Small LaTeX adjustments

CONTRIBUTING.md

@@ -169,9 +169,11 @@ As a practical guideline, avoid introducing more than four properties (or four c
 
 ### Conventions
 
-1. Use `\varnothing` to display the empty set, not `\emptyset`.
-2. Write `non-empty`, not `nonempty`. Same for `non-unital`, `non-expansive`, etc.
-3. For LaTeX symbols that are used repeatedly, in particular category-theoretic notation, define a LaTeX macro in [macros.yaml](databases/catdat/data/macros.yaml).
+1. Write `non-empty`, not `nonempty`. Same for `non-unital`, `non-expansive`, etc.
+2. Use `\varnothing` to display the empty set, not `\emptyset`.
+3. For declarations of functions or morphisms use `f : X \to Y`, not `f \colon X \to Y`.
+4. For definitions use `\coloneqq` instead of `:=`.
+5. For LaTeX symbols that are used repeatedly, in particular category-theoretic notation, define a LaTeX macro in [macros.yaml](databases/catdat/data/macros.yaml).
 
 ### Responsible Use of AI
 

content/cocongruences_of_groups.md

@@ -62,7 +62,7 @@ $$c \circ i_1 = u_1 \circ i_1, \quad c \circ i_2 = u_2 \circ i_2,$$
 
 where $u_1,u_2 : Y \rightrightarrows Y \sqcup_{i_2,X,i_1} Y$ are the pushout inclusions satisfying $u_1 i_2 = u_2 i_1$. We will not use the fact that the cocongruence is cosymmetric; this will follow automatically. Define the monomorphism
 
-$$E := \eq(i_1,i_2) \hookrightarrow X.$$
+$$E \coloneqq \eq(i_1,i_2) \hookrightarrow X.$$
 
 Since $i_1$ and $i_2$ agree on $E$, there exists a unique morphism
 
@@ -74,7 +74,7 @@ We must show that $\varphi$ is an isomorphism. It is clearly an epimorphism, sin
 
 We will show that even the morphism
 
-$$\gamma := c \circ \varphi : X \sqcup_E X \to Y \sqcup_{i_2,X,i_1} Y$$
+$$\gamma \coloneqq c \circ \varphi : X \sqcup_E X \to Y \sqcup_{i_2,X,i_1} Y$$
 
 is a monomorphism. It is characterized by
 

content/comphaus_copresentable.md

@@ -13,16 +13,16 @@ A good overview of some of these results is contained in the introduction of [HN
 We first prove a couple preliminary results.
 
 ::: Lemma 1
-Let $\I$ be a cofiltered category, and let $X : \I \to \CompHaus$ be a cofiltered diagram in which $X_i$ is non-empty for each $i\in \I$. Then $\lim_i X_i$ is also non-empty.
+Let $\I$ be a cofiltered category, and let $X : \I \to \CompHaus$ be a cofiltered diagram in which $X_i$ is non-empty for each $i\in \I$. Then $\lim_{i\in \I} X_i$ is also non-empty.
 :::
 
 _Proof._
 Consider the product space $\prod_{i\in \I} X_i$. Now for each morphism $f : i \to j$ in $\I$, define the subset
 
-$$\textstyle E_f := \bigl\{ x \in \prod_{i \in \I} X_i \mid X_f(x_i) = x_j \bigr\}.$$
+$$\textstyle E_f \coloneqq \bigl\{ x \in \prod_{i \in \I} X_i \mid X_f(x_i) = x_j \bigr\}.$$
 
 Then each $E_f$ is a closed subset.
-Next, we prove that the collection $\{ E_f : f \in \Mor(\I) \}$ has the finite intersection property, i.e. that $\bigcap_{f\in F} E_f$ is non-empty for every finite set $F \subseteq \Mor(\I)$. For $f\in F$ we write $f : i_f \to j_f$. Then the diagram with objects $J := \{ i_f \mid f \in F \} \cup \{ j_f \mid f \in F \}$ and morphisms $\{ f \mid f \in F \}$ has a cone with vertex $k \in \I$ and morphisms $g_i : k \to i$ for each $i \in J$. Now choose $y \in X_k$, and define $x \in \prod_{i \in \I} X_i$ such that $x_i = X_{g_i}(y)$ if $i \in J$, with arbitrary choices of $x_i \in X_i$ for all other $i$. We then see that $x \in \bigcap_{f\in F} E_f$, which finishes the proof of the claim.
+Next, we prove that the collection $\{ E_f : f \in \Mor(\I) \}$ has the finite intersection property, i.e. that $\bigcap_{f\in F} E_f$ is non-empty for every finite set $F \subseteq \Mor(\I)$. For $f\in F$ we write $f : i_f \to j_f$. Then the diagram with objects $J \coloneqq \{ i_f \mid f \in F \} \cup \{ j_f \mid f \in F \}$ and morphisms $\{ f \mid f \in F \}$ has a cone with vertex $k \in \I$ and morphisms $g_i : k \to i$ for each $i \in J$. Now choose $y \in X_k$, and define $x \in \prod_{i \in \I} X_i$ such that $x_i = X_{g_i}(y)$ if $i \in J$, with arbitrary choices of $x_i \in X_i$ for all other $i$. We then see that $x \in \bigcap_{f\in F} E_f$, which finishes the proof of the claim.
 Since $\prod_{i \in \I} X_i$ is compact, that implies that the intersection of all $E_f$ is non-empty. But that intersection is precisely $\lim_{i \in \I} X_i$. <span class="qed">$\square$</span>
 
 ::: Lemma 2
@@ -43,7 +43,7 @@ The functor $\Hom({-}, [0, 1]) : \CompHaus^{\op} \to \Set$ is monadic. (Original
 _Proof._
 We use the crude monadicity theorem (see e.g. [SGL92](#references), Thm. IV.4.2). First, the functor has a left adjoint $S \mapsto [0, 1]^S$ with the evident isomorphism
 
-$$\Hom_{\CompHaus}(X, [0, 1]^S) \simeq \Hom_{\Set}(S, \Hom_{\CompHaus}(X, [0, 1])).$$
+$$\Hom_{\CompHaus}\bigl(X, [0, 1]^S\bigr) \cong \Hom_{\Set}\bigl(S, \Hom_{\CompHaus}(X, [0, 1])\bigr).$$
 
 To see the functor is conservative, suppose we have a continuous function $f : X \to Y$ such that $f^* : \Hom(Y, [0, 1]) \to \Hom(X, [0, 1])$ is a bijection. Then for any $x_1, x_2 \in X$ with $x_1 \ne x_2$, there exists $\varphi : X \to [0, 1]$ with $\varphi(x_1) = 0$ and $\varphi(x_2) = 1$ by Urysohn's lemma. Since $f^*$ is surjective, there exists $\psi : Y \to [0, 1]$ with $\varphi = \psi \circ f$; thus, we must have $f(x_1) \ne f(x_2)$. Likewise, we know the image of $f$ is closed. If this image is not all of $Y$, then by Urysohn's lemma there exists nonzero $\varphi : Y \to [0, 1]$ which is zero on the image. But then $\varphi \circ f = 0 \circ f$, contradicting the injectivity of $f^*$. Thus, $f$ is a bijective continuous function, and therefore a homeomorphism.
 
@@ -57,27 +57,29 @@ $$\Hom(B, [0,1]) ~\overset{f^*}{\underset{g^*}{\rightrightarrows}}~ \Hom(A, [0,
 
 is a coequalizer diagram. We first define $s : \Hom(E,[0,1]) \to \Hom(A,[0,1])$ by choosing a Tietze extension of each continuous function $E \to [0,1]$. Now, for each $\varphi \in \Hom(A,[0,1])$, we can define a continuous function on $\im(f) \cup \im(g) \subseteq B$ to be $\varphi \circ r$ on $\im(f)$, and $s(i^*(\varphi))\circ r$ on $\im(g)$. Note that on the overlap $\im(f)\cap \im(g) = f(E) = g(E)$, the first expression gives $f(e) \mapsto \varphi(e)$, and the second expression gives $g(e) \mapsto s(i^*(\varphi))(e) = \varphi(e)$, so we have indeed given a well-defined function on $\im(f)\cup\im(g)$. Choosing a Tietze extension of this function to a function $B\to [0,1]$ for each $\varphi$, we get a map $t : \Hom(A,[0,1]) \to \Hom(B,[0,1])$. By construction, we have $i^* s = \id$, $f^* t = \id$, and $g^* t = s i^*$, so we have shown that the diagram above is a split coequalizer. <span class="qed">$\square$</span>
 
-This shows that $\CompHaus^{\op}$ is equivalent to the category of algebras over the monad $S \mapsto \Hom_{\CompHaus}([0, 1]^S, [0, 1])$. We may view such algebras as being models of the infinitary algebraic theory of all continuous functions $[0,1]^S \to [0,1]$. In fact, we can show that any such function only depends on countably many coordinates in the domain, so that operations of this theory will be generated by the continuous functions $[0,1]^\omega \to [0,1]$. Indeed, we get the following somewhat stronger result:
+This shows that $\CompHaus^{\op}$ is equivalent to the category of algebras over the monad
+$$S \mapsto \Hom_{\CompHaus}\bigl([0, 1]^S, [0, 1]\bigr).$$
+We may view such algebras as being models of the infinitary algebraic theory of all continuous functions $[0,1]^S \to [0,1]$. In fact, we can show that any such function only depends on countably many coordinates in the domain, so that operations of this theory will be generated by the continuous functions $[0,1]^\omega \to [0,1]$. Indeed, we get the following somewhat stronger result:
 
 ::: Proposition 4
 The object $[0,1]$ of $\CompHaus$ is $\aleph_1$-copresentable. (Originally proved in [GU71](#references), 6.5(c))
 :::
 
 _Proof._
-Suppose we have an $\aleph_1$-cofiltered limit $X = \lim_{i\in \I} X_i$ with projections $p_i : X \to X_i$, and a continuous function $\varphi : X \to [0,1]$. For the time being, fix $n\in \IN_{>0}$. Then for any $x\in X$, there exists an interval neighborhood $N_x$ of $\varphi(x)$ of diameter at most $1/n$ &mdash; for example, we can take $N_x := (\varphi(x) - 1/(2n), \varphi(x) + 1/(2n)) \cap [0,1]$. We can also take a basic open neighborhood whose image is contained in $N_x$; by lemma 2, we can write that basic open neighborhood in the form $p_i^{-1}(V)$ where $V$ is an open subset of $X_i$.
+Suppose we have an $\aleph_1$-cofiltered limit $X = \lim_{i\in \I} X_i$ with projections $p_i : X \to X_i$, and a continuous function $\varphi : X \to [0,1]$. For the time being, fix $n\in \IN_{>0}$. Then for any $x\in X$, there exists an interval neighborhood $N_x$ of $\varphi(x)$ of diameter at most $1/n$ &mdash; for example, we can take $N_x \coloneqq (\varphi(x) - 1/(2n), \varphi(x) + 1/(2n)) \cap [0,1]$. We can also take a basic open neighborhood whose image is contained in $N_x$; by lemma 2, we can write that basic open neighborhood in the form $p_i^{-1}(V)$ where $V$ is an open subset of $X_i$.
 By compactness of $X$, we may take finitely many such basic open neighborhoods of the form $p_i^{-1}(V)$ which cover $X$. Again using the assumption that $\I$ is cofiltered, we may assume that $i$ is the same for each neighborhood. In particular, we see that whenever we have $x, y\in X$ with $p_i(x) = p_i(y)$, then $|\varphi(x) - \varphi(y)| < 1/n$.
 
 To summarize, we have shown that for each $n \in \IN_{>0}$, there exists $i \in \I$ and a finite cover $\{ V_\lambda \mid \lambda \in \Lambda \}$ of $\im(p_i)$ such that whenever $p_i(x), p_i(y) \in V_\lambda$ for some $\lambda$, we have $|\varphi(x) - \varphi(y)| < 1/n$. Now choose such a $i_n$ and associated finite cover of $\im(p_{i_n})$ for each $n$. Then use the fact that $\I$ is $\aleph_1$-cofiltered to take a cone $(j, f_n : j \to i_n)$ of the objects $i_n$. We then see that whenever we have $x, y\in X$ with $p_j(x) = p_j(y)$, then $\varphi(x) = \varphi(y)$. Thus, $\varphi$ induces a well-defined function on the image of $p_j$. This function is continuous, since by construction for any $n\in \IN_{>0}$ and $x \in X$, there is a neighborhood $V$ of $p_j(x)$ such that whenever $p_j(y)\in V$ as well, $|\varphi(y) - \varphi(x)| < 1/n$. By the Tietze extension theorem, this induced function can then be extended to a continuous function $\psi : X_j \to [0,1]$. Then $\varphi = \psi \circ p_j$.
 
 This shows the canonical map
-$$\textstyle \colim_{i\in \I^{\op}} \Hom(X_i, [0,1]) \to \Hom(\lim_{i\in \I} X_i, [0,1])$$
+$$\textstyle \colim_{i\in \I^{\op}} \Hom(X_i, [0,1]) \to \Hom\bigl(\lim_{i\in \I} X_i, [0,1]\bigr)$$
 is surjective.
 
-Likewise, suppose we have $\alpha, \beta : X_i \rightrightarrows [0,1]$ such that $\alpha \circ p_i = \beta \circ p_i$. For each $n\in \IN_{>0}$, consider the set $D_n := \{ x \in X_i \mid |\alpha(x) - \beta(x)| \ge 1/n \}$. Note that $D_n$ is a closed subset of $X_i$, so it is compact. For any $x \in D_n$, we must have $x \notin \im(p_i)$. Using the contrapositive of lemma 1, we can conclude that there exists $f : j \to i$ such that $x \not\in \im(X_f)$. Indeed, suppose not: then the slice category $\I / i$ is cofiltered, with the limit over this slice category being the same as the limit over $\I$. Also, by the contrary assumption, we have that $x \in \im(X_f)$ for any $f : j \to i$, so $X_f^{-1}(\{ x \})$ is non-empty. Therefore, by lemma 1, the limit $p_i^{-1}(\{ x \})$ would be non-empty, contradicting the assumption.
+Likewise, suppose we have $\alpha, \beta : X_i \rightrightarrows [0,1]$ such that $\alpha \circ p_i = \beta \circ p_i$. For each $n\in \IN_{>0}$, consider the set $D_n \coloneqq \{ x \in X_i \mid |\alpha(x) - \beta(x)| \ge 1/n \}$. Note that $D_n$ is a closed subset of $X_i$, so it is compact. For any $x \in D_n$, we must have $x \notin \im(p_i)$. Using the contrapositive of lemma 1, we can conclude that there exists $f : j \to i$ such that $x \not\in \im(X_f)$. Indeed, suppose not: then the slice category $\I / i$ is cofiltered, with the limit over this slice category being the same as the limit over $\I$. Also, by the contrary assumption, we have that $x \in \im(X_f)$ for any $f : j \to i$, so $X_f^{-1}(\{ x \})$ is non-empty. Therefore, by lemma 1, the limit $p_i^{-1}(\{ x \})$ would be non-empty, contradicting the assumption.
 Now each $D_n \setminus \im(X_f)$ is open and we have just shown such sets cover $D_n$; taking a finite subcover and then using the cofiltering assumption again, we conclude that there is a single $f_n : j_n \to i$ such that $\im(X_{f_n})$ is disjoint from $D_n$. Using the $\aleph_1$-cofiltering assumption to take a cone of the $f_n$, we get that there is $f : k \to i$ such that $\im(X_f)$ is disjoint from all $D_n$. This implies that $\alpha \circ X_f = \beta \circ X_f$.
 
 This shows the canonical map
-$$\textstyle \colim_{i\in \I^{\op}} \Hom(X_i, [0,1]) \to \Hom(\lim_{i\in I} X_i, [0,1])$$
+$$\textstyle \colim_{i\in \I^{\op}} \Hom(X_i, [0,1]) \to \Hom\bigl(\lim_{i\in I} X_i, [0,1]\bigr)$$
 is injective. <span class="qed">$\square$</span>
 
 ::: Corollary 5
@@ -86,8 +88,8 @@ The category $\CompHaus$ is locally $\aleph_1$-copresentable.
 
 _Proof._
 It suffices to show that the monad
-$$S \mapsto \Hom_{\CompHaus}([0, 1]^S, [0, 1])$$
-is $\aleph_1$-accessible. This monad is the composition of
+$$S \mapsto \Hom_{\CompHaus}\bigl([0, 1]^S, [0, 1]\bigr)$$
+is $\aleph_1$-accessible. This functor is the composition of
 $$[0, 1]^{-} : \Set \to \CompHaus^{\op}$$
 followed by
 $$\Hom_{\CompHaus}({-}, [0,1]) : \CompHaus^{\op} \to \Set.$$

content/congruences_in_rel.md

@@ -12,7 +12,7 @@ Let $i : E \hookrightarrow X \times X$ be a congruence in $\Rel$. Recall that pr
 $$R_* : P(X) \to P(Y),\, S \mapsto \bigl\{ y\in Y \mid \exists x\in S, (x, y) \in R \bigr\}$$
 is injective, and it is an isomorphism if and only if $R$ is the graph of a bijection $X \to Y$ in $\Set$.
 In particular, we get
-$$i_* : P(E) \to P(X + X) \simeq P(X) \times P(X)$$
+$$i_* : P(E) \to P(X + X) \cong P(X) \times P(X)$$
 which must be injective. It must also preserve arbitrary unions and in particular be inclusion-preserving. From the assumption that $i$ is a congruence, since the functor $(P, {-}_*) : \Rel \to \Set$ is representable by the object 1, we see that $i_*$ must have image given by an equivalence relation $\sim$ on $P(X)$. Note also that since $i_*$ preserves arbitrary unions, we must have that $\sim$ respects arbitrary unions.
 
 Since the symmetry morphism $s : E \to E$ satisfies $s^2 = \id$, it must be the graph of an involution $s_0$ on $E$, where $s_0$ is a morphism in $\Set$.

content/foundations.md

@@ -45,11 +45,11 @@ A _category_ $\C$ consists of a pair of collections $O, M$, whose elements are c
 - $t : M \to O$ (_target_),
 - $c : M \times_O M \to M$ (_composition_),
 
-such that the usual [axioms of a category](<https://en.wikipedia.org/wiki/Category_(mathematics)>) are satisfied. The domain of $c$ consists of all pairs of morphisms $(f,g)$ with $s(f) = t(g)$, and we write $f \circ g := c(f,g)$ for their composition. Instead of $i(X)$ one usually writes $\id_X$ for the identity morphism of $X$. Formally, a category is a tuple
+such that the usual [axioms of a category](<https://en.wikipedia.org/wiki/Category_(mathematics)>) are satisfied. The domain of $c$ consists of all pairs of morphisms $(f,g)$ with $s(f) = t(g)$, and we write $f \circ g \coloneqq c(f,g)$ for their composition. Instead of $i(X)$ one usually writes $\id_X$ for the identity morphism of $X$. Formally, a category is a tuple
 
 $$\C = (O,M,i,s,t,c)$$
 
-of collections (and hence a collection itself). We write $\Ob(\C) := O$ and $\Mor(\C) := M$. Instead of $X \in \Ob(\C)$, we often write $X \in \C$.
+of collections (and hence a collection itself). We write $\Ob(\C) \coloneqq O$ and $\Mor(\C) \coloneqq M$. Instead of $X \in \Ob(\C)$, we often write $X \in \C$.
 
 When $f \in \Mor(\C)$ is a morphism with $s(f) = X$ and $t(f) = Y$, we write
 $$f : X \to Y.$$

content/walking_parallel_pair_sifted_colimit.md

@@ -6,22 +6,22 @@ author: Yuto Kawase
 
 ## The walking parallel pair has sifted colimits
 
-We will prove that the [walking parallel pair](/category/walking_pair) $\{u,v \colon 0 \rightrightarrows 1\}$ has [sifted colimits](/category-property/sifted_colimits).
+We will prove that the [walking parallel pair](/category/walking_pair) $\{u,v : 0 \rightrightarrows 1\}$ has [sifted colimits](/category-property/sifted_colimits).
 
-Let $D \colon \C \to \{u,v \colon 0 \rightrightarrows 1\}$ be a functor from a sifted category. If the object $1$ is not contained in the image under $D$, the object $0$ gives a colimit of $D$ because the sifted category $\C$ is connected.
+Let $D : \C \to \{u,v : 0 \rightrightarrows 1\}$ be a functor from a sifted category. If the object $1$ is not contained in the image under $D$, the object $0$ gives a colimit of $D$ because the sifted category $\C$ is connected.
 In what follows, we suppose that there is an object $c_0 \in \C$ such that $D(c_0)=1$.
 
-We first claim that for every object $c \in \C$ such that $D(c)=0$, there is a morphism $f \colon c \to x$ with $D(x)=1$; moreover, which of $u$ and $v$ such a morphism is sent to by $D$ is independent of the choice of $f$.
+We first claim that for every object $c \in \C$ such that $D(c)=0$, there is a morphism $f : c \to x$ with $D(x)=1$; moreover, which of $u$ and $v$ such a morphism is sent to by $D$ is independent of the choice of $f$.
 The existence of $f$ is easy.
 Indeed, since $\C$ is sifted, there is a cospan $c \rightarrow x \leftarrow c_0$, and $D(x)=1$ follows from $D(c_0)=1$.
 
-To show the independence of the value of $D(f)$, suppose that there are morphisms $f \colon c \to x$ and $g \colon c \to y$ such that $D(f)=u$ and $D(g)=v$.
-Since $\C$ is sifted, there is a cospan consisting of $p\colon x \rightarrow z$ and $q \colon z \leftarrow y$.
+To show the independence of the value of $D(f)$, suppose that there are morphisms $f : c \to x$ and $g : c \to y$ such that $D(f)=u$ and $D(g)=v$.
+Since $\C$ is sifted, there is a cospan consisting of $p : x \rightarrow z$ and $q : z \leftarrow y$.
 Since $\C$ is sifted again, two cospans $(p \circ f, q \circ g)$ and $(\id_c, \id_c)$ are connected to each other, that is, there are a zigzag consisting of
 
-- $s_i \colon d_{i-1} \rightarrow e_i$,
-- $t_i \colon e_i \leftarrow d_i$ $(1 \le i \le n)$, and
-- parallel pairs $(l_i,r_i) \colon c \rightrightarrows d_i$ $(0 \le i \le n)$
+- $s_i : d_{i-1} \rightarrow e_i$,
+- $t_i : e_i \leftarrow d_i$ $(1 \le i \le n)$, and
+- parallel pairs $(l_i,r_i) : c \rightrightarrows d_i$ $(0 \le i \le n)$
 
 such that
 
@@ -44,10 +44,10 @@ By the claim, each object $c \in \C$ can be classified exclusively into the foll
 2. $D(c)=0$ and there is a morphism from itself sent to $u$ by $D$;
 3. $D(c)=0$ and there is a morphism from itself sent to $v$ by $D$.
 
-Now, we have a cocone $(\alpha_c \colon D(c) \to 1)_{c \in \C}$ under $D$ by letting $\alpha_c \coloneqq \id_1$ if $c$ is classified into the first case, $\alpha_c \coloneqq u$ for the second case, and $\alpha_c \coloneqq v$ for the third case.
+Now, we have a cocone $(\alpha_c : D(c) \to 1)_{c \in \C}$ under $D$ by letting $\alpha_c \coloneqq \id_1$ if $c$ is classified into the first case, $\alpha_c \coloneqq u$ for the second case, and $\alpha_c \coloneqq v$ for the third case.
 Moreover, this is a unique cocone under $D$:
 If $\beta$ is another cocone, its vertex should be $1$ by the existence of $c_0$.
 If $c \in \C$ is classified into the first case, $\beta_c$ should be the identity.
-For the second case, taking a morphism $f \colon c \to x$ such that $D(f)=u$, we can obtain $\beta_c = \beta_x \circ D(f) = D(f) = u$.
+For the second case, taking a morphism $f : c \to x$ such that $D(f)=u$, we can obtain $\beta_c = \beta_x \circ D(f) = D(f) = u$.
 Similarly, we have $\beta_c = v$ for the third case.
 This concludes $\beta = \alpha$, and since there is no non-trivial endomorphism on the vertex $1$, $\alpha$ gives a colimit.