm_n(3^n)) = \gcd(2,0) = 2$, but $\coprod_n (2 \times 3^n) = \lcm_n \gcd(2,3^n) = \lcm_n 1 = 1$. - - property_id: countably codistributive + - property: countably codistributive reason: If $p$ runs through all odd primes, we have $2 \sqcup \prod_p p = \lcm(2,\gcd_p p) = \lcm(2,0) = 0$, but $\prod_p (2 \sqcup p) = \gcd_p (\lcm(2,p)) = \gcd_p (2 \cdot p) = 2$. special_objects: diff --git a/databases/catdat/data/categories/real_interval.yaml b/databases/catdat/data/categories/real_interval.yaml index 9b954b76..13a0cf4d 100644 --- a/databases/catdat/data/categories/real_interval.yaml +++ b/databases/catdat/data/categories/real_interval.yaml @@ -14,32 +14,32 @@ related_categories: - N_oo satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: self-dual + - property: self-dual reason: Take $t \mapsto 1-t$. - - property_id: skeletal + - property: skeletal reason: The relation $\leq$ is antisymmetric. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: This is trivial. - - property_id: locally ℵ₁-presentable + - property: locally ℵ₁-presentable reason: See MSE/4481902. unsatisfied_properties: - - property_id: essentially countable + - property: essentially countable reason: This is trivial. - - property_id: direct + - property: direct reason: Consider the strictly decreasing sequence $1/2^n$ for $n \geq 0$. - - property_id: inverse + - property: inverse reason: Consider the strictly increasing sequence $1 - 1/2^n$ for $n \geq 0$. - - property_id: locally finitely presentable + - property: locally finitely presentable reason: It suffices to prove that $0$ (the initial object) is the only finitely presentable object. If $s > 0$, then $s = \sup_{n \in \IN, \, s \geq 1/n } (s - 1/n)$, but there is no $n$ with $s \leq s - 1/n$. special_objects: diff --git a/databases/catdat/data/categories/sSet.yaml b/databases/catdat/data/categories/sSet.yaml index bb9524e8..7b378ddd 100644 --- a/databases/catdat/data/categories/sSet.yaml +++ b/databases/catdat/data/categories/sSet.yaml @@ -15,26 +15,26 @@ related_categories: - Top satisfied_properties: - - property_id: locally small + - property: locally small reason: This follows from the general fact that $[\C,\D]$ is locally small when $\C$ is small and $\D$ is locally small, here applied to $\C = \Delta^{\op}$ and $\D = \Set$. - - property_id: Grothendieck topos + - property: Grothendieck topos reason: This is clear from the definitions. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: Let $X,Y$ be two simplicial sets. Assume that $X_0$ is empty. Then $X_n$ is empty for all $n$ since there is a morphism $[0] \to [n]$, hence a map $X_n \to X_0$. So there is a morphism $X \to Y$ for trivial reasons. If $X_0$ is non-empty, pick an element. By the Yoneda Lemma it corresponds to a morphism $\Delta^0 \to X$. Since $\Delta^0 = 1$ is terminal, there is a morphism $Y \to \Delta^0$, and these compose to a morphism $Y \to X$. - - property_id: generator + - property: generator reason: 'Let $\Delta^n := \Hom([n],-)$ be the standard $n$-simplex for $n \geq 0$. The set $\{\Delta^n : n \geq 0\}$ is a generating set by the Yoneda Lemma. For all $n,m$ there is a morphism $[n] \to [m]$ in $\Delta$ and hence a morphism $\Delta^m \to \Delta^n$ in $\sSet$. Then by this lemma the coproduct $\coprod_{n \geq 0} \Delta^n$ is a generator in $\sSet$.' - - property_id: locally strongly finitely presentable + - property: locally strongly finitely presentable reason: This follows from the fact that every category of presheaves over a small category is locally strongly finitely presentable. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: finitary algebraic + - property: finitary algebraic reason: A one-sorted finitary algebraic category has an object $F$ (the free algebra on one generator) such that $F$ is finitely presentable and every object $X$ admits an epimorphism $\coprod_{s \in S} F \to X$ for some index set $S$. Assume that such a simplicial set $F$ exists. By using the sequence of $n$-skeletons of $F$, we see that there is some $n$ such that every $n$-simplex in $F$ is degenerate. Now take $X = \Delta^n$, which has a non-degenerate $n$-simplex. Then there cannot be an epimorphism $\coprod_{s \in S} F \to X$. special_objects: diff --git a/databases/catdat/data/categories/walking_commutative_square.yaml b/databases/catdat/data/categories/walking_commutative_square.yaml index 8c623584..273cc3ec 100644 --- a/databases/catdat/data/categories/walking_commutative_square.yaml +++ b/databases/catdat/data/categories/walking_commutative_square.yaml @@ -18,29 +18,29 @@ related_categories: - walking_morphism satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: finite + - property: finite reason: This is trivial. - - property_id: skeletal + - property: skeletal reason: The four objects are not isomorphic. - - property_id: self-dual + - property: self-dual reason: This is trivial. - - property_id: locally cartesian closed + - property: locally cartesian closed reason: This is because the walking morphism has this property. - - property_id: locally strongly finitely presentable + - property: locally strongly finitely presentable reason: This is because the walking morphism has this property. Alternatively, we may represent this category as the category of algebras for the finitary algebraic theory with two sorts $S_1,S_2$, the equation $x=y$ for $x,y \in S_1$, and the equation $x=y$ for $x,y \in S_2$. unsatisfied_properties: - - property_id: semi-strongly connected + - property: semi-strongly connected reason: There is no morphism between $b$ and $c$ (resp., between $(0,1)$ and $(1,0)$). - - property_id: finitary algebraic + - property: finitary algebraic reason: This follows from this lemma. special_objects: diff --git a/databases/catdat/data/categories/walking_composable_pair.yaml b/databases/catdat/data/categories/walking_composable_pair.yaml index 6199807a..bbb13bbd 100644 --- a/databases/catdat/data/categories/walking_composable_pair.yaml +++ b/databases/catdat/data/categories/walking_composable_pair.yaml @@ -16,26 +16,26 @@ related_categories: - walking_morphism satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: finite + - property: finite reason: This is trivial. - - property_id: skeletal + - property: skeletal reason: This is trivial. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: This is trivial. - - property_id: self-dual + - property: self-dual reason: This is trivial. - - property_id: locally strongly finitely presentable + - property: locally strongly finitely presentable reason: 'Take the two-sorted (finitary) algebraic theory with exactly one unary operation between them and the equation $x=y$ for each sort. There are exactly three algebras for this theory up to isomorphism: the identities on the empty set and the singleton, the morphism from the empty set to the singleton. Hence we get the equivalence to $\{0 \to 1 \to 2\}$.' unsatisfied_properties: - - property_id: finitary algebraic + - property: finitary algebraic reason: This follows from this lemma. special_objects: diff --git a/databases/catdat/data/categories/walking_coreflexive_pair.yaml b/databases/catdat/data/categories/walking_coreflexive_pair.yaml index f969ed18..0464bd08 100644 --- a/databases/catdat/data/categories/walking_coreflexive_pair.yaml +++ b/databases/catdat/data/categories/walking_coreflexive_pair.yaml @@ -18,56 +18,56 @@ related_categories: - walking_splitting satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: finite + - property: finite reason: This is trivial. - - property_id: strongly connected + - property: strongly connected reason: This is trivial. - - property_id: gaunt + - property: gaunt reason: This is obvious. - - property_id: terminal object + - property: terminal object reason: The object $[0]$ is terminal since it is already terminal in $\Delta$. - - property_id: generator + - property: generator reason: The object $[0]$ is generator since this is already true in $\Delta$. A direct proof is also possible. - - property_id: cogenerator + - property: cogenerator reason: The object $[1]$ is cogenerator since this is already true in $\Delta$. A direct proof is also possible. - - property_id: epi-regular + - property: epi-regular reason: 'The only non-identity epimorphism is $p$, which is the coequalizer of $\id, ip : [1] \rightrightarrows [1]$ (since $pi = \id$).' - - property_id: mono-regular + - property: mono-regular reason: 'The only non-identity monomorphisms are $i$ and $j$. The morphism $i$ is the equalizer of $\id, ip : [1] \rightrightarrows [1]$ (since $pi = \id$), and for $j$ it is the same argument.' - - property_id: coequalizers + - property: coequalizers reason: 'We already know that the $\Delta$ has coequalizers, and the proof has shown that the cardinality does not increase, so we are done. But a direct proof is also possible: There are four non-equal parallel pairs: $(i,j)$, $(ip,jp)$, $(\id,ip)$, and $(\id,jp)$. The first two have the same coequalizer (if it exists) since $p$ is an epimorphism, the last two are symmetric, and we already remarked that $p$ is a coequalizer of $(\id,ip)$. So it suffices to check that $p$ is a coequalizer of $i,j$, which is easy.' - - property_id: cosifted + - property: cosifted reason: Our proof that the $\Delta$ is cosifted has only used $[0],[1]$ as auxiliary objects and therefore also shows that $\Delta^{\leq 1}$ is cosifted. - - property_id: generalized variety + - property: generalized variety reason: This actually holds for every truncated simplex category $\Delta^{\leq n}$. See MO/510760 for a proof that sifted colimits exist. See MO/510827 for a proof that every object is strongly finitely presentable. unsatisfied_properties: - - property_id: strict terminal object + - property: strict terminal object reason: 'The morphism $i : [0] \to [1]$ from the terminal object $[0]$ is a witness.' - - property_id: cofiltered + - property: cofiltered reason: 'The morphisms $i,j : [0] \rightrightarrows [1]$ are not equalized by any morphism.' - - property_id: coreflexive equalizers + - property: coreflexive equalizers reason: 'The coreflexive pair $i,j : [0] \rightrightarrows [1]$ has no equalizer, in fact is not equalized by any morphism.' - - property_id: pushouts + - property: pushouts reason: Assume that $[1] \xleftarrow{i} [0] \xrightarrow{i} [1]$ has a pushout in $\Delta^{\leq 1}$, where $i(0)=0$. This amounts to a universal totally ordered set of cardinality $\leq 2$ with elements $a,b,c$ satisfying $a \leq b$, $a \leq c$. Since a finite totally ordered set has trivial automorphism group, the automorphism defined by $a \mapsto a$, $b \mapsto c$, $c \mapsto b$ must be the identity, i.e., we have $b = c$. However, in $[1]$ the equations $0 \leq 0$, $0 \leq 1$ then show that the universal property fails. - - property_id: multi-complete + - property: multi-complete # TODO: remove this manual proof once locally multi-presentable is dualized, cf. #139 reason: 'This follows directly from existing results: If its dual, the walking reflexive pair, was multi-cocomplete, then since it is also accessible, it would be locally multi-presentable. But then it would have connected limits, in particular pullbacks.' diff --git a/databases/catdat/data/categories/walking_fork.yaml b/databases/catdat/data/categories/walking_fork.yaml index d159c18b..f9ecd66f 100644 --- a/databases/catdat/data/categories/walking_fork.yaml +++ b/databases/catdat/data/categories/walking_fork.yaml @@ -16,44 +16,44 @@ related_categories: - walking_pair satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: finite + - property: finite reason: This is trivial. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: This is obvious. - - property_id: skeletal + - property: skeletal reason: The three objects are clearly not isomorphic. - - property_id: one-way + - property: one-way reason: This is trivial. - - property_id: generator + - property: generator reason: It is easy to check that $1$ is a generator. - - property_id: cogenerator + - property: cogenerator reason: It is easy to check that $2$ is a cogenerator. - - property_id: left cancellative + - property: left cancellative reason: It is easy to check that $i,f,g$ are monomorphisms. - - property_id: equalizers + - property: equalizers reason: The only pair of distinct parallel morphisms is $f,g$, and their equalizer is $i$. - - property_id: locally cartesian closed + - property: locally cartesian closed reason: We need to check that every slice category is cartesian closed. The slice category over $0$ is the trivial category. The slice category over $1$ is the walking morphism. Finally, the slice category over $2$ ist the walking commutative square. All of these are cartesian closed, see their pages for details. unsatisfied_properties: - - property_id: strongly connected + - property: strongly connected reason: There is no morphism $1 \to 0$. - - property_id: balanced + - property: balanced reason: Both $f$ and $g$ are monomorphisms and epimorphisms. - - property_id: binary powers + - property: binary powers reason: 'Assume that $X := 2 \times 2$ exists. Since there is a diagonal morphism $2 \to X$, we must have $X = 2$, and the two projections $p_1,p_2 : X \rightrightarrows 2$ must be equal to the identity. But $f,g$ induce a morphism $(f,g) : 1 \to X$ with $p_1 (f,g) = f$ and $p_2 (f,g) = g$, so that $f=g$, a contradiction.' special_objects: diff --git a/databases/catdat/data/categories/walking_idempotent.yaml b/databases/catdat/data/categories/walking_idempotent.yaml index ef1d1bf0..68b3a491 100644 --- a/databases/catdat/data/categories/walking_idempotent.yaml +++ b/databases/catdat/data/categories/walking_idempotent.yaml @@ -18,35 +18,35 @@ related_categories: - walking_splitting satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: finite + - property: finite reason: This is trivial. - - property_id: gaunt + - property: gaunt reason: This is obvious. - - property_id: self-dual + - property: self-dual reason: This is obvious. - - property_id: generator + - property: generator reason: The unique object is a generator for trivial reasons. - - property_id: subobject-trivial + - property: subobject-trivial reason: This is because the identity is the only monomorphism. - - property_id: preadditive + - property: preadditive reason: The monoid $\{1,e\}$ with $e^2=e$ is the underlying multiplicative monoid of the ring $\IZ/2$, where $e=0$. Thus, the (unique) preadditive structure is given by $1 + e = e + 1 = 1$, $e + e = e$ and $1 + 1 = e$. - - property_id: filtered + - property: filtered reason: The pair $\id,e$ is coequalized by $e$ (non-universally). unsatisfied_properties: - - property_id: terminal object + - property: terminal object reason: This is obvious. - - property_id: Cauchy complete + - property: Cauchy complete reason: The idempotent $e$ does not split. special_objects: {} diff --git a/databases/catdat/data/categories/walking_isomorphism.yaml b/databases/catdat/data/categories/walking_isomorphism.yaml index 75c7612b..e406cd04 100644 --- a/databases/catdat/data/categories/walking_isomorphism.yaml +++ b/databases/catdat/data/categories/walking_isomorphism.yaml @@ -17,17 +17,17 @@ related_categories: - walking_morphism satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: finite + - property: finite reason: This is trivial. - - property_id: trivial + - property: trivial reason: The inclusion $\{0\} \hookrightarrow \{0 \to 1\}$ is an equivalence. unsatisfied_properties: - - property_id: skeletal + - property: skeletal reason: The two objects are isomorphic, but different. special_objects: diff --git a/databases/catdat/data/categories/walking_morphism.yaml b/databases/catdat/data/categories/walking_morphism.yaml index 324254b0..d1525d8a 100644 --- a/databases/catdat/data/categories/walking_morphism.yaml +++ b/databases/catdat/data/categories/walking_morphism.yaml @@ -21,26 +21,26 @@ related_categories: - walking_splitting satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: finite + - property: finite reason: This is trivial. - - property_id: skeletal + - property: skeletal reason: The two objects are not isomorphic. - - property_id: self-dual + - property: self-dual reason: This is trivial. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: This is trivial. - - property_id: finitary algebraic + - property: finitary algebraic reason: Take the algebraic theory with no operations but with the equation $x=y$ that is supposed to hold for all elements $x,y$. The algebras for this theory are the empty set and the singleton set, hence we get the equivalence to $\{0 \to 1\}$. unsatisfied_properties: - - property_id: trivial + - property: trivial reason: This is trivial. special_objects: diff --git a/databases/catdat/data/categories/walking_pair.yaml b/databases/catdat/data/categories/walking_pair.yaml index e11275a4..d01cf598 100644 --- a/databases/catdat/data/categories/walking_pair.yaml +++ b/databases/catdat/data/categories/walking_pair.yaml @@ -16,38 +16,38 @@ related_categories: - walking_morphism satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: finite + - property: finite reason: This is trivial. - - property_id: self-dual + - property: self-dual reason: This is trivial. - - property_id: semi-strongly connected + - property: semi-strongly connected reason: This is trivial. - - property_id: skeletal + - property: skeletal reason: The two objects are not isomorphic. - - property_id: one-way + - property: one-way reason: This is trivial. - - property_id: generator + - property: generator reason: It is easy to check that $0$ is a generator. - - property_id: left cancellative + - property: left cancellative reason: The two morphisms $0 \rightrightarrows 1$ are clearly monomorphisms. - - property_id: sifted colimits + - property: sifted colimits reason: A proof can be found here. unsatisfied_properties: - - property_id: strongly connected + - property: strongly connected reason: There is no morphism $1 \to 0$. - - property_id: pullbacks + - property: pullbacks reason: 'The two morphisms $a,b : 0 \rightrightarrows 1$ have no pullback, since it would have to consist of identities $0 \leftarrow 0 \rightarrow 0$, but $a \neq b$.' special_objects: {} diff --git a/databases/catdat/data/categories/walking_span.yaml b/databases/catdat/data/categories/walking_span.yaml index f0b185e5..2d11940c 100644 --- a/databases/catdat/data/categories/walking_span.yaml +++ b/databases/catdat/data/categories/walking_span.yaml @@ -16,32 +16,32 @@ related_categories: - walking_pair satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: finite + - property: finite reason: This is trivial. - - property_id: thin + - property: thin reason: This is trivial. - - property_id: skeletal + - property: skeletal reason: The three objects are not isomorphic. - - property_id: initial object + - property: initial object reason: $0$ is an initial object. - - property_id: binary products + - property: binary products reason: We have $0 \times x = 0$ for all $x$, $x \times x = x$, and $1 \times 2 = 0$. - - property_id: locally cartesian closed + - property: locally cartesian closed reason: The slice category over $0$ is the trivial category, and the slice category over $1$ is the walking morphism, which is cartesian closed. The same holds for $2$ by symmetry. - - property_id: multi-algebraic + - property: multi-algebraic reason: We first remark that for a set $X$, the identity span $(\id,\id)\colon X \leftarrow X \rightarrow X$ exhibits a product if and only if $X$ is either a singleton or the empty set. Therefore, there is a (finite product, coproduct)-sketch whose $\Set$-model is precisely a pair $(X,Y)$ of sets such that each of $X$ and $Y$ is either a singleton or the empty set and the product $X \times Y$ is the empty set. Any $\Set$-model of such a sketch is isomorphic to either $(\varnothing, \varnothing)$, $(\varnothing, 1)$, or $(1, \varnothing)$; hence the category of models is equivalent to the walking span. unsatisfied_properties: - - property_id: sifted + - property: sifted reason: There is no cospan between $1$ and $2$. special_objects: diff --git a/databases/catdat/data/categories/walking_splitting.yaml b/databases/catdat/data/categories/walking_splitting.yaml index 2b24061e..e1141e9e 100644 --- a/databases/catdat/data/categories/walking_splitting.yaml +++ b/databases/catdat/data/categories/walking_splitting.yaml @@ -17,45 +17,45 @@ related_categories: - walking_isomorphism satisfied_properties: - - property_id: small + - property: small reason: This is trivial. - - property_id: finite + - property: finite reason: This is trivial. - - property_id: gaunt + - property: gaunt reason: This is obvious. - - property_id: self-dual + - property: self-dual reason: There is an isomorphism $\Split^\op \to \Split$ defined by $0 \mapsto 0$, $1 \mapsto 1$, $i \mapsto p$, $p \mapsto i$. - - property_id: pointed + - property: pointed reason: The object $0$ is initial and terminal. This also means that $\id_0, i, p, ip$ are zero morphisms. The only non-zero morphism is $\id_1$. - - property_id: equalizers + - property: equalizers reason: 'The only parallel pair of non-equal morphisms is $\id_1, ip : 1 \rightrightarrows 1$, and their equalizer is $i$.' - - property_id: normal + - property: normal reason: 'The only non-identity monomorphism is $i : 0 \to 1$, which is the kernel of $\id_1$.' - - property_id: generator + - property: generator reason: 'The object $1$ a generator, since the only parallel pair of non-equal morphisms is $\id_1, ip : 1 \rightrightarrows 1$ with domain $1$.' - - property_id: preadditive + - property: preadditive reason: 'We can define $\id_1 + \id_1 := ip$ (and it is clear how to add zero morphisms) and then verify that the axioms of a preadditive category hold. Alternatively, it suffices to find a preadditive category which is isomorphic to the walking splitting: Consider the full subcategory of $\Vect_{\IF_2}$ that consists only of the trivial vector space $\{0\}$ and $\IF_2$. Since $\Vect_{\IF_2}$ is preadditive, it is preadditive as well. It has two objects, two identities, the morphisms $i : \{0\} \to \IF_2$, $p : \IF_2 \to \{0\}$, and the zero morphism $ip : \IF_2 \to \IF_2$. Clearly, $pi$ is the identity.' - - property_id: sifted colimits + - property: sifted colimits reason: |- We work with the representation of the category as $\Vect^{\leq 1}_{\IF_2}$, the category of vector spaces over $\IF_2$ of dimension $\leq 1$. It suffices to show that it is closed under sifted colimits in $\Vect_{\IF_2}$. More generally, we show this for $\Vect^{\leq d}_K \subseteq \Vect_K$, where $d \in \IN$ and $K$ is a field. So let $X : \I \to \Vect_K$ be a sifted diagram with colimit $(u_i : X_i \to X_\infty)_{i \in \I}$. Since $\I$ is sifted, for finitely many objects $i_1,\dotsc,i_n \in \I$ there is an object $k$ that admits morphisms $i_1 \to k, \dotsc, i_n \to k$; this is all we need to know about $\I$. Assume that each $X_i$ is of dimension $\leq d$, we need to show this for $X_\infty$ as well. Every element in $X_\infty$ is a finite sum of elements of the form $u_i(x_i)$ with $x_i \in X_i$. Choose an object $k$ with morphisms $i \to k$ for every occurring $i$. If $y_i \in X_k$ denotes the image of $x_i$, we get $\sum_i u_i(x_i) = \sum_i u_k(y_i) = u_k(\sum_i y_i)$. Therefore, every element of $X_\infty$ has the form $u_i(x_i)$ for some $i \in \I$ and $x_i \in X_i$. Moreover, for finitely many elements in $X_\infty$ the index $i$ may be chosen uniformly. Now, if $X_\infty$ has dimension $> d$, it would have linearly independent vectors $v_0,\dotsc,v_d$, all of which have a preimage in $X_i$ for some $i \in \I$. But then these preimages would be linearly independent as well, which contradicts $\dim(X_i) \leq d$. check_redundancy: false - - property_id: generalized variety + - property: generalized variety reason: Again we work with $\Vect^{\leq 1}_{\IF_2}$. We already know that it has sifted colimits and that the embedding to $\Vect_{\IF_2}$ preserves them. The object $0$ is initial and hence strongly finitely presentable. The object $\IF_2$ is strongly finitely presentable in $\Vect^{\leq 1}_{\IF_2}$ since its hom-functor is the composition of the embedding and the forgetful functor $\Vect_{\IF_2} \to \Set$, and the latter preserves sifted colimits by [AR01, Lemma 3.3] applied to $\IF_2 \in \Vect_{\IF_2}$. unsatisfied_properties: - - property_id: one-way + - property: one-way reason: 'The morphism $ip : 1 \to 1$ provides a counterexample.' special_objects: diff --git a/databases/catdat/data/category-properties/CIP.yaml b/databases/catdat/data/category-properties/CIP.yaml index b3f7ebf0..b83f0a87 100644 --- a/databases/catdat/data/category-properties/CIP.yaml +++ b/databases/catdat/data/category-properties/CIP.yaml @@ -5,7 +5,7 @@ description: >- $$\textstyle \alpha : \coprod_i X_i \to \prod_{i \in I} X_i$$ defined by $p_j \circ \alpha \circ \iota_i = \delta_{i,j}$ is a monomorphism. This is no standard terminology. This property has been added to clarify relationships between other properties, in particular those concerning the commutation between limits and colimits. -dual_property_id: CSP +dual_property: CSP invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/CSP.yaml b/databases/catdat/data/category-properties/CSP.yaml index 4ce32191..0696c77b 100644 --- a/databases/catdat/data/category-properties/CSP.yaml +++ b/databases/catdat/data/category-properties/CSP.yaml @@ -5,7 +5,7 @@ description: >- $$\textstyle \alpha : \coprod_i X_i \to \prod_{i \in I} X_i$$ defined by $p_j \circ \alpha \circ \iota_i = \delta_{i,j}$ is an epimorphism. This is no standard terminology. This property has been added to clarify relationships between other properties, in particular those concerning the commutation between limits and colimits. -dual_property_id: CIP +dual_property: CIP invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/Cauchy complete.yaml b/databases/catdat/data/category-properties/Cauchy complete.yaml index 72f71533..98589161 100644 --- a/databases/catdat/data/category-properties/Cauchy complete.yaml +++ b/databases/catdat/data/category-properties/Cauchy complete.yaml @@ -2,7 +2,7 @@ id: Cauchy complete relation: is description: 'A category is Cauchy complete if every idempotent splits. That is, every endomorphism $e : X \to X$ with $e^2 = e$ may be written as $e = i \circ p$ for some morphisms $p : X \to Y$ and $i : Y \to X$ with $p \circ i = \id_Y$. Equivalently, the pair $e,\id_X : X \rightrightarrows X$ has an equalizer (or an coequalizer).' nlab_link: https://ncatlab.org/nlab/show/Cauchy+complete+category -dual_property_id: Cauchy complete +dual_property: Cauchy complete invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/Malcev.yaml b/databases/catdat/data/category-properties/Malcev.yaml index 2df7eec8..72046f47 100644 --- a/databases/catdat/data/category-properties/Malcev.yaml +++ b/databases/catdat/data/category-properties/Malcev.yaml @@ -2,7 +2,7 @@ id: Malcev relation: is description: A category is Malcev when it has finite limits and every internal reflexive relation is an internal equivalence relation. That is, if $R \subseteq X^2$ is a subobject with $\Delta_X \subseteq R$, then $R$ is symmetric and transitive. nlab_link: https://ncatlab.org/nlab/show/Malcev+category -dual_property_id: co-Malcev +dual_property: co-Malcev invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/abelian.yaml b/databases/catdat/data/category-properties/abelian.yaml index 472b885c..158b2d52 100644 --- a/databases/catdat/data/category-properties/abelian.yaml +++ b/databases/catdat/data/category-properties/abelian.yaml @@ -2,7 +2,7 @@ id: abelian relation: is description: A category is abelian if it is additive, every morphism has a kernel and a cokernel, and every monomorphism and epimorphism is normal. Equivalently, it is additive, has equalizers and coequalizers, and it is mono-regular and epi-regular. As opposed to other types of categories (such as monoidal categories), being abelian turns out to be a mere property. nlab_link: https://ncatlab.org/nlab/show/abelian+category -dual_property_id: abelian +dual_property: abelian invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/accessible.yaml b/databases/catdat/data/category-properties/accessible.yaml index 42049cc6..90ad9c33 100644 --- a/databases/catdat/data/category-properties/accessible.yaml +++ b/databases/catdat/data/category-properties/accessible.yaml @@ -2,7 +2,7 @@ id: accessible relation: is description: Let $\kappa$ be a regular cardinal. A category is $\kappa$-accessible if it has $\kappa$-filtered colimits and there is a (small) set $G$ of $\kappa$-presentable objects such that every object is a $\kappa$-filtered colimit of objects in $G$. A category is accessible if it is $\kappa$-accessible for some regular cardinal $\kappa$. nlab_link: https://ncatlab.org/nlab/show/accessible+category -dual_property_id: coaccessible +dual_property: coaccessible invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/additive.yaml b/databases/catdat/data/category-properties/additive.yaml index 8bf1b7aa..f794892b 100644 --- a/databases/catdat/data/category-properties/additive.yaml +++ b/databases/catdat/data/category-properties/additive.yaml @@ -2,7 +2,7 @@ id: additive relation: is description: A category is additive if it is preadditive and has finite products (equivalently, finite coproducts). Note that in the context of finite products, the preadditive structure is unique. nlab_link: https://ncatlab.org/nlab/show/additive+category -dual_property_id: additive +dual_property: additive invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/balanced.yaml b/databases/catdat/data/category-properties/balanced.yaml index b29c3ef2..b2bda473 100644 --- a/databases/catdat/data/category-properties/balanced.yaml +++ b/databases/catdat/data/category-properties/balanced.yaml @@ -2,7 +2,7 @@ id: balanced relation: is description: A category is balanced if every morphism which is a monomorphism and an epimorphism must be an isomorphism. nlab_link: https://ncatlab.org/nlab/show/balanced+category -dual_property_id: balanced +dual_property: balanced invariant_under_equivalences: true related_properties: [] diff --git a/databases/catdat/data/category-properties/binary copowers.yaml b/databases/catdat/data/category-properties/binary copowers.yaml index e5dea441..e37ff743 100644 --- a/databases/catdat/data/category-properties/binary copowers.yaml +++ b/databases/catdat/data/category-properties/binary copowers.yaml @@ -2,7 +2,7 @@ id: binary copowers relation: has description: A category has binary copowers when for every object $X$ and every binary set $I$ the coproduct $X \sqcup X$ exists. These objects might also be called doubles. nlab_link: https://ncatlab.org/nlab/show/copower -dual_property_id: binary powers +dual_property: binary powers invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/binary coproducts.yaml b/databases/catdat/data/category-properties/binary coproducts.yaml index 24de051c..46376fd2 100644 --- a/databases/catdat/data/category-properties/binary coproducts.yaml +++ b/databases/catdat/data/category-properties/binary coproducts.yaml @@ -2,7 +2,7 @@ id: binary coproducts relation: has description: A category has binary coproducts if every pair $A,B$ of objects has a coproduct $A \sqcup B$. nlab_link: https://ncatlab.org/nlab/show/coproduct -dual_property_id: binary products +dual_property: binary products invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/binary powers.yaml b/databases/catdat/data/category-properties/binary powers.yaml index 9ccb1ad8..b9317197 100644 --- a/databases/catdat/data/category-properties/binary powers.yaml +++ b/databases/catdat/data/category-properties/binary powers.yaml @@ -2,7 +2,7 @@ id: binary powers relation: has description: A category has binary powers when for every object $X$ the product $X \times X$ exists. These objects might also be called squares. nlab_link: https://ncatlab.org/nlab/show/powering -dual_property_id: binary copowers +dual_property: binary copowers invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/binary products.yaml b/databases/catdat/data/category-properties/binary products.yaml index 17e5d8ed..fabae3e2 100644 --- a/databases/catdat/data/category-properties/binary products.yaml +++ b/databases/catdat/data/category-properties/binary products.yaml @@ -2,7 +2,7 @@ id: binary products relation: has description: A category has binary products if every pair $A,B$ of objects has a product $A \times B$. nlab_link: https://ncatlab.org/nlab/show/binary+product -dual_property_id: binary coproducts +dual_property: binary coproducts invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/biproducts.yaml b/databases/catdat/data/category-properties/biproducts.yaml index f3a91826..9d5f5511 100644 --- a/databases/catdat/data/category-properties/biproducts.yaml +++ b/databases/catdat/data/category-properties/biproducts.yaml @@ -6,7 +6,7 @@ description: >- is an isomorphism. Such a category is also called semi-additive, and it is automatically enriched over commutative monoids: the sum of $f,g : A \rightrightarrows B$ is defined as: $$A \xrightarrow{(f,g)} B \times B \xrightarrow{\mu^{-1}} B \oplus B \xrightarrow{\nabla} B$$ nlab_link: https://ncatlab.org/nlab/show/biproduct -dual_property_id: biproducts +dual_property: biproducts invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/cartesian closed.yaml b/databases/catdat/data/category-properties/cartesian closed.yaml index 0405b62b..18fc27df 100644 --- a/databases/catdat/data/category-properties/cartesian closed.yaml +++ b/databases/catdat/data/category-properties/cartesian closed.yaml @@ -2,7 +2,7 @@ id: cartesian closed relation: is description: A category is cartesian closed if all finite products and exponentials $[X,Y]$ exist, defined by the adjunction $\Hom(T,[X,Y]) \cong \Hom(T \times X,Y)$. nlab_link: https://ncatlab.org/nlab/show/cartesian+closed+category -dual_property_id: cocartesian coclosed +dual_property: cocartesian coclosed invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/cartesian filtered colimits.yaml b/databases/catdat/data/category-properties/cartesian filtered colimits.yaml index 6fb671c0..519928b7 100644 --- a/databases/catdat/data/category-properties/cartesian filtered colimits.yaml +++ b/databases/catdat/data/category-properties/cartesian filtered colimits.yaml @@ -3,7 +3,7 @@ relation: has description: |- In a category $\C$, which we assume to have filtered colimits and finite products, we say that filtered colimits are cartesian if for every finite set $I$ the product functor $\prod : \C^I \to \C$ preserves filtered colimits. Equivalently, for every $X \in \C$ the functor $X \times - : \C \to \C$ preserves filtered colimits. This is no standard terminology, it has been suggested in MO/510240. We have added it to the database since it clarifies the relationship between many related properties. -dual_property_id: cocartesian cofiltered limits +dual_property: cocartesian cofiltered limits invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/co-Malcev.yaml b/databases/catdat/data/category-properties/co-Malcev.yaml index b52e94f3..d3c56a89 100644 --- a/databases/catdat/data/category-properties/co-Malcev.yaml +++ b/databases/catdat/data/category-properties/co-Malcev.yaml @@ -4,7 +4,7 @@ description: |- A category is co-Malcev when its dual is Malcev, i.e., it has finite colimits and if $X \sqcup X \twoheadrightarrow R$ is a coreflexive corelation, then it is cosymmetric and cotransitive. This terminology is not standard, but we have added it to properly formulate the interesting theorem that the dual of an elementary topos is Malcev, i.e., that every elementary topos is co-Malcev. o settle this property, we often use that $\C$ is co-Malcev if and only if the category of representable functors $\C \to \Set^+$ is Malcev. -dual_property_id: Malcev +dual_property: Malcev invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/coaccessible.yaml b/databases/catdat/data/category-properties/coaccessible.yaml index 3ef75db6..ef434d95 100644 --- a/databases/catdat/data/category-properties/coaccessible.yaml +++ b/databases/catdat/data/category-properties/coaccessible.yaml @@ -1,7 +1,7 @@ id: coaccessible relation: is description: A category is coaccessible if its opposite category is accessible. -dual_property_id: accessible +dual_property: accessible invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/cocartesian coclosed.yaml b/databases/catdat/data/category-properties/cocartesian coclosed.yaml index 33d20d5b..dd20312b 100644 --- a/databases/catdat/data/category-properties/cocartesian coclosed.yaml +++ b/databases/catdat/data/category-properties/cocartesian coclosed.yaml @@ -2,7 +2,7 @@ id: cocartesian coclosed relation: is description: A category is cocartesian coclosed if its dual category is cartesian closed, i.e. if all finite coproducts and coexponentials $\Coexp(X,Y)$ exist, defined by the adjunction $\Hom(\Coexp[X,Y],T) \cong \Hom(Y,T \sqcup X)$. nlab_link: https://ncatlab.org/nlab/show/cocartesian+coclosed+category -dual_property_id: cartesian closed +dual_property: cartesian closed invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/cocartesian cofiltered limits.yaml b/databases/catdat/data/category-properties/cocartesian cofiltered limits.yaml index 57223334..be2cdc96 100644 --- a/databases/catdat/data/category-properties/cocartesian cofiltered limits.yaml +++ b/databases/catdat/data/category-properties/cocartesian cofiltered limits.yaml @@ -3,7 +3,7 @@ relation: has description: |- In a category $\C$, which we assume to have cofiltered limits and finite coproducts, we say that cofiltered limits are cocartesian if for every finite set $I$ the coproduct functor $\coprod : \C^I \to \C$ preserves cofiltered limits. Equivalently, for every $X \in \C$ the functor $X \sqcup - : \C \to \C$ preserves cofiltered limits. This is no standard terminology, its dual has been suggested in MO/510240. We have added it to the database since it clarifies the relationship between many related properties. -dual_property_id: cartesian filtered colimits +dual_property: cartesian filtered colimits invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/cocomplete.yaml b/databases/catdat/data/category-properties/cocomplete.yaml index 85aeda66..487bd32b 100644 --- a/databases/catdat/data/category-properties/cocomplete.yaml +++ b/databases/catdat/data/category-properties/cocomplete.yaml @@ -2,7 +2,7 @@ id: cocomplete relation: is description: A category is cocomplete when every small diagram in the category has a colimit. nlab_link: https://ncatlab.org/nlab/show/cocomplete+category -dual_property_id: complete +dual_property: complete invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/codistributive.yaml b/databases/catdat/data/category-properties/codistributive.yaml index c6c165b6..2f7dd68d 100644 --- a/databases/catdat/data/category-properties/codistributive.yaml +++ b/databases/catdat/data/category-properties/codistributive.yaml @@ -1,7 +1,7 @@ id: codistributive relation: is description: A category is codistributive if it has finite coproducts, finite products, and for every object $A$ the functor $- \sqcup A$ preserves finite products. Concretely, for every finite family of objects $(B_i)$ the canonical morphism $A \sqcup \prod_i B_i \to \prod_i (A \sqcup B_i)$ must be an isomorphism. -dual_property_id: distributive +dual_property: distributive invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/coequalizers.yaml b/databases/catdat/data/category-properties/coequalizers.yaml index 615585b0..d09e6c15 100644 --- a/databases/catdat/data/category-properties/coequalizers.yaml +++ b/databases/catdat/data/category-properties/coequalizers.yaml @@ -2,7 +2,7 @@ id: coequalizers relation: has description: 'A coequalizer of a pair of morphisms $f,g : A \rightrightarrows B$ is an object $C$ with a morphism $c : B \to C$ such that $c \circ f = c \circ g$ and which is universal with respect to this property. This property refers to the existence of coequalizers.' nlab_link: https://ncatlab.org/nlab/show/coequalizer -dual_property_id: equalizers +dual_property: equalizers invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/coextensive.yaml b/databases/catdat/data/category-properties/coextensive.yaml index 05a61549..96a0f985 100644 --- a/databases/catdat/data/category-properties/coextensive.yaml +++ b/databases/catdat/data/category-properties/coextensive.yaml @@ -1,7 +1,7 @@ id: coextensive relation: is description: A category $\C$ is coextensive when it has finite products and for all objects $A,B \in \C$ the product functor $A/\C \times B/\C \to (A \times B)/\C$ is an equivalence of categories. The prototypical example is the category of commutative rings. -dual_property_id: extensive +dual_property: extensive invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/cofiltered limits.yaml b/databases/catdat/data/category-properties/cofiltered limits.yaml index c58a0151..9f840030 100644 --- a/databases/catdat/data/category-properties/cofiltered limits.yaml +++ b/databases/catdat/data/category-properties/cofiltered limits.yaml @@ -2,7 +2,7 @@ id: cofiltered limits relation: has description: A category has cofiltered limits if it has limits of diagrams indexed by small cofiltered categories. This is actually equivalent to having directed limits. nlab_link: https://ncatlab.org/nlab/show/filtered+limit -dual_property_id: filtered colimits +dual_property: filtered colimits invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/cofiltered-limit-stable epimorphisms.yaml b/databases/catdat/data/category-properties/cofiltered-limit-stable epimorphisms.yaml index 2954c122..dca20585 100644 --- a/databases/catdat/data/category-properties/cofiltered-limit-stable epimorphisms.yaml +++ b/databases/catdat/data/category-properties/cofiltered-limit-stable epimorphisms.yaml @@ -1,7 +1,7 @@ id: cofiltered-limit-stable epimorphisms relation: has description: A category has cofiltered-limit-stable epimorphisms if it has cofiltered limits and for every cofiltered diagram of epimorphisms $(X_i \to Y_i)$ also their limit $\lim_i X_i \to \lim_i Y_i$ is an epimorphism. -dual_property_id: filtered-colimit-stable monomorphisms +dual_property: filtered-colimit-stable monomorphisms invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/cofiltered.yaml b/databases/catdat/data/category-properties/cofiltered.yaml index 081dc6d3..c0f4dc19 100644 --- a/databases/catdat/data/category-properties/cofiltered.yaml +++ b/databases/catdat/data/category-properties/cofiltered.yaml @@ -2,7 +2,7 @@ id: cofiltered relation: is description: A category is cofiltered if every finite diagram admits a cone. Equivalently, it is inhabited, for every two objects $x,y$ there is a span $x \leftarrow p \rightarrow y$ (not necessarily universal), and every parallel pair $x \rightrightarrows y$ is equalized by some morphism $e \to x$ (not necessarily universal). nlab_link: https://ncatlab.org/nlab/show/cofiltered+category -dual_property_id: filtered +dual_property: filtered invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/cogenerating set.yaml b/databases/catdat/data/category-properties/cogenerating set.yaml index 78669c49..73bac907 100644 --- a/databases/catdat/data/category-properties/cogenerating set.yaml +++ b/databases/catdat/data/category-properties/cogenerating set.yaml @@ -2,7 +2,7 @@ id: cogenerating set relation: has a description: 'A set of objects $S$ is called a cogenerating set if for every pair of parallel morphisms $f,g : A \rightrightarrows B$, $f = g$ holds if and only if for every morphism $h : B \to G$ with $G \in S$ we have $h \circ f = h \circ g$. Equivalently, the functor $(\Hom(-,G))_{G \in S} : \C^{\op} \to (\Set^+)^S$ is faithful. This property refers to the existence of a cogenerating set.' nlab_link: https://ncatlab.org/nlab/show/cogenerator -dual_property_id: generating set +dual_property: generating set invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/cogenerator.yaml b/databases/catdat/data/category-properties/cogenerator.yaml index 68ab7f00..e111ba14 100644 --- a/databases/catdat/data/category-properties/cogenerator.yaml +++ b/databases/catdat/data/category-properties/cogenerator.yaml @@ -2,7 +2,7 @@ id: cogenerator relation: has a description: 'An object $Q$ of a category is called a cogenerator if for every pair of parallel morphisms $f,g : A \rightrightarrows B$ the equation $f = g$ holds if for every morphism $h : B \to Q$ we have $h \circ f = h \circ g$. Equivalently, the functor $\Hom(-,Q) : \C^{\op} \to \Set^+$ is faithful. This property refers to the existence of a cogenerator. By definition, $Q$ is a cogenerator if and only if $\{Q\}$ is a cogenerating set.' nlab_link: https://ncatlab.org/nlab/show/cogenerator -dual_property_id: generator +dual_property: generator invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/cokernels.yaml b/databases/catdat/data/category-properties/cokernels.yaml index 2e20bf61..49282222 100644 --- a/databases/catdat/data/category-properties/cokernels.yaml +++ b/databases/catdat/data/category-properties/cokernels.yaml @@ -2,7 +2,7 @@ id: cokernels relation: has description: 'A category has cokernels if it has zero morphisms and every morphism $f : A \to B$ has a cokernel, i.e. a coequalizer of $f$ with the zero morphism $0_{A,B} : A \to B$.' nlab_link: https://ncatlab.org/nlab/show/cokernel -dual_property_id: kernels +dual_property: kernels invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/complete.yaml b/databases/catdat/data/category-properties/complete.yaml index 93ff4cce..9b164df5 100644 --- a/databases/catdat/data/category-properties/complete.yaml +++ b/databases/catdat/data/category-properties/complete.yaml @@ -2,7 +2,7 @@ id: complete relation: is description: A category is complete when every small diagram in the category has a limit. nlab_link: https://ncatlab.org/nlab/show/complete+category -dual_property_id: cocomplete +dual_property: cocomplete invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/connected colimits.yaml b/databases/catdat/data/category-properties/connected colimits.yaml index 6a94e6c0..c1b0328f 100644 --- a/databases/catdat/data/category-properties/connected colimits.yaml +++ b/databases/catdat/data/category-properties/connected colimits.yaml @@ -2,7 +2,7 @@ id: connected colimits relation: has description: A category has connected colimits if it has colimits of diagrams indexed by connected small categories. nlab_link: https://ncatlab.org/nlab/show/connected+colimit -dual_property_id: connected limits +dual_property: connected limits invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/connected limits.yaml b/databases/catdat/data/category-properties/connected limits.yaml index 62691b3f..3a4f36bf 100644 --- a/databases/catdat/data/category-properties/connected limits.yaml +++ b/databases/catdat/data/category-properties/connected limits.yaml @@ -2,7 +2,7 @@ id: connected limits relation: has description: A category has connected limits if it has limits of diagrams indexed by connected small categories. nlab_link: https://ncatlab.org/nlab/show/connected+limit -dual_property_id: connected colimits +dual_property: connected colimits invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/connected.yaml b/databases/catdat/data/category-properties/connected.yaml index aa329f26..8d8a8eb4 100644 --- a/databases/catdat/data/category-properties/connected.yaml +++ b/databases/catdat/data/category-properties/connected.yaml @@ -2,7 +2,7 @@ id: connected relation: is description: A category is connected if it is inhabited and every two objects can be joined via a zig-zag path of morphisms. Equivalently, $\C$ is connected if $\C \simeq \coprod_{i \in I} \C_i$ implies $\C_i \simeq 0$ for some $i$. nlab_link: https://ncatlab.org/nlab/show/connected+category -dual_property_id: connected +dual_property: connected invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/conormal.yaml b/databases/catdat/data/category-properties/conormal.yaml index 1a271c00..97855b14 100644 --- a/databases/catdat/data/category-properties/conormal.yaml +++ b/databases/catdat/data/category-properties/conormal.yaml @@ -2,7 +2,7 @@ id: conormal relation: is description: A category is conormal if it has zero morphisms and every epimorphism is a cokernel of some morphism (in which case case it is also called a normal epimorphism). The assumption of having zero morphisms makes it possible to talk about cokernels. nlab_link: https://ncatlab.org/nlab/show/normal+epimorphism -dual_property_id: normal +dual_property: normal invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/copowers.yaml b/databases/catdat/data/category-properties/copowers.yaml index 3ee4e067..e3af72e9 100644 --- a/databases/catdat/data/category-properties/copowers.yaml +++ b/databases/catdat/data/category-properties/copowers.yaml @@ -2,7 +2,7 @@ id: copowers relation: has description: If $X$ is an object and $I$ is a set, the copower is defined as the coproduct $I \otimes X := \coprod_{i \in I} X$. This property refers to the existence of copowers. nlab_link: https://ncatlab.org/nlab/show/copower -dual_property_id: powers +dual_property: powers invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/coproducts.yaml b/databases/catdat/data/category-properties/coproducts.yaml index b32fba73..dd948e8b 100644 --- a/databases/catdat/data/category-properties/coproducts.yaml +++ b/databases/catdat/data/category-properties/coproducts.yaml @@ -2,7 +2,7 @@ id: coproducts relation: has description: 'Given a family of objects $(A_i)_{i \in I}$, a coproduct $\coprod_{i \in I} A_i$ is defined as an object with morphisms $i_i : A_i \to \coprod_{i \in I} A_i$ satisfying the following universal property: For every object $T$ and every family of morphisms $(f_i : A_i \to T)_{i \in I}$ there is a unique morphism $f : \coprod_{i \in I} A_i \to T$ such that $f \circ i_i = f_i$ for all $i \in I$. This property refers to the existence of small coproducts, i.e., coproducts of small families of objects.' nlab_link: https://ncatlab.org/nlab/show/coproduct -dual_property_id: products +dual_property: products invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/coquotients of cocongruences.yaml b/databases/catdat/data/category-properties/coquotients of cocongruences.yaml index 095c7fa2..203730b3 100644 --- a/databases/catdat/data/category-properties/coquotients of cocongruences.yaml +++ b/databases/catdat/data/category-properties/coquotients of cocongruences.yaml @@ -1,7 +1,7 @@ id: coquotients of cocongruences relation: has description: 'A cocongruence (or internal equivalence corelation) on an object $X$ of a category is a parallel pair $i_1, i_2 : X \rightrightarrows E$ which is jointly epimorphic, and such that for every object $T$, the image of $({-} \circ i_1, {-} \circ i_2) : \Hom(E, T) \to \Hom(X, T)^2$ is an equivalence relation. The category has coquotients of cocongruences if for each such cocongruence, there exists an equalizer of $i_1$ and $i_2$. Note that in the case of a category with binary copowers, the corresponding quotients of $X + X$ are also commonly referred to as cocongruences, or as internal equivalence corelations.' -dual_property_id: quotients of congruences +dual_property: quotients of congruences invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/core-thin.yaml b/databases/catdat/data/category-properties/core-thin.yaml index 9efda471..111407dc 100644 --- a/databases/catdat/data/category-properties/core-thin.yaml +++ b/databases/catdat/data/category-properties/core-thin.yaml @@ -8,7 +8,7 @@ description: >-
  • Every automorphism is the identity.
    • nlab_link: https://ncatlab.org/nlab/show/gaunt+category -dual_property_id: core-thin +dual_property: core-thin invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/coreflexive equalizers.yaml b/databases/catdat/data/category-properties/coreflexive equalizers.yaml index 06cae65a..1508fc34 100644 --- a/databases/catdat/data/category-properties/coreflexive equalizers.yaml +++ b/databases/catdat/data/category-properties/coreflexive equalizers.yaml @@ -2,7 +2,7 @@ id: coreflexive equalizers relation: has description: A coreflexive equalizer is a limit of a diagram consisting of a parallel pair of morphisms with a common retraction (left inverse), which is the same concept as an equalizer of such a parallel pair. This property refers to the existence of coreflexive equalizers. nlab_link: https://ncatlab.org/nlab/show/reflexive+coequalizer -dual_property_id: reflexive coequalizers +dual_property: reflexive coequalizers invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/coregular.yaml b/databases/catdat/data/category-properties/coregular.yaml index 861373ca..b5b98413 100644 --- a/databases/catdat/data/category-properties/coregular.yaml +++ b/databases/catdat/data/category-properties/coregular.yaml @@ -1,7 +1,7 @@ id: coregular relation: is description: A category is coregular when its dual is regular, i.e. it is finitely cocomplete, for every morphism $Y \to X$ its cokernel pair $X \rightrightarrows X \sqcup_Y X$ has an equalizer, and regular monomorphisms are stable under pushouts. -dual_property_id: regular +dual_property: regular invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/cosifted limits.yaml b/databases/catdat/data/category-properties/cosifted limits.yaml index 0eb08615..ba0314e9 100644 --- a/databases/catdat/data/category-properties/cosifted limits.yaml +++ b/databases/catdat/data/category-properties/cosifted limits.yaml @@ -2,7 +2,7 @@ id: cosifted limits relation: has description: A category has cosifted limits if it has limits of diagrams indexed by small cosifted categories. nlab_link: https://ncatlab.org/nlab/show/sifted+colimit -dual_property_id: sifted colimits +dual_property: sifted colimits invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/cosifted.yaml b/databases/catdat/data/category-properties/cosifted.yaml index bf62a03d..ac78af88 100644 --- a/databases/catdat/data/category-properties/cosifted.yaml +++ b/databases/catdat/data/category-properties/cosifted.yaml @@ -2,7 +2,7 @@ id: cosifted relation: is description: 'A category $\C$ is cosifted if it is inhabited and the diagonal functor $\Delta : \C \to \C \times \C$ is initial, i.e. if it is non-empty and for any two objects $X,Y \in \C$ the category of spans $$X \leftarrow Z \rightarrow Y$$ is connected. Equivalently, a small category $\C$ is cosifted if $\colim : \Set^{{\C}^\op} \to \Set$ preserves finite products. This property is a weaker notion than being cofiltered.' nlab_link: https://ncatlab.org/nlab/show/sifted+category -dual_property_id: sifted +dual_property: sifted invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/counital.yaml b/databases/catdat/data/category-properties/counital.yaml index ce9740fe..b71b7cdb 100644 --- a/databases/catdat/data/category-properties/counital.yaml +++ b/databases/catdat/data/category-properties/counital.yaml @@ -1,7 +1,7 @@ id: counital relation: is description: 'A category is counital if its dual is unital, i.e., it has a zero object, finite colimits, and for all objects $X,Y$ the two morphisms $(\id_X;0) : X \sqcup Y \twoheadrightarrow X$ and $(0;\id_Y) : X \sqcup Y \twoheadrightarrow Y$ are jointly strongly monomorphic. When products exist, the canonical morphism $X \sqcup Y \to X \times Y$ therefore must be a strong monomorphism.' -dual_property_id: unital +dual_property: unital invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/countable copowers.yaml b/databases/catdat/data/category-properties/countable copowers.yaml index ef4a3b3d..df31f0bc 100644 --- a/databases/catdat/data/category-properties/countable copowers.yaml +++ b/databases/catdat/data/category-properties/countable copowers.yaml @@ -2,7 +2,7 @@ id: countable copowers relation: has description: A category has countable copowers when for every object $X$ and every countable set $I$ the copower $I \otimes X$ exists. nlab_link: https://ncatlab.org/nlab/show/copower -dual_property_id: countable powers +dual_property: countable powers invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/countable coproducts.yaml b/databases/catdat/data/category-properties/countable coproducts.yaml index 8f00fc8f..9af28aed 100644 --- a/databases/catdat/data/category-properties/countable coproducts.yaml +++ b/databases/catdat/data/category-properties/countable coproducts.yaml @@ -1,7 +1,7 @@ id: countable coproducts relation: has description: A category has countable coproducts if it has coproducts for countable families of objects. -dual_property_id: countable products +dual_property: countable products invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/countable powers.yaml b/databases/catdat/data/category-properties/countable powers.yaml index 7f570b45..6edefee3 100644 --- a/databases/catdat/data/category-properties/countable powers.yaml +++ b/databases/catdat/data/category-properties/countable powers.yaml @@ -2,7 +2,7 @@ id: countable powers relation: has description: A category has countable powers when for every object $X$ and every countable set $I$ the power $X^I$ exists. nlab_link: https://ncatlab.org/nlab/show/powering -dual_property_id: countable copowers +dual_property: countable copowers invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/countable products.yaml b/databases/catdat/data/category-properties/countable products.yaml index 1be31289..fe9b60c2 100644 --- a/databases/catdat/data/category-properties/countable products.yaml +++ b/databases/catdat/data/category-properties/countable products.yaml @@ -1,7 +1,7 @@ id: countable products relation: has description: A category has countable products if it has products for countable families of objects. -dual_property_id: countable coproducts +dual_property: countable coproducts invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/countable.yaml b/databases/catdat/data/category-properties/countable.yaml index d82fbe6c..4c0ef80a 100644 --- a/databases/catdat/data/category-properties/countable.yaml +++ b/databases/catdat/data/category-properties/countable.yaml @@ -1,7 +1,7 @@ id: countable relation: is description: A category is countable if it has countably many objects and morphisms. -dual_property_id: countable +dual_property: countable invariant_under_equivalences: false related_properties: diff --git a/databases/catdat/data/category-properties/countably codistributive.yaml b/databases/catdat/data/category-properties/countably codistributive.yaml index 23a1d63b..90ac5e0a 100644 --- a/databases/catdat/data/category-properties/countably codistributive.yaml +++ b/databases/catdat/data/category-properties/countably codistributive.yaml @@ -1,7 +1,7 @@ id: countably codistributive relation: is description: A category is countably codistributive if it has finite coproducts, countable products, and for every object $A$ the functor $A \sqcup -$ preserves countable products. Concretely, for every countable family of objects $(B_i)$ the canonical morphism $A \sqcup \prod_i B_i \to \prod_i (A \sqcup B_i)$ must be an isomorphism. -dual_property_id: countably distributive +dual_property: countably distributive invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/countably distributive.yaml b/databases/catdat/data/category-properties/countably distributive.yaml index 66cb4892..b4b0d85d 100644 --- a/databases/catdat/data/category-properties/countably distributive.yaml +++ b/databases/catdat/data/category-properties/countably distributive.yaml @@ -1,7 +1,7 @@ id: countably distributive relation: is description: A category is countably distributive if it has finite products, countable coproducts, and for every object $A$ the functor $A \times -$ preserves countable coproducts. Concretely, for every countable family of objects $(B_i)$ the canonical morphism $\coprod_i (A \times B_i) \to A \times \coprod_i B_i$ must be an isomorphism. -dual_property_id: countably codistributive +dual_property: countably codistributive invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/direct.yaml b/databases/catdat/data/category-properties/direct.yaml index 301fb353..eb66caa2 100644 --- a/databases/catdat/data/category-properties/direct.yaml +++ b/databases/catdat/data/category-properties/direct.yaml @@ -5,7 +5,7 @@ description: >- $$\cdots \to A_2 \to A_1 \to A_0.$$ For example, a poset is direct iff it is well-founded. nlab_link: https://ncatlab.org/nlab/show/direct+category -dual_property_id: inverse +dual_property: inverse invariant_under_equivalences: false related_properties: diff --git a/databases/catdat/data/category-properties/directed colimits.yaml b/databases/catdat/data/category-properties/directed colimits.yaml index 1259fca8..531cf388 100644 --- a/databases/catdat/data/category-properties/directed colimits.yaml +++ b/databases/catdat/data/category-properties/directed colimits.yaml @@ -2,7 +2,7 @@ id: directed colimits relation: has description: A category has directed colimits if it has colimits of diagrams indexed by directed (small) posets. This is actually equivalent to having filtered colimits. Directed colimits are (somewhat confusingly) also known as inverse limits. nlab_link: https://ncatlab.org/nlab/show/directed+colimit -dual_property_id: directed limits +dual_property: directed limits invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/directed limits.yaml b/databases/catdat/data/category-properties/directed limits.yaml index c49d04fc..7d1d9665 100644 --- a/databases/catdat/data/category-properties/directed limits.yaml +++ b/databases/catdat/data/category-properties/directed limits.yaml @@ -2,7 +2,7 @@ id: directed limits relation: has description: A category has directed limits if it has limits of diagrams indexed by codirected (small) posets. This is actually equivalent to having cofiltered limits. nlab_link: https://ncatlab.org/nlab/show/directed+limit -dual_property_id: directed colimits +dual_property: directed colimits invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/discrete.yaml b/databases/catdat/data/category-properties/discrete.yaml index 16df2d1a..01a1a28b 100644 --- a/databases/catdat/data/category-properties/discrete.yaml +++ b/databases/catdat/data/category-properties/discrete.yaml @@ -2,7 +2,7 @@ id: discrete relation: is description: A category is discrete when every morphism is an identity morphism. Thus, a discrete category is merely a collection of objects. nlab_link: https://ncatlab.org/nlab/show/discrete+category -dual_property_id: discrete +dual_property: discrete invariant_under_equivalences: false related_properties: diff --git a/databases/catdat/data/category-properties/disjoint coproducts.yaml b/databases/catdat/data/category-properties/disjoint coproducts.yaml index b8a1e477..1e864a92 100644 --- a/databases/catdat/data/category-properties/disjoint coproducts.yaml +++ b/databases/catdat/data/category-properties/disjoint coproducts.yaml @@ -2,7 +2,7 @@ id: disjoint coproducts relation: has description: A category has disjoint coproducts if it has coproducts, the coproduct inclusions $A_i \to \coprod_{i \in I} A_i$ are monomorphisms, and the pullback of the inclusions $A_i \to \coprod_{i \in I} A_i$ and $A_j \to \coprod_{i \in I} A_i$ for $i \neq j$ exists and is given by the initial object $0$. nlab_link: https://ncatlab.org/nlab/show/disjoint+coproduct -dual_property_id: disjoint products +dual_property: disjoint products invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/disjoint finite coproducts.yaml b/databases/catdat/data/category-properties/disjoint finite coproducts.yaml index 77f6b46d..cbae37fa 100644 --- a/databases/catdat/data/category-properties/disjoint finite coproducts.yaml +++ b/databases/catdat/data/category-properties/disjoint finite coproducts.yaml @@ -2,7 +2,7 @@ id: disjoint finite coproducts relation: has description: A category has disjoint finite coproducts if it has finite coproducts, for every pair of objects $A,B$ the coproduct inclusions $A \rightarrow A+B \leftarrow B$ are monomorphisms, and the pullback $A \times_{A + B} B$ exists and is given by the initial object $0$. nlab_link: https://ncatlab.org/nlab/show/disjoint+coproduct -dual_property_id: disjoint finite products +dual_property: disjoint finite products invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/disjoint finite products.yaml b/databases/catdat/data/category-properties/disjoint finite products.yaml index f804ff12..899648ca 100644 --- a/databases/catdat/data/category-properties/disjoint finite products.yaml +++ b/databases/catdat/data/category-properties/disjoint finite products.yaml @@ -3,7 +3,7 @@ relation: has description: |- A category has disjoint finite products if it has finite products, for every pair of objects $A,B$ the product projections $A \leftarrow A \times B \rightarrow B$ are epimorphisms, and the pushout $A \sqcup_{A \times B} B$ exists and is given by the terminal object $1$. This terminology does not seem to be common, but we have added it as a dual for the more commonly known property of having disjoint finite coproducts. -dual_property_id: disjoint finite coproducts +dual_property: disjoint finite coproducts invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/disjoint products.yaml b/databases/catdat/data/category-properties/disjoint products.yaml index f84fa39f..75e6cdc2 100644 --- a/databases/catdat/data/category-properties/disjoint products.yaml +++ b/databases/catdat/data/category-properties/disjoint products.yaml @@ -3,7 +3,7 @@ relation: has description: |- A category has disjoint products if it has products, the product projections $\prod_{i \in I} A_i \to A_i$ are epimorphisms, and the pushout of the projections $\prod_{i \in I} A_i \to A_i$ and $\prod_{i \in I} A_i \to A_j$ for $i \neq j$ exists and is given by the terminal object $1$. This terminology does not seem to be common, but we have added it as a dual for the more commonly known property of having disjoint coproducts. -dual_property_id: disjoint coproducts +dual_property: disjoint coproducts invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/distributive.yaml b/databases/catdat/data/category-properties/distributive.yaml index ae7b4015..e2a49126 100644 --- a/databases/catdat/data/category-properties/distributive.yaml +++ b/databases/catdat/data/category-properties/distributive.yaml @@ -2,7 +2,7 @@ id: distributive relation: is description: A category is distributive if it has finite products, finite coproducts, and for every object $A$ the functor $A \times -$ preserves finite coproducts. Concretely, for every finite family of objects $(B_i)$ the canonical morphism $\coprod_i (A \times B_i) \to A \times \coprod_i B_i$ must be an isomorphism. nlab_link: https://ncatlab.org/nlab/show/distributive+category -dual_property_id: codistributive +dual_property: codistributive invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/effective cocongruences.yaml b/databases/catdat/data/category-properties/effective cocongruences.yaml index 43b2c0d1..a0c00327 100644 --- a/databases/catdat/data/category-properties/effective cocongruences.yaml +++ b/databases/catdat/data/category-properties/effective cocongruences.yaml @@ -4,7 +4,7 @@ description: >- A cocongruence $f, g : X \rightrightarrows E$ (see definition here) is effective if it is the cokernel pair of some morphism, i.e. if there is a morphism $h : Y \to X$ such that we have a cocartesian square $$\begin{CD} Y @> h >> X \\ @V h VV @VV f V \\ X @>> g > E. \end{CD}$$ A category has effective cocongruences if every cocongruence in the category is effective. -dual_property_id: effective congruences +dual_property: effective congruences invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/effective congruences.yaml b/databases/catdat/data/category-properties/effective congruences.yaml index fe6bc845..004d99ee 100644 --- a/databases/catdat/data/category-properties/effective congruences.yaml +++ b/databases/catdat/data/category-properties/effective congruences.yaml @@ -5,7 +5,7 @@ description: >- $$\begin{CD} E @> f >> X \\ @V g VV @VV h V \\ X @>> h > Y. \end{CD}$$ A category has effective congruences if every congruence in the category is effective. nlab_link: https://ncatlab.org/nlab/show/congruence -dual_property_id: effective cocongruences +dual_property: effective cocongruences invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/epi-regular.yaml b/databases/catdat/data/category-properties/epi-regular.yaml index f2a828ee..91d0b17a 100644 --- a/databases/catdat/data/category-properties/epi-regular.yaml +++ b/databases/catdat/data/category-properties/epi-regular.yaml @@ -2,7 +2,7 @@ id: epi-regular relation: is description: A category is epi-regular when every epimorphism is regular, i.e. the coequalizer of a pair of morphisms. Notice that this is not standard terminology, apparently the literature has no name for this yet. A preadditive category is epi-regular iff it is conormal. The notion of a conormal category is reserved for categories with zero morphisms, while epi-regular applies to all categories. nlab_link: https://ncatlab.org/nlab/show/regular+epimorphism -dual_property_id: mono-regular +dual_property: mono-regular invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/equalizers.yaml b/databases/catdat/data/category-properties/equalizers.yaml index baf95a3d..a7e56e30 100644 --- a/databases/catdat/data/category-properties/equalizers.yaml +++ b/databases/catdat/data/category-properties/equalizers.yaml @@ -2,7 +2,7 @@ id: equalizers relation: has description: 'An equalizer of a pair of morphisms $f,g : A \rightrightarrows B$ is an object $E$ with a morphism $e : E \to A$ such that $f \circ e = g \circ e$ and which is universal with respect to this property. This property refers to the existence of equalizers.' nlab_link: https://ncatlab.org/nlab/show/equalizer -dual_property_id: coequalizers +dual_property: coequalizers invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/essentially countable.yaml b/databases/catdat/data/category-properties/essentially countable.yaml index 94519b38..2d24c3d6 100644 --- a/databases/catdat/data/category-properties/essentially countable.yaml +++ b/databases/catdat/data/category-properties/essentially countable.yaml @@ -1,7 +1,7 @@ id: essentially countable relation: is description: A category is essentially countable if it is equivalent to a countable category. -dual_property_id: essentially countable +dual_property: essentially countable invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/essentially discrete.yaml b/databases/catdat/data/category-properties/essentially discrete.yaml index 03a4dce7..e81567f7 100644 --- a/databases/catdat/data/category-properties/essentially discrete.yaml +++ b/databases/catdat/data/category-properties/essentially discrete.yaml @@ -2,7 +2,7 @@ id: essentially discrete relation: is description: A category is essentially discrete if it is equivalent to a discrete category. Equivalently, it is a thin groupoid. Notice that the nLab calls this property simply "discrete". In contrast to being discrete, this property is invariant under equivalences of categories. An essentially discrete category is the same as a setoid (a set equipped with an equivalence relation). nlab_link: https://ncatlab.org/nlab/show/discrete+category -dual_property_id: essentially discrete +dual_property: essentially discrete invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/essentially finite.yaml b/databases/catdat/data/category-properties/essentially finite.yaml index d7ed4a1d..0323dfa6 100644 --- a/databases/catdat/data/category-properties/essentially finite.yaml +++ b/databases/catdat/data/category-properties/essentially finite.yaml @@ -1,7 +1,7 @@ id: essentially finite relation: is description: A category is essentially finite if it is equivalent to a finite category. Equivalently, there are only finitely many objects up to isomorphism, and the collection of morphisms between any two objects is isomorphic to a finite set. In contrast to being finite, this property is invariant under equivalences of categories. -dual_property_id: essentially finite +dual_property: essentially finite invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/essentially small.yaml b/databases/catdat/data/category-properties/essentially small.yaml index 670304a2..10e292a9 100644 --- a/databases/catdat/data/category-properties/essentially small.yaml +++ b/databases/catdat/data/category-properties/essentially small.yaml @@ -2,7 +2,7 @@ id: essentially small relation: is description: A category is essentially small when it is equivalent to a small category. In particular, there is a set of objects such that every object is isomorphic to an object in this set. In contrast to the property of being small, being essentially small is invariant under equivalences of categories. nlab_link: https://ncatlab.org/nlab/show/small+category -dual_property_id: essentially small +dual_property: essentially small invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/exact cofiltered limits.yaml b/databases/catdat/data/category-properties/exact cofiltered limits.yaml index fcfdefd4..34a63abc 100644 --- a/databases/catdat/data/category-properties/exact cofiltered limits.yaml +++ b/databases/catdat/data/category-properties/exact cofiltered limits.yaml @@ -8,7 +8,7 @@ description: >-
  • For every diagram $X : \I \times \J \to \C$, where $\I$ is finite and $\J$ is small cofiltered, the canonical morphism $\colim_i \lim_j X(i,j) \to \lim_j \colim_i X(i,j)$ is an isomorphism.
    • nlab_link: https://ncatlab.org/nlab/show/commutativity+of+limits+and+colimits -dual_property_id: exact filtered colimits +dual_property: exact filtered colimits invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/exact filtered colimits.yaml b/databases/catdat/data/category-properties/exact filtered colimits.yaml index e442faa5..8d83e983 100644 --- a/databases/catdat/data/category-properties/exact filtered colimits.yaml +++ b/databases/catdat/data/category-properties/exact filtered colimits.yaml @@ -8,7 +8,7 @@ description: >-
  • For every diagram $X : \I \times \J \to \C$, where $\I$ is finite and $\J$ is small filtered, the canonical morphism $\colim_j \lim_i X(i,j) \to \lim_i \colim_j X(i,j)$ is an isomorphism.
    • nlab_link: https://ncatlab.org/nlab/show/commutativity+of+limits+and+colimits -dual_property_id: exact cofiltered limits +dual_property: exact cofiltered limits invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/extensive.yaml b/databases/catdat/data/category-properties/extensive.yaml index 5146b55d..6300bc3f 100644 --- a/databases/catdat/data/category-properties/extensive.yaml +++ b/databases/catdat/data/category-properties/extensive.yaml @@ -2,7 +2,7 @@ id: extensive relation: is description: A category $\C$ is extensive when it has finite coproducts and for all objects $A,B \in \C$ the coproduct functor $\C/A \times \C/B \to \C/(A+B)$ is an equivalence of categories. Equivalently, pullbacks of finite coproduct inclusions along arbitrary morphisms exist and finite coproducts are disjoint and stable under pullback. nlab_link: https://ncatlab.org/nlab/show/extensive+category -dual_property_id: coextensive +dual_property: coextensive invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/filtered colimits.yaml b/databases/catdat/data/category-properties/filtered colimits.yaml index a9cb9627..ebab748e 100644 --- a/databases/catdat/data/category-properties/filtered colimits.yaml +++ b/databases/catdat/data/category-properties/filtered colimits.yaml @@ -2,7 +2,7 @@ id: filtered colimits relation: has description: A category has filtered colimits if it has colimits of diagrams indexed by small filtered categories. This is actually equivalent to having directed colimits. nlab_link: https://ncatlab.org/nlab/show/filtered+colimit -dual_property_id: cofiltered limits +dual_property: cofiltered limits invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/filtered-colimit-stable monomorphisms.yaml b/databases/catdat/data/category-properties/filtered-colimit-stable monomorphisms.yaml index a5d3d9bb..d3c18515 100644 --- a/databases/catdat/data/category-properties/filtered-colimit-stable monomorphisms.yaml +++ b/databases/catdat/data/category-properties/filtered-colimit-stable monomorphisms.yaml @@ -1,7 +1,7 @@ id: filtered-colimit-stable monomorphisms relation: has description: A category has filtered-colimit-stable monomorphisms if it has filtered colimits and for every filtered diagram of monomorphisms $(X_i \to Y_i)$ also their colimit $\colim_i X_i \to \colim_i Y_i$ is a monomorphism. -dual_property_id: cofiltered-limit-stable epimorphisms +dual_property: cofiltered-limit-stable epimorphisms invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/filtered.yaml b/databases/catdat/data/category-properties/filtered.yaml index 9465d23d..938bdf4a 100644 --- a/databases/catdat/data/category-properties/filtered.yaml +++ b/databases/catdat/data/category-properties/filtered.yaml @@ -2,7 +2,7 @@ id: filtered relation: is description: A category is filtered if every finite diagram admits a cocone. Equivalently, it is inhabited, for every two objects $x,y$ there is a cospan $x \rightarrow s \leftarrow y$ (not necessarily universal), and every parallel pair $x \rightrightarrows y$ is coequalized by some morphism $y \to c$ (not necessarily universal). nlab_link: https://ncatlab.org/nlab/show/filtered+category -dual_property_id: cofiltered +dual_property: cofiltered invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/finite copowers.yaml b/databases/catdat/data/category-properties/finite copowers.yaml index 9f00fca8..e662ba43 100644 --- a/databases/catdat/data/category-properties/finite copowers.yaml +++ b/databases/catdat/data/category-properties/finite copowers.yaml @@ -2,7 +2,7 @@ id: finite copowers relation: has description: A category has finite copowers when for every object $X$ and every finite set $I$ the copower $I \otimes X$ exists. Equivalently, for every $n \in \IN$ the copower $n \otimes X$ exists. nlab_link: https://ncatlab.org/nlab/show/copower -dual_property_id: finite powers +dual_property: finite powers invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/finite coproducts.yaml b/databases/catdat/data/category-properties/finite coproducts.yaml index 5e5ef46e..76b342f6 100644 --- a/databases/catdat/data/category-properties/finite coproducts.yaml +++ b/databases/catdat/data/category-properties/finite coproducts.yaml @@ -2,7 +2,7 @@ id: finite coproducts relation: has description: A category has finite coproducts if it has coproducts for finite families of objects. Equivalently, it has an initial object and binary coproducts. nlab_link: https://ncatlab.org/nlab/show/finite+coproduct -dual_property_id: finite products +dual_property: finite products invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/finite powers.yaml b/databases/catdat/data/category-properties/finite powers.yaml index cb0130eb..0f06315a 100644 --- a/databases/catdat/data/category-properties/finite powers.yaml +++ b/databases/catdat/data/category-properties/finite powers.yaml @@ -2,7 +2,7 @@ id: finite powers relation: has description: A category has finite powers when for every object $X$ and every finite set $I$ the power $X^I$ exists. Equivalently, for every $n \in \IN$ the power $X^n$ exists. nlab_link: https://ncatlab.org/nlab/show/powering -dual_property_id: finite copowers +dual_property: finite copowers invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/finite products.yaml b/databases/catdat/data/category-properties/finite products.yaml index 0fd80c30..fbcc8670 100644 --- a/databases/catdat/data/category-properties/finite products.yaml +++ b/databases/catdat/data/category-properties/finite products.yaml @@ -2,7 +2,7 @@ id: finite products relation: has description: A category has finite products if it has products for finite families of objects. Equivalently, it has a terminal object and binary products. nlab_link: https://ncatlab.org/nlab/show/finite+product -dual_property_id: finite coproducts +dual_property: finite coproducts invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/finite.yaml b/databases/catdat/data/category-properties/finite.yaml index 816f8922..b7a9e792 100644 --- a/databases/catdat/data/category-properties/finite.yaml +++ b/databases/catdat/data/category-properties/finite.yaml @@ -2,7 +2,7 @@ id: finite relation: is description: A category is finite if it has finitely many objects and morphisms. nlab_link: https://ncatlab.org/nlab/show/finite+category -dual_property_id: finite +dual_property: finite invariant_under_equivalences: false related_properties: diff --git a/databases/catdat/data/category-properties/finitely cocomplete.yaml b/databases/catdat/data/category-properties/finitely cocomplete.yaml index 5bf7307a..593348d6 100644 --- a/databases/catdat/data/category-properties/finitely cocomplete.yaml +++ b/databases/catdat/data/category-properties/finitely cocomplete.yaml @@ -2,7 +2,7 @@ id: finitely cocomplete relation: is description: A category is finitely cocomplete when every finite diagram has a colimit. nlab_link: https://ncatlab.org/nlab/show/finitely+cocomplete+category -dual_property_id: finitely complete +dual_property: finitely complete invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/finitely complete.yaml b/databases/catdat/data/category-properties/finitely complete.yaml index a79d5245..932f2a44 100644 --- a/databases/catdat/data/category-properties/finitely complete.yaml +++ b/databases/catdat/data/category-properties/finitely complete.yaml @@ -2,7 +2,7 @@ id: finitely complete relation: is description: A category is finitely complete when every finite diagram has a limit. nlab_link: https://ncatlab.org/nlab/show/finitely+complete+category -dual_property_id: finitely cocomplete +dual_property: finitely cocomplete invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/gaunt.yaml b/databases/catdat/data/category-properties/gaunt.yaml index d0be5b1e..563bc17c 100644 --- a/databases/catdat/data/category-properties/gaunt.yaml +++ b/databases/catdat/data/category-properties/gaunt.yaml @@ -2,7 +2,7 @@ id: gaunt relation: is description: 'A category is gaunt when every isomorphism $f : A \to B$ must be the identity (in particular, $A = B$). This is the "skeletal variant" of being core-thin.' nlab_link: https://ncatlab.org/nlab/show/gaunt+category -dual_property_id: gaunt +dual_property: gaunt invariant_under_equivalences: false related_properties: diff --git a/databases/catdat/data/category-properties/generating set.yaml b/databases/catdat/data/category-properties/generating set.yaml index 8d6e1f8e..4588065c 100644 --- a/databases/catdat/data/category-properties/generating set.yaml +++ b/databases/catdat/data/category-properties/generating set.yaml @@ -2,7 +2,7 @@ id: generating set relation: has a description: 'A set of objects $S$ is called a generating set if for every pair of parallel morphisms $f,g : A \rightrightarrows B$, $f = g$ holds if and only if for every morphism $h : G \to A$ with $G \in S$ we have $f \circ h = g \circ h$. Equivalently, the functor $(\Hom(G,-))_{G \in S} : \C \to (\Set^+)^S$ is faithful. This property refers to the existence of a generating set.' nlab_link: https://ncatlab.org/nlab/show/separator -dual_property_id: cogenerating set +dual_property: cogenerating set invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/generator.yaml b/databases/catdat/data/category-properties/generator.yaml index 65f922c6..51b95d34 100644 --- a/databases/catdat/data/category-properties/generator.yaml +++ b/databases/catdat/data/category-properties/generator.yaml @@ -2,7 +2,7 @@ id: generator relation: has a description: 'An object $G$ of a category is called a generator if for every pair of parallel morphisms $f,g : A \rightrightarrows B$ the equation $f = g$ holds if for every morphism $h : G \to A$ we have $f \circ h = g \circ h$. Equivalently, the functor $\Hom(G,-) : \C \to \Set^+$ is faithful. This property refers to the existence of a generator. By definition, $G$ is a generator if and only if $\{G\}$ is a generating set.' nlab_link: https://ncatlab.org/nlab/show/separator -dual_property_id: cogenerator +dual_property: cogenerator invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/groupoid.yaml b/databases/catdat/data/category-properties/groupoid.yaml index f43b125d..706c0dfb 100644 --- a/databases/catdat/data/category-properties/groupoid.yaml +++ b/databases/catdat/data/category-properties/groupoid.yaml @@ -2,7 +2,7 @@ id: groupoid relation: is a description: A groupoid is a category in which every morphism is an isomorphism. nlab_link: https://ncatlab.org/nlab/show/groupoid -dual_property_id: groupoid +dual_property: groupoid invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/infinitary codistributive.yaml b/databases/catdat/data/category-properties/infinitary codistributive.yaml index 0bdcf94f..aa8d3936 100644 --- a/databases/catdat/data/category-properties/infinitary codistributive.yaml +++ b/databases/catdat/data/category-properties/infinitary codistributive.yaml @@ -1,7 +1,7 @@ id: infinitary codistributive relation: is description: A category is infinitary codistributive if it has finite coproducts, all products, and for every object $A$ the functor $A \sqcup -$ preserves all products. Concretely, for every family of objects $(B_i)$ the canonical morphism $A \sqcup \prod_i B_i \to \prod_i (A \sqcup B_i)$ must be an isomorphism. -dual_property_id: infinitary distributive +dual_property: infinitary distributive invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/infinitary coextensive.yaml b/databases/catdat/data/category-properties/infinitary coextensive.yaml index 2436a256..0af9ba6c 100644 --- a/databases/catdat/data/category-properties/infinitary coextensive.yaml +++ b/databases/catdat/data/category-properties/infinitary coextensive.yaml @@ -3,7 +3,7 @@ relation: is description: |- A category $\C$ is infinitary coextensive when it has products and for all families of objects $(A_i)_{i \in I}$ the product functor $\prod_{i \in I} A_i / \C/A_i \to \prod_{i \in I} A_i / \C$ is an equivalence of categories. This terminology does not seem to be common, but we have added it as a dual for the more commonly known property of being infinitary extensive. -dual_property_id: infinitary extensive +dual_property: infinitary extensive invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/infinitary distributive.yaml b/databases/catdat/data/category-properties/infinitary distributive.yaml index 6d67b601..c4388dbe 100644 --- a/databases/catdat/data/category-properties/infinitary distributive.yaml +++ b/databases/catdat/data/category-properties/infinitary distributive.yaml @@ -2,7 +2,7 @@ id: infinitary distributive relation: is description: A category is infinitary distributive if it has finite products, all coproducts, and for every object $A$ the functor $A \times -$ preserves all coproducts. Concretely, for every family of objects $(B_i)$ the canonical morphism $\coprod_i (A \times B_i) \to A \times \coprod_i B_i$ must be an isomorphism. nlab_link: https://ncatlab.org/nlab/show/distributive+category -dual_property_id: infinitary codistributive +dual_property: infinitary codistributive invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/infinitary extensive.yaml b/databases/catdat/data/category-properties/infinitary extensive.yaml index dbb5e8f2..ca81ddec 100644 --- a/databases/catdat/data/category-properties/infinitary extensive.yaml +++ b/databases/catdat/data/category-properties/infinitary extensive.yaml @@ -2,7 +2,7 @@ id: infinitary extensive relation: is description: A category $\C$ is infinitary extensive when it has coproducts and for all families of objects $(A_i)_{i \in I}$ the coproduct functor $\prod_{i \in I} \C/A_i \to \C/(\coprod_{i \in I} A_i)$ is an equivalence of categories. Equivalently, pullbacks of coproduct inclusions along arbitrary morphisms exist and coproducts are disjoint and stable under pullback. nlab_link: https://ncatlab.org/nlab/show/extensive+category -dual_property_id: infinitary coextensive +dual_property: infinitary coextensive invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/inhabited.yaml b/databases/catdat/data/category-properties/inhabited.yaml index 37a3add9..4d93cd40 100644 --- a/databases/catdat/data/category-properties/inhabited.yaml +++ b/databases/catdat/data/category-properties/inhabited.yaml @@ -2,7 +2,7 @@ id: inhabited relation: is description: A category is inhabited if it has at least one object. In classical logic, this is equivalent to being non-empty (which is a double negation). nlab_link: https://ncatlab.org/nlab/show/inhabited+set -dual_property_id: inhabited +dual_property: inhabited invariant_under_equivalences: true related_properties: [] diff --git a/databases/catdat/data/category-properties/initial object.yaml b/databases/catdat/data/category-properties/initial object.yaml index 83001b6d..e7a482bf 100644 --- a/databases/catdat/data/category-properties/initial object.yaml +++ b/databases/catdat/data/category-properties/initial object.yaml @@ -2,7 +2,7 @@ id: initial object relation: has an description: An initial object is an object that has a unique morphism to every object in the category. This property refers to the existence of an initial object. nlab_link: https://ncatlab.org/nlab/show/initial+object -dual_property_id: terminal object +dual_property: terminal object invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/inverse.yaml b/databases/catdat/data/category-properties/inverse.yaml index f55b3e2c..766dae97 100644 --- a/databases/catdat/data/category-properties/inverse.yaml +++ b/databases/catdat/data/category-properties/inverse.yaml @@ -4,7 +4,7 @@ description: >- A category is inverse if its dual is direct, i.e., if it contains no infinite sequence of non-identity morphisms of the form $$A_0 \to A_1 \to A_2 \to \cdots.$$ nlab_link: https://ncatlab.org/nlab/show/inverse+category -dual_property_id: direct +dual_property: direct invariant_under_equivalences: false related_properties: diff --git a/databases/catdat/data/category-properties/kernels.yaml b/databases/catdat/data/category-properties/kernels.yaml index d7ea10d8..5373d2be 100644 --- a/databases/catdat/data/category-properties/kernels.yaml +++ b/databases/catdat/data/category-properties/kernels.yaml @@ -2,7 +2,7 @@ id: kernels relation: has description: 'A category has kernels if it has zero morphisms and every morphism $f : A \to B$ has a kernel, i.e. an equalizer of $f$ with the zero morphism $0_{A,B} : A \to B$.' nlab_link: https://ncatlab.org/nlab/show/kernel -dual_property_id: cokernels +dual_property: cokernels invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/left cancellative.yaml b/databases/catdat/data/category-properties/left cancellative.yaml index 0db6ed98..ad9ce7fe 100644 --- a/databases/catdat/data/category-properties/left cancellative.yaml +++ b/databases/catdat/data/category-properties/left cancellative.yaml @@ -2,7 +2,7 @@ id: left cancellative relation: is description: 'A category is left cancellative if for every morphism $f : A \to B$ and every parallel pair of morphisms $g,h : T \rightrightarrows A$ with $f \circ g = f \circ h$ we have $g = h$. Equivalently, every morphism is a monomorphism.' nlab_link: https://ncatlab.org/nlab/show/cancellative+category -dual_property_id: right cancellative +dual_property: right cancellative invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/locally cartesian closed.yaml b/databases/catdat/data/category-properties/locally cartesian closed.yaml index 4afbc11b..b98679b3 100644 --- a/databases/catdat/data/category-properties/locally cartesian closed.yaml +++ b/databases/catdat/data/category-properties/locally cartesian closed.yaml @@ -2,7 +2,7 @@ id: locally cartesian closed relation: is description: A category is locally cartesian closed if each of its slice categories is cartesian closed. nlab_link: https://ncatlab.org/nlab/show/locally+cartesian+closed+category -dual_property_id: locally cocartesian coclosed +dual_property: locally cocartesian coclosed invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/locally cocartesian coclosed.yaml b/databases/catdat/data/category-properties/locally cocartesian coclosed.yaml index 32954404..f8c53e8c 100644 --- a/databases/catdat/data/category-properties/locally cocartesian coclosed.yaml +++ b/databases/catdat/data/category-properties/locally cocartesian coclosed.yaml @@ -2,7 +2,7 @@ id: locally cocartesian coclosed relation: is description: A category is locally cocartesian coclosed if its dual is locally cartesian closed, i.e. if each of its coslice categories is cocartesian coclosed. nlab_link: https://ncatlab.org/nlab/show/locally+cocartesian+coclosed+category -dual_property_id: locally cartesian closed +dual_property: locally cartesian closed invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/locally copresentable.yaml b/databases/catdat/data/category-properties/locally copresentable.yaml index c5d770d7..12563b17 100644 --- a/databases/catdat/data/category-properties/locally copresentable.yaml +++ b/databases/catdat/data/category-properties/locally copresentable.yaml @@ -1,7 +1,7 @@ id: locally copresentable relation: is description: A category is locally copresentable if its opposite category is locally presentable. -dual_property_id: locally presentable +dual_property: locally presentable invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/locally essentially small.yaml b/databases/catdat/data/category-properties/locally essentially small.yaml index 7fa39aac..59915cdf 100644 --- a/databases/catdat/data/category-properties/locally essentially small.yaml +++ b/databases/catdat/data/category-properties/locally essentially small.yaml @@ -1,7 +1,7 @@ id: locally essentially small relation: is description: A category is locally essentially small when for every pair of objects $A,B$ the collection of morphisms $A \to B$ is isomorphic to a set. (Here, we work with a set-theoretic foundation in which there are sets and collections. Categories are based on collections of objects and morphisms.) Equivalently, the category is equivalent to a locally small category. In contrast to being locally small, this condition is invariant under equivalences of categories. This is why we have added it to the database. For instance, every algebraic category is locally essentially small, but not necessarily locally small. This indicates that this is the "right" notion to work with. -dual_property_id: locally essentially small +dual_property: locally essentially small invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/locally finite.yaml b/databases/catdat/data/category-properties/locally finite.yaml index 09b693f7..2b031889 100644 --- a/databases/catdat/data/category-properties/locally finite.yaml +++ b/databases/catdat/data/category-properties/locally finite.yaml @@ -2,7 +2,7 @@ id: locally finite relation: is description: A category is locally finite or Hom-finite when for all objects $A,B$ the collection $\Hom(A,B)$ is finite. (We do not assume that this collection is an actual set. Therefore, this property is invariant under equivalences of categories.) nlab_link: https://ncatlab.org/nlab/show/locally+finite+category -dual_property_id: locally finite +dual_property: locally finite invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/locally presentable.yaml b/databases/catdat/data/category-properties/locally presentable.yaml index 0523abd6..53c681d2 100644 --- a/databases/catdat/data/category-properties/locally presentable.yaml +++ b/databases/catdat/data/category-properties/locally presentable.yaml @@ -11,7 +11,7 @@ description: >- For equivalence of conditions above, see Cor. 2.47, Thm. 1.46, and Cor. 1.52 in Adamek-Rosicky. A category is locally presentable if it is locally $\kappa$-presentable for some regular cardinal $\kappa$. nlab_link: https://ncatlab.org/nlab/show/locally+presentable+category -dual_property_id: locally copresentable +dual_property: locally copresentable invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/locally small.yaml b/databases/catdat/data/category-properties/locally small.yaml index 0156f3ca..d0a38605 100644 --- a/databases/catdat/data/category-properties/locally small.yaml +++ b/databases/catdat/data/category-properties/locally small.yaml @@ -2,7 +2,7 @@ id: locally small relation: is description: A category is locally small when for every pair of objects $A,B$ the collection of morphisms $A \to B$ is a set. Here, we work with a set-theoretic foundation in which there are sets and collections. Categories are based on collections of objects and morphisms. nlab_link: https://ncatlab.org/nlab/show/locally+small+category -dual_property_id: locally small +dual_property: locally small invariant_under_equivalences: false related_properties: diff --git a/databases/catdat/data/category-properties/mono-regular.yaml b/databases/catdat/data/category-properties/mono-regular.yaml index c9aaf2ec..3602d4b6 100644 --- a/databases/catdat/data/category-properties/mono-regular.yaml +++ b/databases/catdat/data/category-properties/mono-regular.yaml @@ -2,7 +2,7 @@ id: mono-regular relation: is description: A category is mono-regular when every monomorphism is regular, i.e. the equalizer of a pair of morphisms. Notice that this is not standard terminology, apparently the literature has no name for this yet. A preadditive category is mono-regular iff it is normal. The notion of a normal category is reserved for categories with zero morphisms, while mono-regular applies to all categories. nlab_link: https://ncatlab.org/nlab/show/regular+monomorphism -dual_property_id: epi-regular +dual_property: epi-regular invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/multi-cocomplete.yaml b/databases/catdat/data/category-properties/multi-cocomplete.yaml index 5c3a0743..69665468 100644 --- a/databases/catdat/data/category-properties/multi-cocomplete.yaml +++ b/databases/catdat/data/category-properties/multi-cocomplete.yaml @@ -2,7 +2,7 @@ id: multi-cocomplete relation: is description: A multi-colimit of a diagram $D\colon \S \to \C$ is a set $I$ of cocones under $D$ such that every cocone under $D$ uniquely factors through a unique cocone belonging to $I$. This property refers to the existence of multi-colimits of small diagrams. Note that any diagram with no cocone admits a multi-colimit, which is the empty set of cocones. nlab_link: https://ncatlab.org/nlab/show/multilimit -dual_property_id: multi-complete +dual_property: multi-complete invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/multi-complete.yaml b/databases/catdat/data/category-properties/multi-complete.yaml index 3a6899d9..6dfc881b 100644 --- a/databases/catdat/data/category-properties/multi-complete.yaml +++ b/databases/catdat/data/category-properties/multi-complete.yaml @@ -2,7 +2,7 @@ id: multi-complete relation: is description: A multi-limit of a diagram $D\colon \S \to \C$ is a set $I$ of cones over $D$ such that every cone over $D$ uniquely factors through a unique cone belonging to $I$. This property refers to the existence of multi-limits of small diagrams. Note that any diagram with no cone admits a multi-limit, which is the empty set of cones. nlab_link: https://ncatlab.org/nlab/show/multilimit -dual_property_id: multi-cocomplete +dual_property: multi-cocomplete invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/multi-initial object.yaml b/databases/catdat/data/category-properties/multi-initial object.yaml index 792fb925..e6e7d93b 100644 --- a/databases/catdat/data/category-properties/multi-initial object.yaml +++ b/databases/catdat/data/category-properties/multi-initial object.yaml @@ -2,7 +2,7 @@ id: multi-initial object relation: has a description: This property refers to the existence of a multi-colimit of the empty diagram. A category has a multi-initial object if and only if the collection of all connected components is isomorphic to a set, and each connected component has a initial object. nlab_link: https://ncatlab.org/nlab/show/multilimit -dual_property_id: multi-terminal object +dual_property: multi-terminal object invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/multi-terminal object.yaml b/databases/catdat/data/category-properties/multi-terminal object.yaml index a1e0d5cd..8f878ebd 100644 --- a/databases/catdat/data/category-properties/multi-terminal object.yaml +++ b/databases/catdat/data/category-properties/multi-terminal object.yaml @@ -2,7 +2,7 @@ id: multi-terminal object relation: has a description: This property refers to the existence of a multi-limit of the empty diagram. A category has a multi-terminal object if and only if the collection of all connected components is isomorphic to a set, and each connected component has a terminal object. nlab_link: https://ncatlab.org/nlab/show/multilimit -dual_property_id: multi-initial object +dual_property: multi-initial object invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/normal.yaml b/databases/catdat/data/category-properties/normal.yaml index cc3b3d75..0d998244 100644 --- a/databases/catdat/data/category-properties/normal.yaml +++ b/databases/catdat/data/category-properties/normal.yaml @@ -2,7 +2,7 @@ id: normal relation: is description: A category is normal if it has zero morphisms and every monomorphism is a kernel of some morphism (in which case case it is also called a normal monomorphism). The assumption of having zero morphisms makes it possible to talk about kernels. nlab_link: https://ncatlab.org/nlab/show/normal+monomorphism -dual_property_id: conormal +dual_property: conormal invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/one-way.yaml b/databases/catdat/data/category-properties/one-way.yaml index bad84b6c..8a79d149 100644 --- a/databases/catdat/data/category-properties/one-way.yaml +++ b/databases/catdat/data/category-properties/one-way.yaml @@ -2,7 +2,7 @@ id: one-way relation: is description: A category is one-way if every endomorphism in it is equal to the identity. nlab_link: https://ncatlab.org/nlab/show/one-way+category -dual_property_id: one-way +dual_property: one-way invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/pointed.yaml b/databases/catdat/data/category-properties/pointed.yaml index 12c7843f..8063da3e 100644 --- a/databases/catdat/data/category-properties/pointed.yaml +++ b/databases/catdat/data/category-properties/pointed.yaml @@ -2,7 +2,7 @@ id: pointed relation: is description: A category is pointed when it has a zero object, i.e. an object which is both initial and terminal. nlab_link: https://ncatlab.org/nlab/show/pointed+category -dual_property_id: pointed +dual_property: pointed invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/powers.yaml b/databases/catdat/data/category-properties/powers.yaml index a198a37c..6e90b71a 100644 --- a/databases/catdat/data/category-properties/powers.yaml +++ b/databases/catdat/data/category-properties/powers.yaml @@ -2,7 +2,7 @@ id: powers relation: has description: If $X$ is an object and $I$ is a set, the power is defined as the product $X^I := \prod_{i \in I} X$. This property refers to the existence of powers. nlab_link: https://ncatlab.org/nlab/show/powering -dual_property_id: copowers +dual_property: copowers invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/preadditive.yaml b/databases/catdat/data/category-properties/preadditive.yaml index ccb2ca4d..f14f9072 100644 --- a/databases/catdat/data/category-properties/preadditive.yaml +++ b/databases/catdat/data/category-properties/preadditive.yaml @@ -4,7 +4,7 @@ description: |- A category is preadditive when it is locally essentially small* and each hom-set carries the structure of an abelian group such that the composition is bilinear. Notice that "preadditive" is an extra structure. The property here just says that some preadditive structure exists. *We demand this instead of the more common "locally small" to ensure that preadditive categories are invariant under equivalences of categories. nlab_link: https://ncatlab.org/nlab/show/Ab-enriched+category -dual_property_id: preadditive +dual_property: preadditive invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/products.yaml b/databases/catdat/data/category-properties/products.yaml index 05d5e89f..71ac979e 100644 --- a/databases/catdat/data/category-properties/products.yaml +++ b/databases/catdat/data/category-properties/products.yaml @@ -2,7 +2,7 @@ id: products relation: has description: 'Given a family of objects $(A_i)_{i \in I}$, a product $\prod_{i \in I} A_i$ is defined as an object with morphisms $p_i : \prod_{i \in I} A_i \to A_i$ satisfying the following universal property: For every object $T$ and every family of morphisms $(f_i : T \to A_i)_{i \in I}$ there is a unique morphism $f : T \to \prod_{i \in I} A_i$ such that $p_i \circ f = f_i$ for all $i \in I$. This property refers to the existence of small products, i.e., products of small families of objects.' nlab_link: https://ncatlab.org/nlab/show/cartesian+product -dual_property_id: coproducts +dual_property: coproducts invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/pullbacks.yaml b/databases/catdat/data/category-properties/pullbacks.yaml index 7d89f3c7..05b5bbdf 100644 --- a/databases/catdat/data/category-properties/pullbacks.yaml +++ b/databases/catdat/data/category-properties/pullbacks.yaml @@ -2,7 +2,7 @@ id: pullbacks relation: has description: A category $\C$ has pullbacks if every cospan of morphisms $X \rightarrow S \leftarrow Y$ has a pullback $X \times_S Y$. This is also known as a fiber product. Equivalently, the slice category $\C/S$ has binary products. nlab_link: https://ncatlab.org/nlab/show/pullback -dual_property_id: pushouts +dual_property: pushouts invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/pushouts.yaml b/databases/catdat/data/category-properties/pushouts.yaml index a275762f..60a55438 100644 --- a/databases/catdat/data/category-properties/pushouts.yaml +++ b/databases/catdat/data/category-properties/pushouts.yaml @@ -2,7 +2,7 @@ id: pushouts relation: has description: A category $\C$ has pushouts if every span of morphisms $X \leftarrow S \rightarrow Y$ has a pushout $X \sqcup_S Y$. This is also known as a fiber coproduct. Equivalently, the coslice category $S/\C$ has binary coproducts. nlab_link: https://ncatlab.org/nlab/show/pushout -dual_property_id: pullbacks +dual_property: pullbacks invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/quotient object classifier.yaml b/databases/catdat/data/category-properties/quotient object classifier.yaml index fe682d6a..cbf3c0cc 100644 --- a/databases/catdat/data/category-properties/quotient object classifier.yaml +++ b/databases/catdat/data/category-properties/quotient object classifier.yaml @@ -5,7 +5,7 @@ description: >- $$\begin{CD} \Psi @>{\top}>> 0 \\ @V{\psi_e}VV @VV{!}V \\ A @>>{e}> B \end{CD}$$ is a pushout diagram. Equivalently, the functor $\Quot : \C \to \Set^+$ is representable. *Every morphism $\Psi \to 0$ is a split epimorphism anyway. -dual_property_id: subobject classifier +dual_property: subobject classifier invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/quotient-trivial.yaml b/databases/catdat/data/category-properties/quotient-trivial.yaml index 96a6df2c..67d64811 100644 --- a/databases/catdat/data/category-properties/quotient-trivial.yaml +++ b/databases/catdat/data/category-properties/quotient-trivial.yaml @@ -1,7 +1,7 @@ id: quotient-trivial relation: is description: A category is quotient-trivial if every epimorphism is an isomorphism. Equivalently, the poset of quotients of any object is trivial. This is no standard terminology. We have added it to the database since it clarifies the relationship between several related properties. -dual_property_id: subobject-trivial +dual_property: subobject-trivial invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/quotients of congruences.yaml b/databases/catdat/data/category-properties/quotients of congruences.yaml index 33072fa1..c17dfa64 100644 --- a/databases/catdat/data/category-properties/quotients of congruences.yaml +++ b/databases/catdat/data/category-properties/quotients of congruences.yaml @@ -2,7 +2,7 @@ id: quotients of congruences relation: has description: 'A congruence (or internal equivalence relation) on an object $X$ of a category is a parallel pair $p_1, p_2 : E \rightrightarrows X$ which is jointly monomorphic, and such that for every object $T$, the image of $(p_1 \circ {-}, p_2 \circ {-}) : \Hom(T, E) \to \Hom(T, X)^2$ is an equivalence relation. The category has quotients of congruences if for each such congruence, there exists a coequalizer of $p_1$ and $p_2$. Note that in the case of a category with binary powers, the corresponding subobjects of $X \times X$ are also commonly referred to as congruences, or as internal equivalence relations.' nlab_link: https://ncatlab.org/nlab/show/congruence -dual_property_id: coquotients of cocongruences +dual_property: coquotients of cocongruences invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/reflexive coequalizers.yaml b/databases/catdat/data/category-properties/reflexive coequalizers.yaml index f8dacbd7..b8aee672 100644 --- a/databases/catdat/data/category-properties/reflexive coequalizers.yaml +++ b/databases/catdat/data/category-properties/reflexive coequalizers.yaml @@ -2,7 +2,7 @@ id: reflexive coequalizers relation: has description: A reflexive coequalizer is a colimit of a diagram consisting of a parallel pair of morphisms with a common section (right inverse), which is the same concept as a coequalizer of such a parallel pair. This property refers to the existence of reflexive coequalizers. nlab_link: https://ncatlab.org/nlab/show/reflexive+coequalizer -dual_property_id: coreflexive equalizers +dual_property: coreflexive equalizers invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/regular quotient object classifier.yaml b/databases/catdat/data/category-properties/regular quotient object classifier.yaml index c3db87b4..29100512 100644 --- a/databases/catdat/data/category-properties/regular quotient object classifier.yaml +++ b/databases/catdat/data/category-properties/regular quotient object classifier.yaml @@ -5,7 +5,7 @@ description: >- $$\begin{CD} \Psi @>{\top}>> 0 \\ @V{\psi_e}VV @VV{!}V \\ A @>>{e}> B \end{CD}$$ is a pushout diagram. Equivalently, the functor $\Quot_{\reg} : \C \to \Set^+$ is representable. *Every morphism $\Psi \to 0$ is a split epimorphism anyway. -dual_property_id: regular subobject classifier +dual_property: regular subobject classifier invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/regular subobject classifier.yaml b/databases/catdat/data/category-properties/regular subobject classifier.yaml index efd70a13..fda2e901 100644 --- a/databases/catdat/data/category-properties/regular subobject classifier.yaml +++ b/databases/catdat/data/category-properties/regular subobject classifier.yaml @@ -6,7 +6,7 @@ description: >- is a pullback diagram. Equivalently, the functor $\Sub_{\reg} : \C^{\op} \to \Set^+$ is representable. *Every morphism $1 \to \Omega$ is a split monomorphism and hence regular anyway. nlab_link: https://ncatlab.org/nlab/show/subobject+classifier -dual_property_id: regular quotient object classifier +dual_property: regular quotient object classifier invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/regular.yaml b/databases/catdat/data/category-properties/regular.yaml index 25bda660..641a7517 100644 --- a/databases/catdat/data/category-properties/regular.yaml +++ b/databases/catdat/data/category-properties/regular.yaml @@ -2,7 +2,7 @@ id: regular relation: is description: A category is regular when it is finitely complete, for every morphism $X \to Y$ its kernel pair $X \times_Y X \rightrightarrows X$ has a coequalizer, and regular epimorphisms are stable under pullbacks. nlab_link: https://ncatlab.org/nlab/show/regular+category -dual_property_id: coregular +dual_property: coregular invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/right cancellative.yaml b/databases/catdat/data/category-properties/right cancellative.yaml index 29d2eb3c..72b4046f 100644 --- a/databases/catdat/data/category-properties/right cancellative.yaml +++ b/databases/catdat/data/category-properties/right cancellative.yaml @@ -2,7 +2,7 @@ id: right cancellative relation: is description: 'A category is right cancellative if for every morphism $f : A \to B$ and every parallel pair of morphisms $g,h : B \rightrightarrows T$ with $g \circ f = h \circ f$ we have $g = h$. Equivalently, every morphism is an epimorphism.' nlab_link: https://ncatlab.org/nlab/show/cancellative+category -dual_property_id: left cancellative +dual_property: left cancellative invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/self-dual.yaml b/databases/catdat/data/category-properties/self-dual.yaml index 889c9940..bd4d772c 100644 --- a/databases/catdat/data/category-properties/self-dual.yaml +++ b/databases/catdat/data/category-properties/self-dual.yaml @@ -2,7 +2,7 @@ id: self-dual relation: is description: A category is self-dual if it is equivalent to its opposite (or dual) category. nlab_link: https://ncatlab.org/nlab/show/opposite+category -dual_property_id: self-dual +dual_property: self-dual invariant_under_equivalences: true related_properties: [] diff --git a/databases/catdat/data/category-properties/semi-strongly connected.yaml b/databases/catdat/data/category-properties/semi-strongly connected.yaml index bc561087..4f1fb35d 100644 --- a/databases/catdat/data/category-properties/semi-strongly connected.yaml +++ b/databases/catdat/data/category-properties/semi-strongly connected.yaml @@ -2,7 +2,7 @@ id: semi-strongly connected relation: is description: A category is semi-strongly connected if it is inhabited and for every two objects $A,B$ there is a morphism $A \to B$ or there is a morphism $B \to A$. Notice that this is stronger than being connected, and that posets with this property are precisely the inhabited totally ordered sets. nlab_link: https://ncatlab.org/nlab/show/strongly+connected+category -dual_property_id: semi-strongly connected +dual_property: semi-strongly connected invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/sequential colimits.yaml b/databases/catdat/data/category-properties/sequential colimits.yaml index 903aa26d..20db002a 100644 --- a/databases/catdat/data/category-properties/sequential colimits.yaml +++ b/databases/catdat/data/category-properties/sequential colimits.yaml @@ -2,7 +2,7 @@ id: sequential colimits relation: has description: 'A category has sequential colimits if it has colimits of diagrams of the form: $\bullet \to \bullet \to \bullet \to \cdots$.' nlab_link: https://ncatlab.org/nlab/show/sequential+colimit -dual_property_id: sequential limits +dual_property: sequential limits invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/sequential limits.yaml b/databases/catdat/data/category-properties/sequential limits.yaml index 9a917102..6ecb32ff 100644 --- a/databases/catdat/data/category-properties/sequential limits.yaml +++ b/databases/catdat/data/category-properties/sequential limits.yaml @@ -2,7 +2,7 @@ id: sequential limits relation: has description: A category has sequential limits if it has limits of diagrams of the form $\cdots \bullet \to \bullet \to \bullet$. nlab_link: https://ncatlab.org/nlab/show/sequential+limit -dual_property_id: sequential colimits +dual_property: sequential colimits invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/sifted colimits.yaml b/databases/catdat/data/category-properties/sifted colimits.yaml index 5ddfc679..acf99709 100644 --- a/databases/catdat/data/category-properties/sifted colimits.yaml +++ b/databases/catdat/data/category-properties/sifted colimits.yaml @@ -2,7 +2,7 @@ id: sifted colimits relation: has description: A category has sifted colimits if it has colimits of diagrams indexed by small sifted categories. nlab_link: https://ncatlab.org/nlab/show/sifted+colimit -dual_property_id: cosifted limits +dual_property: cosifted limits invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/sifted.yaml b/databases/catdat/data/category-properties/sifted.yaml index 0e7b7293..d9da6430 100644 --- a/databases/catdat/data/category-properties/sifted.yaml +++ b/databases/catdat/data/category-properties/sifted.yaml @@ -5,7 +5,7 @@ description: >- $$X \rightarrow Z \leftarrow Y$$ is connected. Equivalently, a small category $\C$ is sifted if $\colim : \Set^{\C} \to \Set$ preserves finite products. This property is a weaker notion than being filtered. nlab_link: https://ncatlab.org/nlab/show/sifted+category -dual_property_id: cosifted +dual_property: cosifted invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/skeletal.yaml b/databases/catdat/data/category-properties/skeletal.yaml index 2aecfb13..0ba73830 100644 --- a/databases/catdat/data/category-properties/skeletal.yaml +++ b/databases/catdat/data/category-properties/skeletal.yaml @@ -2,7 +2,7 @@ id: skeletal relation: is description: A category is skeletal when isomorphic objects are already equal. Every category is equivalent to a skeletal category (using the axiom of choice). nlab_link: https://ncatlab.org/nlab/show/skeleton -dual_property_id: skeletal +dual_property: skeletal invariant_under_equivalences: false related_properties: diff --git a/databases/catdat/data/category-properties/small.yaml b/databases/catdat/data/category-properties/small.yaml index ea608433..866b44a3 100644 --- a/databases/catdat/data/category-properties/small.yaml +++ b/databases/catdat/data/category-properties/small.yaml @@ -2,7 +2,7 @@ id: small relation: is description: A category is small when the collection of objects and the collection of morphisms are sets, i.e. small. nlab_link: https://ncatlab.org/nlab/show/small+category -dual_property_id: small +dual_property: small invariant_under_equivalences: false related_properties: diff --git a/databases/catdat/data/category-properties/split abelian.yaml b/databases/catdat/data/category-properties/split abelian.yaml index 8276cd9c..91bc4664 100644 --- a/databases/catdat/data/category-properties/split abelian.yaml +++ b/databases/catdat/data/category-properties/split abelian.yaml @@ -2,7 +2,7 @@ id: split abelian relation: is description: A category is split abelian if it is abelian and every short exact sequence splits. Equivalently, every object is projective. Equivalently, every object is injective. nlab_link: https://ncatlab.org/nlab/show/split+exact+sequence -dual_property_id: split abelian +dual_property: split abelian invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/strict initial object.yaml b/databases/catdat/data/category-properties/strict initial object.yaml index 8dfe51d5..96e50c35 100644 --- a/databases/catdat/data/category-properties/strict initial object.yaml +++ b/databases/catdat/data/category-properties/strict initial object.yaml @@ -2,7 +2,7 @@ id: strict initial object relation: has a description: A strict initial object is an initial object $0$ such that every morphism $A \to 0$ is an isomorphism. This property refers to the existence of a strict initial object. nlab_link: https://ncatlab.org/nlab/show/strict+initial+object -dual_property_id: strict terminal object +dual_property: strict terminal object invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/strict terminal object.yaml b/databases/catdat/data/category-properties/strict terminal object.yaml index a2c1bcb9..cb72965b 100644 --- a/databases/catdat/data/category-properties/strict terminal object.yaml +++ b/databases/catdat/data/category-properties/strict terminal object.yaml @@ -2,7 +2,7 @@ id: strict terminal object relation: has a description: A strict terminal object is a terminal object $1$ such that every morphism $1 \to A$ is an isomorphism. This property refers to the existence of a strict terminal object. nlab_link: https://ncatlab.org/nlab/show/strict+terminal+object -dual_property_id: strict initial object +dual_property: strict initial object invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/strongly connected.yaml b/databases/catdat/data/category-properties/strongly connected.yaml index 563ce106..aad8550d 100644 --- a/databases/catdat/data/category-properties/strongly connected.yaml +++ b/databases/catdat/data/category-properties/strongly connected.yaml @@ -2,7 +2,7 @@ id: strongly connected relation: is description: A category is strongly connected if it is inhabited and for every two objects $A,B$ there is a morphism $A \to B$. In other words, each hom-set is inhabited. Notice that when a terminal object $1$ exists, this property means that every object $A$ admits a morphism $1 \to A$. nlab_link: https://ncatlab.org/nlab/show/strongly+connected+category -dual_property_id: strongly connected +dual_property: strongly connected invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/subobject classifier.yaml b/databases/catdat/data/category-properties/subobject classifier.yaml index 6d0175b7..d19b772a 100644 --- a/databases/catdat/data/category-properties/subobject classifier.yaml +++ b/databases/catdat/data/category-properties/subobject classifier.yaml @@ -6,7 +6,7 @@ description: >- is a pullback diagram. Equivalently, the functor $\Sub : \C^{\op} \to \Set^+$ is representable. *Every morphism $1 \to \Omega$ is a split monomorphism anyway. nlab_link: https://ncatlab.org/nlab/show/subobject+classifier -dual_property_id: quotient object classifier +dual_property: quotient object classifier invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/subobject-trivial.yaml b/databases/catdat/data/category-properties/subobject-trivial.yaml index f5f523f5..4802d42d 100644 --- a/databases/catdat/data/category-properties/subobject-trivial.yaml +++ b/databases/catdat/data/category-properties/subobject-trivial.yaml @@ -1,7 +1,7 @@ id: subobject-trivial relation: is description: A category is subobject-trivial if every monomorphism is an isomorphism. Equivalently, the poset of subobjects of any object is trivial. This is no standard terminology. We have added it to the database since it clarifies the relationship between several related properties. -dual_property_id: quotient-trivial +dual_property: quotient-trivial invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/terminal object.yaml b/databases/catdat/data/category-properties/terminal object.yaml index 53d10f30..09e95c3e 100644 --- a/databases/catdat/data/category-properties/terminal object.yaml +++ b/databases/catdat/data/category-properties/terminal object.yaml @@ -2,7 +2,7 @@ id: terminal object relation: has a description: A terminal object (or final object) is an object that has a unique morphism from every object in the category. This property refers to the existence of a terminal object. nlab_link: https://ncatlab.org/nlab/show/terminal+object -dual_property_id: initial object +dual_property: initial object invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/thin.yaml b/databases/catdat/data/category-properties/thin.yaml index d070c834..dacb9d73 100644 --- a/databases/catdat/data/category-properties/thin.yaml +++ b/databases/catdat/data/category-properties/thin.yaml @@ -2,7 +2,7 @@ id: thin relation: is description: A category is thin when between any pair of objects there is at most one morphism. Such categories correspond to preordered collections. nlab_link: https://ncatlab.org/nlab/show/thin+category -dual_property_id: thin +dual_property: thin invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/trivial.yaml b/databases/catdat/data/category-properties/trivial.yaml index 55715880..188dba46 100644 --- a/databases/catdat/data/category-properties/trivial.yaml +++ b/databases/catdat/data/category-properties/trivial.yaml @@ -2,7 +2,7 @@ id: trivial relation: is description: A category is trivial if it is equivalent to the trivial category. Equivalently, there is an initial object $0$ such that for every object $A$ the unique morphism $0 \to A$ is an isomorphism. Notice that we do not demand that the category is isomorphic to the trivial category. As a consequence, every inhabited indiscrete category is trivial in our sense. nlab_link: https://ncatlab.org/nlab/show/terminal+category -dual_property_id: trivial +dual_property: trivial invariant_under_equivalences: true related_properties: [] diff --git a/databases/catdat/data/category-properties/unital.yaml b/databases/catdat/data/category-properties/unital.yaml index 76d11f71..acd194b3 100644 --- a/databases/catdat/data/category-properties/unital.yaml +++ b/databases/catdat/data/category-properties/unital.yaml @@ -2,7 +2,7 @@ id: unital relation: is description: 'A category is unital if it has a zero object, finite limits, and for all objects $X,Y$ the two morphisms $(\id_X,0) : X \hookrightarrow X \times Y$ and $(0,\id_Y) : Y \hookrightarrow X \times Y$ are jointly strongly epimorphic. This means: there is no proper subobject of $X \times Y$ that contains $X$ and $Y$. When coproducts exist, the canonical morphism $X \sqcup Y \to X \times Y$ therefore must be a strong epimorphism.' nlab_link: https://ncatlab.org/nlab/show/unital+category -dual_property_id: counital +dual_property: counital invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/well-copowered.yaml b/databases/catdat/data/category-properties/well-copowered.yaml index 0818ae8d..03065a85 100644 --- a/databases/catdat/data/category-properties/well-copowered.yaml +++ b/databases/catdat/data/category-properties/well-copowered.yaml @@ -2,7 +2,7 @@ id: well-copowered relation: is description: A category is well-copowered if the collection of quotients of any object is isomorphic to a set. nlab_link: https://ncatlab.org/nlab/show/well-powered+category -dual_property_id: well-powered +dual_property: well-powered invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/well-powered.yaml b/databases/catdat/data/category-properties/well-powered.yaml index ffb34d5b..c7353d62 100644 --- a/databases/catdat/data/category-properties/well-powered.yaml +++ b/databases/catdat/data/category-properties/well-powered.yaml @@ -2,7 +2,7 @@ id: well-powered relation: is description: A category is well-powered if the collection of subobjects of any object is isomorphic to a set. nlab_link: https://ncatlab.org/nlab/show/well-powered+category -dual_property_id: well-copowered +dual_property: well-copowered invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/wide pullbacks.yaml b/databases/catdat/data/category-properties/wide pullbacks.yaml index e1540893..a4312308 100644 --- a/databases/catdat/data/category-properties/wide pullbacks.yaml +++ b/databases/catdat/data/category-properties/wide pullbacks.yaml @@ -2,7 +2,7 @@ id: wide pullbacks relation: has description: A category $\C$ has wide pullbacks if for every object $S$ the slice category $\C/S$ has arbitrary small products. nlab_link: https://ncatlab.org/nlab/show/wide+pullback -dual_property_id: wide pushouts +dual_property: wide pushouts invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/wide pushouts.yaml b/databases/catdat/data/category-properties/wide pushouts.yaml index 60a3128a..ce2b274a 100644 --- a/databases/catdat/data/category-properties/wide pushouts.yaml +++ b/databases/catdat/data/category-properties/wide pushouts.yaml @@ -2,7 +2,7 @@ id: wide pushouts relation: has description: A category $\C$ has wide pushouts if for every object $S$ the coslice category $S/\C$ has arbitrary small coproducts. nlab_link: https://ncatlab.org/nlab/show/wide+pushout -dual_property_id: wide pullbacks +dual_property: wide pullbacks invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/category-properties/zero morphisms.yaml b/databases/catdat/data/category-properties/zero morphisms.yaml index 1e67af92..34f26ea2 100644 --- a/databases/catdat/data/category-properties/zero morphisms.yaml +++ b/databases/catdat/data/category-properties/zero morphisms.yaml @@ -2,7 +2,7 @@ id: zero morphisms relation: has description: 'A category has zero morphisms if for every pair of objects $A,B$ there is a distinguished morphism $0_{A,B} : A \to B$, called the zero morphism, such that we have $f \circ 0_{A,B} = 0_{A,C}$ and $0_{B,C} \circ g = 0_{A,C}$ for all morphisms $f : B \to C$ and $g : A \to B$. The zero morphisms are unique if they exist, hence this is actually a property of the category.' nlab_link: https://ncatlab.org/nlab/show/zero+morphism -dual_property_id: zero morphisms +dual_property: zero morphisms invariant_under_equivalences: true related_properties: diff --git a/databases/catdat/data/functor-properties/cocontinuous.yaml b/databases/catdat/data/functor-properties/cocontinuous.yaml index 68230019..487a8d0e 100644 --- a/databases/catdat/data/functor-properties/cocontinuous.yaml +++ b/databases/catdat/data/functor-properties/cocontinuous.yaml @@ -3,4 +3,4 @@ relation: is description: A functor is cocontinuous when it preserves all small colimits. nlab_link: https://ncatlab.org/nlab/show/cocontinuous+functor invariant_under_equivalences: true -dual_property_id: continuous +dual_property: continuous diff --git a/databases/catdat/data/functor-properties/coequalizer-preserving.yaml b/databases/catdat/data/functor-properties/coequalizer-preserving.yaml index b7f79580..2eb95bc6 100644 --- a/databases/catdat/data/functor-properties/coequalizer-preserving.yaml +++ b/databases/catdat/data/functor-properties/coequalizer-preserving.yaml @@ -3,4 +3,4 @@ relation: is description: 'A functor $F$ preserves coequalizers when for every parallel pair of morphisms $f,g : A \rightrightarrows B$ whose coequalizer $p : B \to Q$ exists, also $F(p) : F(B) \to F(Q)$ is an coequalizer of $F(f),F(g) : F(A) \rightrightarrows F(B)$.' nlab_link: null invariant_under_equivalences: true -dual_property_id: equalizer-preserving +dual_property: equalizer-preserving diff --git a/databases/catdat/data/functor-properties/cofinitary.yaml b/databases/catdat/data/functor-properties/cofinitary.yaml index 22354368..4fb5c7ee 100644 --- a/databases/catdat/data/functor-properties/cofinitary.yaml +++ b/databases/catdat/data/functor-properties/cofinitary.yaml @@ -3,4 +3,4 @@ relation: is description: A functor is cofinitary when it preserves cofiltered limits. nlab_link: null invariant_under_equivalences: true -dual_property_id: finitary +dual_property: finitary diff --git a/databases/catdat/data/functor-properties/comonadic.yaml b/databases/catdat/data/functor-properties/comonadic.yaml index b66fb321..2a64f9fd 100644 --- a/databases/catdat/data/functor-properties/comonadic.yaml +++ b/databases/catdat/data/functor-properties/comonadic.yaml @@ -3,4 +3,4 @@ relation: is description: 'A functor $F : \C \to \D$ is comonadic when there is a comonad $T$ on $\D$ such that $F$ is equivalent to the forgetful functor $U^T : \CoAlg(T) \to \D$.' nlab_link: https://ncatlab.org/nlab/show/comonadic+functor invariant_under_equivalences: true -dual_property_id: monadic +dual_property: monadic diff --git a/databases/catdat/data/functor-properties/conservative.yaml b/databases/catdat/data/functor-properties/conservative.yaml index a0616ae7..3f05f486 100644 --- a/databases/catdat/data/functor-properties/conservative.yaml +++ b/databases/catdat/data/functor-properties/conservative.yaml @@ -3,4 +3,4 @@ relation: is description: 'A functor $F : \C \to \D$ is conservative when it is isomorphic-reflecting: If $f$ is a morphism in $\C$ such that $F(f)$ is an isomorphism, then $f$ is an isomorphism.' nlab_link: https://ncatlab.org/nlab/show/conservative+functor invariant_under_equivalences: true -dual_property_id: conservative +dual_property: conservative diff --git a/databases/catdat/data/functor-properties/continuous.yaml b/databases/catdat/data/functor-properties/continuous.yaml index a2ae9810..73c9586d 100644 --- a/databases/catdat/data/functor-properties/continuous.yaml +++ b/databases/catdat/data/functor-properties/continuous.yaml @@ -3,4 +3,4 @@ relation: is description: A functor is continuous when it preserves all small limits. nlab_link: https://ncatlab.org/nlab/show/continuous+functor invariant_under_equivalences: true -dual_property_id: cocontinuous +dual_property: cocontinuous diff --git a/databases/catdat/data/functor-properties/coproduct-preserving.yaml b/databases/catdat/data/functor-properties/coproduct-preserving.yaml index 1d99d609..3d9f681f 100644 --- a/databases/catdat/data/functor-properties/coproduct-preserving.yaml +++ b/databases/catdat/data/functor-properties/coproduct-preserving.yaml @@ -3,4 +3,4 @@ relation: is description: A functor $F$ preserves coproducts when for every family of objects $(A_i)$ in the source whose coproduct $\prod_i A_i$ exists, also the coproduct $\coprod_i F(A_i)$ exists in the target and such that the canonical morphism $\coprod_i F(A_i) \to F(\coprod_i A_i)$ is an isomorphism. nlab_link: null invariant_under_equivalences: true -dual_property_id: product-preserving +dual_property: product-preserving diff --git a/databases/catdat/data/functor-properties/epimorphism-preserving.yaml b/databases/catdat/data/functor-properties/epimorphism-preserving.yaml index ff3f7639..e800de97 100644 --- a/databases/catdat/data/functor-properties/epimorphism-preserving.yaml +++ b/databases/catdat/data/functor-properties/epimorphism-preserving.yaml @@ -5,4 +5,4 @@ description: |- This property is useful to rule out some adjunctions. nlab_link: https://ncatlab.org/nlab/show/epimorphism invariant_under_equivalences: true -dual_property_id: monomorphism-preserving +dual_property: monomorphism-preserving diff --git a/databases/catdat/data/functor-properties/equalizer-preserving.yaml b/databases/catdat/data/functor-properties/equalizer-preserving.yaml index f2799db9..5447dea2 100644 --- a/databases/catdat/data/functor-properties/equalizer-preserving.yaml +++ b/databases/catdat/data/functor-properties/equalizer-preserving.yaml @@ -3,4 +3,4 @@ relation: is description: 'A functor $F$ preserves equalizers when for every parallel pair of morphisms $f,g : A \rightrightarrows B$ whose equalizer $i : E \to A$ exists, also $F(i) : F(E) \to F(A)$ is an equalizer of $F(f),F(g) : F(A) \rightrightarrows F(B)$.' nlab_link: null invariant_under_equivalences: true -dual_property_id: coequalizer-preserving +dual_property: coequalizer-preserving diff --git a/databases/catdat/data/functor-properties/equivalence.yaml b/databases/catdat/data/functor-properties/equivalence.yaml index 75eea815..da972413 100644 --- a/databases/catdat/data/functor-properties/equivalence.yaml +++ b/databases/catdat/data/functor-properties/equivalence.yaml @@ -3,4 +3,4 @@ relation: is an description: A functor is an equivalence if it has a pseudo-inverse functor. nlab_link: https://ncatlab.org/nlab/show/equivalence+of+categories invariant_under_equivalences: true -dual_property_id: equivalence +dual_property: equivalence diff --git a/databases/catdat/data/functor-properties/essentially surjective.yaml b/databases/catdat/data/functor-properties/essentially surjective.yaml index 97a45ecc..8304fd31 100644 --- a/databases/catdat/data/functor-properties/essentially surjective.yaml +++ b/databases/catdat/data/functor-properties/essentially surjective.yaml @@ -3,4 +3,4 @@ relation: is description: 'A functor $F : \C \to \D$ is essentially surjective when every object $Y \in \D$ is isomorphic to $F(X)$ for some $X \in \C$.' nlab_link: https://ncatlab.org/nlab/show/essentially+surjective+functor invariant_under_equivalences: true -dual_property_id: essentially surjective +dual_property: essentially surjective diff --git a/databases/catdat/data/functor-properties/exact.yaml b/databases/catdat/data/functor-properties/exact.yaml index 25c964c4..fba1cf05 100644 --- a/databases/catdat/data/functor-properties/exact.yaml +++ b/databases/catdat/data/functor-properties/exact.yaml @@ -3,4 +3,4 @@ relation: is description: A functor is exact when it is left exact and right exact. nlab_link: https://ncatlab.org/nlab/show/exact+functor invariant_under_equivalences: true -dual_property_id: exact +dual_property: exact diff --git a/databases/catdat/data/functor-properties/faithful.yaml b/databases/catdat/data/functor-properties/faithful.yaml index 9f0e7130..9582fb29 100644 --- a/databases/catdat/data/functor-properties/faithful.yaml +++ b/databases/catdat/data/functor-properties/faithful.yaml @@ -3,4 +3,4 @@ relation: is description: 'A functor is faithful when it is injective on Hom-sets: If $F(f)=F(g)$, then $f=g$.' nlab_link: https://ncatlab.org/nlab/show/faithful+functor invariant_under_equivalences: true -dual_property_id: faithful +dual_property: faithful diff --git a/databases/catdat/data/functor-properties/finitary.yaml b/databases/catdat/data/functor-properties/finitary.yaml index 61bdc304..840fa308 100644 --- a/databases/catdat/data/functor-properties/finitary.yaml +++ b/databases/catdat/data/functor-properties/finitary.yaml @@ -3,4 +3,4 @@ relation: is description: A functor is finitary when it preserves filtered colimits. nlab_link: https://ncatlab.org/nlab/show/finitary+functor invariant_under_equivalences: true -dual_property_id: cofinitary +dual_property: cofinitary diff --git a/databases/catdat/data/functor-properties/finite-coproduct-preserving.yaml b/databases/catdat/data/functor-properties/finite-coproduct-preserving.yaml index 6a129fa5..5dc526e2 100644 --- a/databases/catdat/data/functor-properties/finite-coproduct-preserving.yaml +++ b/databases/catdat/data/functor-properties/finite-coproduct-preserving.yaml @@ -3,4 +3,4 @@ relation: is description: A functor $F$ preserves finite coproducts when for every family of objects $(A_i)$ in the source whose coproduct $\prod_i A_i$ exists, also the coproduct $\coprod_i F(A_i)$ exists in the target and such that the canonical morphism $\coprod_i F(A_i) \to F(\coprod_i A_i)$ is an isomorphism. nlab_link: null invariant_under_equivalences: true -dual_property_id: finite-product-preserving +dual_property: finite-product-preserving diff --git a/databases/catdat/data/functor-properties/finite-product-preserving.yaml b/databases/catdat/data/functor-properties/finite-product-preserving.yaml index 3623a3cc..874285c1 100644 --- a/databases/catdat/data/functor-properties/finite-product-preserving.yaml +++ b/databases/catdat/data/functor-properties/finite-product-preserving.yaml @@ -3,4 +3,4 @@ relation: is description: A functor $F$ preserves finite products when for every finite family of objects $(A_i)$ in the source whose product $\prod_i A_i$ exists, also the product $\prod_i F(A_i)$ exists in the target and such that the canonical morphism $F(\prod_i A_i) \to \prod_i F(A_i)$ is an isomorphism. nlab_link: null invariant_under_equivalences: true -dual_property_id: finite-coproduct-preserving +dual_property: finite-coproduct-preserving diff --git a/databases/catdat/data/functor-properties/full.yaml b/databases/catdat/data/functor-properties/full.yaml index cfbf7569..1507d88f 100644 --- a/databases/catdat/data/functor-properties/full.yaml +++ b/databases/catdat/data/functor-properties/full.yaml @@ -3,4 +3,4 @@ relation: is description: 'A functor is full when it is surjective on Hom-sets: Every morphism $F(A) \to F(B)$ is induced by a morphism $A \to B$.' nlab_link: https://ncatlab.org/nlab/show/full+functor invariant_under_equivalences: true -dual_property_id: full +dual_property: full diff --git a/databases/catdat/data/functor-properties/initial-object-preserving.yaml b/databases/catdat/data/functor-properties/initial-object-preserving.yaml index 39a13bf4..cd64792a 100644 --- a/databases/catdat/data/functor-properties/initial-object-preserving.yaml +++ b/databases/catdat/data/functor-properties/initial-object-preserving.yaml @@ -3,4 +3,4 @@ relation: is description: A functor $F$ preserves initial objects when it maps every initial object to an initial object. It is not assumed that the source category has a initial object. nlab_link: null invariant_under_equivalences: true -dual_property_id: terminal-object-preserving +dual_property: terminal-object-preserving diff --git a/databases/catdat/data/functor-properties/left adjoint.yaml b/databases/catdat/data/functor-properties/left adjoint.yaml index 8538f110..58ce7413 100644 --- a/databases/catdat/data/functor-properties/left adjoint.yaml +++ b/databases/catdat/data/functor-properties/left adjoint.yaml @@ -3,4 +3,4 @@ relation: is a description: 'A functor $F : \C \to \D$ is a left adjoint when there is a functor $G : \D \to \C$ such that there are natural bijections $\Hom(F(A),B) \cong \Hom(A,G(B))$.' nlab_link: https://ncatlab.org/nlab/show/left+adjoint invariant_under_equivalences: true -dual_property_id: right adjoint +dual_property: right adjoint diff --git a/databases/catdat/data/functor-properties/left exact.yaml b/databases/catdat/data/functor-properties/left exact.yaml index a5ab5e04..44f9c15d 100644 --- a/databases/catdat/data/functor-properties/left exact.yaml +++ b/databases/catdat/data/functor-properties/left exact.yaml @@ -3,4 +3,4 @@ relation: is description: A functor is left exact when it preserves finite limits. nlab_link: https://ncatlab.org/nlab/show/exact+functor invariant_under_equivalences: true -dual_property_id: right exact +dual_property: right exact diff --git a/databases/catdat/data/functor-properties/monadic.yaml b/databases/catdat/data/functor-properties/monadic.yaml index 0408de01..7404ccb3 100644 --- a/databases/catdat/data/functor-properties/monadic.yaml +++ b/databases/catdat/data/functor-properties/monadic.yaml @@ -3,4 +3,4 @@ relation: is description: 'A functor $F : \C \to \D$ is monadic when there is a monad $T$ on $\D$ such that $F$ is equivalent to the forgetful functor $U^T : \Alg(T) \to \D$.' nlab_link: https://ncatlab.org/nlab/show/monadic+functor invariant_under_equivalences: true -dual_property_id: comonadic +dual_property: comonadic diff --git a/databases/catdat/data/functor-properties/monomorphism-preserving.yaml b/databases/catdat/data/functor-properties/monomorphism-preserving.yaml index f7ed1cd0..dfe4c1c5 100644 --- a/databases/catdat/data/functor-properties/monomorphism-preserving.yaml +++ b/databases/catdat/data/functor-properties/monomorphism-preserving.yaml @@ -5,4 +5,4 @@ description: |- This property is useful to rule out some adjunctions. nlab_link: https://ncatlab.org/nlab/show/monomorphism invariant_under_equivalences: true -dual_property_id: epimorphism-preserving +dual_property: epimorphism-preserving diff --git a/databases/catdat/data/functor-properties/product-preserving.yaml b/databases/catdat/data/functor-properties/product-preserving.yaml index 65880664..6de48e43 100644 --- a/databases/catdat/data/functor-properties/product-preserving.yaml +++ b/databases/catdat/data/functor-properties/product-preserving.yaml @@ -3,7 +3,7 @@ relation: is description: A functor $F$ preserves products when for every family of objects $(A_i)$ in the source whose product $\prod_i A_i$ exists, also the product $\prod_i F(A_i)$ exists in the target and such that the canonical morphism $F(\prod_i A_i) \to \prod_i F(A_i)$ is an isomorphism. nlab_link: null invariant_under_equivalences: true -dual_property_id: coproduct-preserving +dual_property: coproduct-preserving # Here is why we do not call this property "preserves products": # Either we name the property "preserves products" and choose the diff --git a/databases/catdat/data/functor-properties/representable.yaml b/databases/catdat/data/functor-properties/representable.yaml index 79eda781..b1e517ec 100644 --- a/databases/catdat/data/functor-properties/representable.yaml +++ b/databases/catdat/data/functor-properties/representable.yaml @@ -3,4 +3,4 @@ relation: is description: 'A functor $F : \C \to \D$ is representable if $\C$ is locally small, $\D = \Set$, and there is an object $A \in \C$ with $F \cong \Hom(A,-)$.' nlab_link: https://ncatlab.org/nlab/show/representable+functor invariant_under_equivalences: true -dual_property_id: null +dual_property: null diff --git a/databases/catdat/data/functor-properties/right adjoint.yaml b/databases/catdat/data/functor-properties/right adjoint.yaml index b2976915..6f3c4940 100644 --- a/databases/catdat/data/functor-properties/right adjoint.yaml +++ b/databases/catdat/data/functor-properties/right adjoint.yaml @@ -3,4 +3,4 @@ relation: is a description: 'A functor $F : \C \to \D$ is a right adjoint when there is a functor $G : \D \to \C$ such that there are natural bijections $\Hom(G(A),B) \cong \Hom(A,F(B))$.' nlab_link: https://ncatlab.org/nlab/show/right+adjoint invariant_under_equivalences: true -dual_property_id: left adjoint +dual_property: left adjoint diff --git a/databases/catdat/data/functor-properties/right exact.yaml b/databases/catdat/data/functor-properties/right exact.yaml index 753dd070..dce36f2f 100644 --- a/databases/catdat/data/functor-properties/right exact.yaml +++ b/databases/catdat/data/functor-properties/right exact.yaml @@ -3,4 +3,4 @@ relation: is description: A functor is right exact when it preserves finite colimits. nlab_link: https://ncatlab.org/nlab/show/exact+functor invariant_under_equivalences: true -dual_property_id: left exact +dual_property: left exact diff --git a/databases/catdat/data/functor-properties/terminal-object-preserving.yaml b/databases/catdat/data/functor-properties/terminal-object-preserving.yaml index 48e501db..7775658c 100644 --- a/databases/catdat/data/functor-properties/terminal-object-preserving.yaml +++ b/databases/catdat/data/functor-properties/terminal-object-preserving.yaml @@ -3,4 +3,4 @@ relation: is description: A functor $F$ preserves terminal objects when it maps every terminal object to a terminal object. It is not assumed that the source category has a terminal object. nlab_link: null invariant_under_equivalences: true -dual_property_id: initial-object-preserving +dual_property: initial-object-preserving diff --git a/databases/catdat/data/functors/abelianization.yaml b/databases/catdat/data/functors/abelianization.yaml index fda28d30..37bae25d 100644 --- a/databases/catdat/data/functors/abelianization.yaml +++ b/databases/catdat/data/functors/abelianization.yaml @@ -6,21 +6,21 @@ description: This functor maps a group $G$ to its abelianization $G^{\ab} := G/[ nlab_link: https://ncatlab.org/nlab/show/abelianization satisfied_properties: - - property_id: essentially surjective + - property: essentially surjective reason: For abelian groups $G$ we have $G \cong G^{\ab}$. - - property_id: finite-product-preserving + - property: finite-product-preserving reason: See MO/386144. - - property_id: left adjoint + - property: left adjoint reason: This functor is left adjoint to the forgetful functor. unsatisfied_properties: - - property_id: conservative + - property: conservative reason: The proper inclusion $S_3 \hookrightarrow S_4$ gets mapped to the trivial homomorphism $1 \to 1$, which is an isomorphism. - - property_id: faithful + - property: faithful reason: Both the inclusion $A_3 \hookrightarrow S_3$ and the trivial homomorphism $A_3 \to S_3$ are mapped to the trivial homomorphism $A_3 \to 1$. - - property_id: full + - property: full reason: See MSE/716686. - - property_id: monomorphism-preserving + - property: monomorphism-preserving reason: The monomorphism $A_3 \hookrightarrow S_3$ is mapped to $A_3 \to 1$. - - property_id: product-preserving + - property: product-preserving reason: 'If $G$ is a group, the canonical homomorphism $(G^{\IN})^{\ab} \to (G^{\ab})^{\IN}$ is surjective, but does not need to be an isomorphism: otherwise, the inclusion $[G^{\IN}, G^{\IN}] \subseteq [G,G]^{\IN}$ would be an equality. But this requires the commutator width of $G$ to be finite, which fails for $G = F_2$ for instance. See also The abelianization of inverse limits of groups, Remark 0.0.7.' diff --git a/databases/catdat/data/functors/forget_vector.yaml b/databases/catdat/data/functors/forget_vector.yaml index 54004565..537dd6f2 100644 --- a/databases/catdat/data/functors/forget_vector.yaml +++ b/databases/catdat/data/functors/forget_vector.yaml @@ -6,21 +6,21 @@ description: This functor $U$ maps a vector space $V$ (over a fixed field $K$) t nlab_link: https://ncatlab.org/nlab/show/forgetful+functor satisfied_properties: - - property_id: conservative + - property: conservative reason: It is standard that the inverse of a bijective linear map is also linear. - - property_id: epimorphism-preserving + - property: epimorphism-preserving reason: This follows from the classifications of epimorphisms in $\Vect$ and in $\Set$. - - property_id: finitary + - property: finitary reason: For every algebraic category the forgetful functor to the category of sets preserves filtered colimits. - - property_id: monadic + - property: monadic reason: For every algebraic category the forgetful functor to the category of sets is monadic. - - property_id: representable + - property: representable reason: This functor is represented by any $1$-dimensional vector space. unsatisfied_properties: - - property_id: coequalizer-preserving + - property: coequalizer-preserving reason: 'The coequalizer of $0,i_1 : K \to K^2$ in $\Vect$ is $p_2 : K^2 \to K$, but the coequalizer in $\Set$ is $(K \times K^*) \cup \{(0,0)\}$.' - - property_id: essentially surjective + - property: essentially surjective reason: The empty has has no vector space structure. - - property_id: initial-object-preserving + - property: initial-object-preserving reason: The underlying set of a trivial vector space is not empty. diff --git a/databases/catdat/data/functors/free_group.yaml b/databases/catdat/data/functors/free_group.yaml index a02f1cc4..7bf04ac2 100644 --- a/databases/catdat/data/functors/free_group.yaml +++ b/databases/catdat/data/functors/free_group.yaml @@ -6,21 +6,21 @@ description: This functor maps a set $X$ to the free group $F(X)$ on that set. nlab_link: https://ncatlab.org/nlab/show/free+functor satisfied_properties: - - property_id: conservative + - property: conservative reason: 'Let $f : X \to Y$ be a map of sets such that $F(f) : F(X) \to F(Y)$ is an isomorphism of groups. We know that $F$ is faithful, so that it reflects monomorphisms. Thus, $f$ is injective. Choose a complement $U \subseteq Y$ of $f(X) \subseteq Y$. Then $F(X) \to F(Y) = F(X) \sqcup F(U)$ is an isomorphism. This implies $F(U)=1$ and hence $U = \varnothing$.' - - property_id: faithful + - property: faithful reason: A left adjoint is faithful if and only if its unit consists of monomorphisms. So we only need to check that every set embeds into (the underlying set of) its free group. But this is clear since $X$ already embeds into the free abelian group $\IZ^{\oplus X}$, which is a quotient of the free group. - - property_id: left adjoint + - property: left adjoint reason: This functor is left adjoint to the forgetful functor. - - property_id: monomorphism-preserving + - property: monomorphism-preserving reason: 'This can be deduced from the description of the elements of a free group, but here is an abstract argument: Split monomorphisms are preserved by any functor. The only injective maps in $\Set$ that are not split are $\varnothing \hookrightarrow X$ (for non-empty $X$), and $F(\varnothing) \to F(X)$ is injective since $F(\varnothing)$ is the trivial group.' unsatisfied_properties: - - property_id: equalizer-preserving + - property: equalizer-preserving reason: 'Let $f,g : \{a,b\} \rightrightarrows \{c,d\}$ be the two constant maps, $f \equiv c$, $g \equiv d$. Their equalizer is empty, and the free group on that equalizer is trivial. Now consider the induced group homomorphisms $F(f), F(g) : F(\{a,b\}) \rightrightarrows F(\{c,d\})$ and observe that $a \cdot b^{-1}$ lies in the kernel of both homomorphisms, hence in their equalizer.' - - property_id: essentially surjective + - property: essentially surjective reason: Not every group is free (consider $\IZ/2$). - - property_id: full + - property: full reason: The map $1 = \Hom(1,1) \to \Hom(F(1),F(1)) = \Hom(\IZ,\IZ) = \IZ$ is not surjective. - - property_id: terminal-object-preserving + - property: terminal-object-preserving reason: The free group of rank $1$ is not the trivial group. diff --git a/databases/catdat/data/functors/id_Set.yaml b/databases/catdat/data/functors/id_Set.yaml index bce366bc..017ec767 100644 --- a/databases/catdat/data/functors/id_Set.yaml +++ b/databases/catdat/data/functors/id_Set.yaml @@ -6,9 +6,9 @@ description: Every category $\C$ has an identity functor $\id_{\C}$. Here, we sp nlab_link: https://ncatlab.org/nlab/show/identity+functor satisfied_properties: - - property_id: equivalence + - property: equivalence reason: This is trivial. - - property_id: representable + - property: representable reason: This functor is represented by any singleton set. unsatisfied_properties: [] diff --git a/databases/catdat/data/functors/power_set_contravariant.yaml b/databases/catdat/data/functors/power_set_contravariant.yaml index c1f71ab7..80a0f4a5 100644 --- a/databases/catdat/data/functors/power_set_contravariant.yaml +++ b/databases/catdat/data/functors/power_set_contravariant.yaml @@ -6,21 +6,21 @@ description: 'This functor maps a set $X$ to its power set $P(X)$ and a map of s nlab_link: https://ncatlab.org/nlab/show/power+set satisfied_properties: - - property_id: epimorphism-preserving + - property: epimorphism-preserving reason: 'If $f : X \to Y$ is injective, then $f^* : P(Y) \to P(X)$ is surjective, since for all $A \subseteq X$ we have $A = f^*(f_*(A))$.' - - property_id: monadic + - property: monadic reason: See Johnstone, Theorem 2.2.7. - - property_id: representable + - property: representable reason: This is because there are natural bijections $P(X) \cong \Hom(X,2)$, sending a subset to its characteristic function. unsatisfied_properties: - - property_id: coequalizer-preserving + - property: coequalizer-preserving reason: 'The power set functor preserves reflexive coequalizers, but not all coequalizers, i.e. there are maps $f,g : Y \to X$ with equalizer $E \subseteq Y$ such that $P(Y) \twoheadrightarrow P(E)$ is not the coequalizer of $f^*,g^* : P(X) \to P(Y)$: Let $X=\{0,1\}$ and $Y=\{a,b,c\}$. Define maps $f,g:Y\to X$ by $f(a)=0$, $f(b)=0$, $f(c)=1$ and $g(a)=0$, $g(b)=1$, $g(c)=0$. Their equalizer is $E = \{a\}$, so that $P(E)$ has $2$ elements. For $S=\{0\}$ one has $f^*(S)=\{a,b\}$, $g^*(S)=\{a,c\}$, and for $S=\{1\}$ one has $f^*(S)=\{c\}$, $g^*(S)=\{b\}$. Thus the coequalizer of $f^*,g^*$ is obtained from $P(Y)$ by imposing the two relations $\{a,b\} \sim \{a,c\}$ and $\{c\} \sim \{b\}$. But then it as $8-2 = 6$ elements.' - - property_id: essentially surjective + - property: essentially surjective reason: The initial object does not lie in the essential image since $P(X) \neq 0$ for all $X$. - - property_id: finitary + - property: finitary reason: Consider the sequence of projections $\cdots \twoheadrightarrow \{0,1\}^2 \twoheadrightarrow \{0,1\}^1$. Its limit in $\Set$ (i.e. colimit in $\Set^{\op}$) is $\{0,1\}^{\IN}$, which is uncountable, so that $P(\{0,1\}^{\IN})$ is also uncountable. But the colimit of the induced diagram $P(\{0,1\}^1) \hookrightarrow P(\{0,1\}^2) \hookrightarrow \cdots$ is countable since each $P(\{0,1\}^n)$ is finite. - - property_id: full + - property: full reason: 'The maps $f^* : P(Y) \to P(X)$ preserve the empty set, so take the constant map $P(X) \to P(X)$, $T \mapsto X$ for instance (where $X$ is non-empty).' - - property_id: initial-object-preserving + - property: initial-object-preserving reason: In fact, the initial object does not even lie in the essential image since $P(X) \neq 0$ for all $X$. diff --git a/databases/catdat/data/functors/power_set_covariant.yaml b/databases/catdat/data/functors/power_set_covariant.yaml index a7e1f2f9..bd7c6448 100644 --- a/databases/catdat/data/functors/power_set_covariant.yaml +++ b/databases/catdat/data/functors/power_set_covariant.yaml @@ -6,27 +6,27 @@ description: 'This functor maps a set $X$ to its power set $P(X)$ and a map of s nlab_link: https://ncatlab.org/nlab/show/power+set satisfied_properties: - - property_id: conservative + - property: conservative reason: 'Assume that $f : X \to Y$ is a map such that $f_* : P(X) \to P(Y)$ is an isomorphism. There is some $A \subseteq X$ with $Y = f_*(A)$, this proves that $f$ is surjective. It is also injective: If $x,y \in X$ satisfy $f(x) = f(y)$, then $f_*(\{x\}) = f_*(\{y\})$, and hence $\{x\} = \{y\}$, i.e. $x = y$.' - - property_id: epimorphism-preserving + - property: epimorphism-preserving reason: 'If $f : X \to Y$ is surjective, then $f_* \circ f^* = \id_{P(Y)}$, so that $f^*$ is surjective.' - - property_id: faithful + - property: faithful reason: 'Let $f,g : X \rightrightarrows Y$ be two maps with $f_* = g_* : P(X) \rightrightarrows P(Y)$. Then $\{f(x)\} = f_*(\{x\}) = g_*(\{x\}) = \{g(x)\}$ and hence $f(x) = g(x)$ for all $x \in X$.' - - property_id: monomorphism-preserving + - property: monomorphism-preserving reason: 'If $f : X \to Y$ is injective, then $f^* \circ f_* = \id_{P(X)}$, so that $f_*$ is injective.' unsatisfied_properties: - - property_id: coequalizer-preserving + - property: coequalizer-preserving reason: 'Let $X := \{x,y\}$. Consider the two maps $x,y : \{0\} \rightrightarrows X$. Their coequalizer $Q = X / (x = y)$ has just one element, so that $P(Q)$ has two elements. The induced maps $x_*,y_* : P(\{0\}) \rightrightarrows P(X)$ (which already agree on the empty set) have coequalizer $P(X) / (\{x\} = \{y\})$, which has $3$ elements. So it cannot be $P(Q)$.' - - property_id: equalizer-preserving + - property: equalizer-preserving reason: 'Any pair of distinct surjective maps $f,g : X \rightrightarrows Y$ provides a counterexample: Their equalizer $E$ is a proper subset of $X$, so that $P(E)$ cannot contain $X$. But $f_*(X) = Y = g_*(X)$ shows that $X$ is contained in the equalizer of $f_*,g_* : P(X) \rightrightarrows P(Y)$.' - - property_id: essentially surjective + - property: essentially surjective reason: Every power set is non-empty. - - property_id: finitary + - property: finitary reason: The filtered colimit $\IN = \bigcup_{n \geq 0} \IN_{\leq n}$ is not preserved by $P$, since $\bigcup_{n \geq 0} P(\IN_{\leq n})$ just consists of the finite subsets of $\IN$. - - property_id: full + - property: full reason: Take any map $P(X) \to P(X)$ that does not preserve the empty set, say the constant map with value $X$ (for $X \neq \varnothing$). - - property_id: initial-object-preserving + - property: initial-object-preserving reason: We have $2^0 \neq 0$. - - property_id: terminal-object-preserving + - property: terminal-object-preserving reason: We have $2^1 \neq 1$. diff --git a/databases/catdat/scripts/seed.ts b/databases/catdat/scripts/seed.ts index c053348b..c7c00bdc 100644 --- a/databases/catdat/scripts/seed.ts +++ b/databases/catdat/scripts/seed.ts @@ -206,7 +206,7 @@ function seed_categories() { category.morphisms, category.description, category.nlab_link, - category.dual_category_id || null, + category.dual_category || null, ) for (const tag of category.tags) { @@ -239,7 +239,7 @@ function seed_categories() { for (const entry of category.satisfied_properties) { property_assignment_insert.run( category.id, - entry.property_id, + entry.property, 1, entry.reason, entry.check_redundancy === false ? 0 : 1, @@ -249,7 +249,7 @@ function seed_categories() { for (const entry of category.unsatisfied_properties) { property_assignment_insert.run( category.id, - entry.property_id, + entry.property, 0, entry.reason, entry.check_redundancy === false ? 0 : 1, @@ -259,7 +259,7 @@ function seed_categories() { for (const comment_obj of category.category_property_comments ?? []) { property_comment_insert.run( category.id, - comment_obj.property_id, + comment_obj.property, comment_obj.comment, ) } @@ -306,7 +306,7 @@ function seed_category_properties() { property.relation, property.description, property.nlab_link || null, - property.dual_property_id || null, + property.dual_property || null, Number(property.invariant_under_equivalences), ) @@ -442,7 +442,7 @@ function seed_functor_properties() { property.relation, property.description, property.nlab_link || null, - property.dual_property_id || null, + property.dual_property || null, Number(property.invariant_under_equivalences), ) } @@ -574,11 +574,11 @@ function seed_functors() { ) for (const entry of functor.satisfied_properties) { - property_assignment_insert.run(functor.id, entry.property_id, 1, entry.reason) + property_assignment_insert.run(functor.id, entry.property, 1, entry.reason) } for (const entry of functor.unsatisfied_properties) { - property_assignment_insert.run(functor.id, entry.property_id, 0, entry.reason) + property_assignment_insert.run(functor.id, entry.property, 0, entry.reason) } } diff --git a/databases/catdat/scripts/seed.types.ts b/databases/catdat/scripts/seed.types.ts index 2d95964d..6502df3a 100644 --- a/databases/catdat/scripts/seed.types.ts +++ b/databases/catdat/scripts/seed.types.ts @@ -23,7 +23,7 @@ export type CategoryYaml = { morphisms: string description: string | null nlab_link: string | null - dual_category_id?: string | null + dual_category?: string | null tags: string[] related_categories: string[] satisfied_properties: PropertyEntry[] @@ -32,13 +32,13 @@ export type CategoryYaml = { special_morphisms: Record comments?: string[] category_property_comments?: { - property_id: string + property: string comment: string }[] } type PropertyEntry = { - property_id: string + property: string reason: string check_redundancy?: boolean } @@ -57,7 +57,7 @@ export type CategoryPropertyYaml = { relation: string description: string nlab_link?: string | null - dual_property_id?: string | null + dual_property?: string | null invariant_under_equivalences: boolean related_properties: string[] } @@ -92,7 +92,7 @@ export type FunctorPropertyYaml = { relation: string description: string nlab_link?: string | null - dual_property_id?: string | null + dual_property?: string | null invariant_under_equivalences: boolean }