Add an example of a cocomplete category without equalizers

.cspell.json

@@ -65,6 +65,7 @@
 		"cocomplete",
 		"cocompleteness",
 		"cocompletion",
+		"cocompletions",
 		"cocone",
 		"cocones",
 		"cocongruence",
@@ -215,6 +216,7 @@
 		"Kerodon",
 		"Kolmogorov",
 		"Kunen",
+		"Kuratowski",
 		"Lawvere",
 		"libsql",
 		"Lindelöf",

content/free-cocompletion.md

@@ -0,0 +1,165 @@
+---
+title: The free cocompletion of a locally small category
+description: We investigate the properties of the free cocompletion of a locally small category.
+---
+
+## The free cocompletion of a locally small category
+
+Let $\C$ be a locally small category. All results here can easily be adapted to the case that $\C$ is locally essentially small, and we do not assume that $\C$ is small. Then $\widehat{\C}$ denotes its _free cocompletion_ (often called $P\C$ in the literature when $\C$ is not assumed to be small), which is the full subcategory of $[\C^{\op},\Set]$ consisting presheaves
+$$F : \C^{\op} \to \Set$$
+that are _small_. This condition can be described in many equivalent ways:
+
+1. $F$ is a small colimit of representable functors.
+2. There is a small category $\I$ such that $F$ is the left Kan extension of a presheaf on $\I$ along a functor $\I \to \C$.
+3. There is small subcategory $\I \subseteq \C$ such that $F$ is the left Kan extension of its restriction to $\I$.
+4. The category of elements $\int F$ is [finally small](https://ncatlab.org/nlab/show/finally+small).
+
+Here, the objects of $\int F$ are pairs $(X,a)$, where $X \in \C$ and $a \in F(X)$, and a morphism $(X,a) \to (Y,b)$ is a morphism $f : X \to Y$ with $F(f)(b) = a$. The equivalence of the conditions (1), (2), (3) is proven as Proposition 4.83 in Kelly's book [Basic Concepts of Enriched Category Theory](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.html). The implication (1) $\implies$ (4) is proven as Proposition 3.7 in <a href="https://doi.org/10.1007/s10485-021-09671-9">Kan Extensions are Partial Colimits</a> by Perrone-Tholen (but there must be earlier references). The implication (4) $\implies$ (1) follows from the [co-Yoneda Lemma](https://ncatlab.org/nlab/show/co-Yoneda+lemma)
+$$F \cong \colim_{(X,a) \in \int F} \Hom(-,X)$$
+and the fact that final functors do not "change" colimits; see Proposition 2.5.2 in <a href="https://ncatlab.org/nlab/show/Categories+and+Sheaves" target="_blank">Kashiwara-Schapira</a>.
+
+In contrast to the full presheaf category $[\C^{\op},\Set]$, its subcategory $\widehat{\C}$ of small presheaves is always locally essentially small:
+
+::: Lemma 1
+If $\C$ is a locally small category, then $\widehat{\C}$ is locally essentially small.
+:::
+
+::: Proof
+Let $F : \C^{\op} \to \Set$ be a small presheaf, so that $F \cong \colim_i \Hom(-,X_i)$ for a small diagram $X : \I \to \C$. For every other (small) presheaf $G : \C^{\op} \to \Set$ we compute, using the Yoneda Lemma,
+$$\textstyle \Hom(F,G) \cong \lim_i \Hom(\Hom(-,X_i),G) \cong \lim_i G(X_i),$$
+and the latter is a set.
+:::
+
+But it is usually not locally small:
+
+::: Lemma 2
+If $\widehat{\C}$ is locally small, then $\C$ is small.
+:::
+
+Disclaimer: This result and its proof are not relevant for category theory and are also depending on implementation details of set theory. That $\widehat{\C}$ is locally essentially small is only what matters.
+
+::: Proof
+If $\C$ is empty, there is nothing to prove. Otherwise, choose an object $X \in \C$. Consider the collection of morphisms $\Hom(-,X) \to \Hom(-,X)$, which is surely isomorphic to the set $\Hom(X,X)$. By assumption, it actually _is_ a set. It follows that $\{\id_{\Hom(-,X)}\}$ is a set, and therefore also that $\id_{\Hom(-,X)}$ is a set. This natural transformation is a map that associates to every object $Y \in \Ob(\C)$ the map $\id_{\Hom(Y,X)}$. If we model a map as a set of ordered pairs and ordered pairs as Kuratowski pairs, we get
+
+$$
+\begin{align*}
+\id_{\Hom(-,X)} & = \bigl\{(Y,\id_{\Hom(Y,X)}) : Y \in \Ob(\C)\bigr\} \\
+& = \bigl\{\{\{Y\},\{Y,\id_{\Hom(Y,X)}\}\} : Y \in \Ob(\C)\bigr\}
+\end{align*}
+$$
+
+This construction shows $\Ob(\C) \subseteq \bigcup \bigcup \id_{\Hom(-,X)}$, so that $\Ob(\C)$ is indeed a set.
+:::
+
+::: Lemma 3
+If $\C$ is a locally small category, then $\widehat{\C}$ is cocomplete. Colimits can be constructed objectwise.
+:::
+
+::: Proof
+This follows from cocompleteness of $[\C^{\op},\Set]$ with objectwise constructed colimits and the third characterization of small presheaves above. Details can be found as Proposition 5.34 in Kelly's book.
+:::
+
+The existence of limits in $\widehat{\C}$ is a much more complicated issue, see the paper [_Limits of small functors_](https://arxiv.org/pdf/math/0610439) by Day-Lack. The following result is useful in this regard. Namely, it shows that $\widehat{\C}$ has limits of a given type if and only if small functors are closed under these limits taken in the category of all presheaves.
+
+::: Lemma 4
+For every $X \in \C$ the evaluation functor $\ev_X : \widehat{\C} \to \Set$, $F \mapsto F(X)$ is continuous. In particular, the inclusion functor $\widehat{\C} \hookrightarrow [\C^{\op},\Set]$ is continuous, and every limit that exists in $\widehat{\C}$ is an objectwise limit.
+:::
+
+::: Proof
+By the Yoneda Lemma, the evaluation functor is represented by $\Hom(-,X)$. Thus, it is continuous.
+:::
+
+::: Lemma 5
+A morphism $\alpha : F \to G$ in $\widehat{\C}$ is a monomorphism (resp. epimorphism) if and only if for every $X \in \C$ the map $\alpha(X) : F(X) \to G(X)$ injective (resp. surjective).
+:::
+
+::: Proof
+The direction $\impliedby$ is trivial in each case. For the direction $\implies$, the evaluation functor $\ev_X : \widehat{\C} \to \Set$ is continuous by Lemma 4 and therefore preserves monomorphisms. Furthermore, it is also cocontinuous by Lemma 3 and therefore preserves epimorphisms.
+:::
+
+::: Lemma 6
+If $\C$ is a locally small category, then $\widehat{\C}$ is mono-regular. Actually, every monomorphism is an effective monomorphism. Moreover, monomorphisms are stable under filtered colimits.
+:::
+
+::: Proof
+The first statement is a formal consequence of the fact that every monomorphism in $\Set$ is effective and the already established facts that monomorphisms and pushouts can be understood objectwise. For similar reasons, the second statement is a formal consequence of the corresponding fact for $\Set$.
+:::
+
+::: Lemma 7
+If $\C$ is a locally small category, then $\widehat{\C}$ is infinitary extensive.
+:::
+
+::: Proof
+We need to prove that for a family of small presheaves $(P_i)_{i \in I}$ the coproduct functor
+$$\textstyle \prod_{i \in I} \widehat{\C} / P_i \to \widehat{\C}/\coprod_{i \in I} P_i$$
+is an equivalence of categories. Since $\Set$ is infinitary extensive, also $[\C^{\op},\Set]$ is infinitary extensive, so that the coproduct functor
+$$\textstyle \prod_{i \in I} [\C^{\op},\Set] / P_i \to [\C^{\op},\Set]/\coprod_{i \in I} P_i$$
+is an equivalence of categories. Since $\widehat{\C}$ is a full subcategory of $[\C^{\op},\Set]$ that is closed under coproducts, it remains to prove that if a coproduct of presheaves $\coprod_{i \in I} F_i$ is small, then each $F_i$ is small. For this, it suffices to prove for two presheaves $F,G$ for which $F+G$ is small, that $F$ is small. The category of elements $\int (F+G)$ identifies with $\int F + \int G$. Thus, the claim follows from the next lemma.
+:::
+
+::: Lemma 8
+Let $\C,\D$ be two categories. Assume that the coproduct $\C + \D$ is finally small. Then $\C$ is finally small.
+:::
+
+::: Proof
+Assume that $\I \to \C + \D$ is a final functor, where $\I$ is small. Since $\Cat$ is extensive, we get a decomposition $\I = \I_\C + \I_\D$ with two functors $\I_\C \to \C$ and $\I_\D \to \D$. For every $X \in \C$ the comma category $X \downarrow I_\C$ identifies with the comma category $X \downarrow I$, which is connected. Therefore, $I_\C \to \C$ is final.
+:::
+
+::: Lemma 9
+Let $\C$ be a locally small category. Then $\widehat{\C}$ is co-Malcev.
+:::
+
+::: Proof
+This follows since $\Set$ is co-Malcev and since finite colimits are objectwise.
+:::
+
+::: Proposition 10
+Let $\C$ be a locally small category. Then $\widehat{\C}$ is epi-regular.
+:::
+
+Notice that this would be easy if $\widehat{\C}$ has pullbacks. In that case, every epimorphism would even be effective since this is the case for $\Set$. But in general, $\widehat{\C}$ may fail to have pullbacks. This is why the proof is more complicated.
+
+::: Proof
+First, notice that the Yoneda Lemma and the description of epimorphisms (see Lemma 5) implies that representable functors are [projective objects](https://ncatlab.org/nlab/show/projective+object). Therefore, also coproducts of representable functors are projective.
+
+Now let $\eta : F \to G$ be an epimorphism of small presheaves. Since $F$ is small, there is an epimorphism
+$$F_0 \xrightarrow{~ \pi ~} F,$$
+where $F_0$ is a coproduct of representable functors. Since $G$ is small, there is a coequalizer diagram
+
+$$
+G_1
+\begin{array}{c}
+\xrightarrow{~ \alpha ~ }\\[-1.25ex] \xrightarrow[~ \beta ~ ]{}
+\end{array}
+G_0 \xrightarrow{~ \psi ~} G,
+$$
+
+where $G_0$ and $G_1$ are coproducts of representable functors. Since $G_0$ is projective, there is a morphism $\lambda : G_0 \to F$ such that $\eta \circ \lambda = \psi$. Since $F_0$ is projective, there is a morphism $\mu : F_0 \to G_0$ such that $\psi \circ \mu = \eta \circ \pi$. We get the following diagram, where the outer square and the lower triangle commutes, but not necessarily the upper triangle.
+
+$$
+\begin{CD}
+F_0 @>{\pi}>> F \\
+@V{\mu}VV \,  \, \nearrow{\scriptsize \, \lambda}  @VV{\eta}V \\
+G_0 @>>{\psi}> G
+\end{CD}
+$$
+
+Define the morphisms $\gamma,\delta : G_1 \sqcup F_0 \rightrightarrows F$ by
+$$\gamma|_{G_1} = \lambda \circ \alpha, \quad \delta|_{G_1} = \lambda \circ \beta,$$
+$$\gamma|_{F_0} = \lambda \circ \mu, \quad \delta|_{F_0} = \pi.$$
+We will prove that $\eta : F \to G$ is a coequalizer of $\gamma$ and $\delta$. First, $\eta$ coequalizes these because
+$$\eta \circ \gamma|_{G_1} = \eta \circ \lambda \circ \alpha = \psi \circ \alpha = \psi \circ \beta = \eta \circ \lambda \circ \beta = \eta \circ \delta|_{G_1}$$
+and
+$$\eta \circ \gamma|_{F_0} = \eta \circ \lambda \circ \mu = \psi \circ \mu = \eta \circ \pi = \eta \circ \delta|_{F_0}.$$
+Conversely, suppose that $\vartheta : F \to H$ is a morphism that coequalizes these morphisms, meaning that $\vartheta \circ \lambda \circ \alpha = \vartheta \circ \lambda \circ \beta$ and $\vartheta \circ \lambda \circ \mu = \vartheta \circ \pi$. The first equation means that there is a morphism $\vartheta' : G \to H$ such that $\vartheta' \circ \psi = \vartheta \circ \lambda$. The second equation then becomes
+$$\vartheta \circ \pi = \vartheta' \circ \psi \circ \mu = \vartheta' \circ \eta \circ \pi,$$
+which is equivalent to $\vartheta = \vartheta' \circ \eta$. We have thus shown that every morphism that coequalizes $\alpha$ and $\beta$ factors through $\eta$, and uniqueness is clear since $\eta$ is an epimorphism.
+:::
+
+::: Lemma 11
+Let $\C$ be a locally small category. Then $\widehat{\C}$ has effective congruences.
+:::
+
+::: Proof
+Let $f,g : F \rightrightarrows G$ be a congruence in $\widehat{\C}$. Let $p : G \twoheadrightarrow Q$ be its quotient (i.e. coequalizer) in $\widehat{\C}$, which is constructed objectwise. Applying the functorial definition of a congruence to representable functors in $\widehat{\C}$, we see that for every object $X \in \C$ that $f(X),g(X) : F(X) \rightrightarrows G(X)$ is a congruence in $\Set$. Since congruences in $\Set$ are effective, $f(X),g(X)$ is the kernel pair of $p(X)$; we are also using [this result](/content/effective-congruence-quotients). Therefore, $f,g$ is the kernel pair of $p$ in the category of all presheaves, _a fortiori_ in the category of small presheaves.
+:::

content/nice-and-small-presheaves.md

@@ -0,0 +1,70 @@
+---
+title: Nice and small presheaves
+description: We give a concrete description of small presheaves on a specific locally small category.
+---
+
+## Nice and small presheaves
+
+Let $\C$ be the locally small category defined in [this entry](/category/example-cocomplete-no-equalizers). Its objects are $A$, $B$, and every set $X$. The non-identity morphisms are $f,g : A \rightrightarrows B$, $u_X : X \to A$ and $v_X : X \to B$ for every set $X$, and they satisfy the relation $f \circ u_X = g \circ u_X = v_X$.
+
+Let $F$ be a presheaf on $\C$. Concretely, this means that we are given sets $F(A)$, $F(B)$, and $F(X)$ for every set $X$, two maps $f^*,g^* : F(B) \rightrightarrows F(A)$ and, for every set $X$, a map $u_X^* : F(A) \to F(X)$ satisfying
+$$u_X^* \circ f^* = u_X^* \circ g^* = v_X^*,$$
+where $v_X^* : F(B) \to F(X)$. Thus, we have a commutative diagram
+$$F(B) ~\overset{f^*}{\underset{g^*}{\rightrightarrows}}~ F(A) \xrightarrow{u_X^*} F(X).$$
+If this is a coequalizer diagram, we say that $F$ is _nice at_ $X$. In particular, $u_X^*$ must be surjective. We say that $F$ is _nice_ if there is a set $S_F$ of sets such that $F$ is nice at every set $X \notin S_F$. We then call $S_F$ an _exceptional set_ for $F$. Intuitively, this means that $F$ is nice "almost everywhere".
+
+Let us determine which representable presheaves are nice.
+
+1. When $F = \Hom(-,A)$, the diagram evaluates to
+   $$\varnothing ~\overset{f^*}{\underset{g^*}{\rightrightarrows}}~ \{\id_A\} \xrightarrow{u_X^*} \{u_X\},$$
+   which is clearly a coequalizer diagram. Thus, this presheaf is nice everywhere.
+
+2. When $F = \Hom(-,B)$, the diagram evaluates to
+   $$\{\id_B\} ~\overset{f^*}{\underset{g^*}{\rightrightarrows}}~ \{f,g\} \xrightarrow{u_X^*} \{v_X\},$$
+   which is again clearly a coequalizer diagram. Thus, this presheaf is nice everywhere.
+
+3. When $F = \Hom(-,X)$ for a set $X$, the diagram at $X$ is
+   $$\varnothing ~\overset{f^*}{\underset{g^*}{\rightrightarrows}}~ \varnothing \xrightarrow{u_X^*} \{\id_X\},$$
+   which is not a coequalizer diagram. At $Y \neq X$, however, the diagram evaluates to
+   $$\varnothing ~\overset{f^*}{\underset{g^*}{\rightrightarrows}}~ \varnothing \xrightarrow{u_Y^*} \varnothing,$$
+   and is therefore a coequalizer diagram. Hence, the set $\{X\}$ is an exceptional set for $\Hom(-,X)$.
+
+::: Lemma
+A presheaf $F$ on $\C$ is small if and only if it is nice.
+:::
+
+::: Proof
+The collection of nice presheaves is clearly closed under small colimits of presheaves, since colimits commute with colimits and colimits of presheaves are computed objectwise. Furthermore, we have seen above that representable presheaves are nice. It follows that every small presheaf is nice.
+
+Conversely, assume that $F$ is a nice presheaf and choose an exceptional set $S_F$. To show that $F$ is small, we will show that its category of elements $\int F$ has a small final subcategory. Let $\D$ be the full subcategory of $\int F$ consisting of the objects
+
+- $(A,a)$ for $a \in F(A)$,
+- $(B,b)$ for $b \in F(B)$,
+- $(X,x)$ for $x \in F(X)$ and $X \in S_F$.
+
+This is a small category since $S_F$ is a set.
+
+We need to show that, for every object $T \in \int F$, the comma category $T \downarrow \D$ is connected. This is trivial for objects $T$ of $\D$. It remains to check this for $(X,x)$, where $x \in F(X)$ and $X \notin S_F$. By the definition of $S_F$, the diagram
+$$F(B) ~\overset{f^*}{\underset{g^*}{\rightrightarrows}}~ F(A) \xrightarrow{u_X^*} F(X) \tag{1}$$
+is a coequalizer diagram. In particular, $u_X^*$ is surjective, so there is some $a \in F(A)$ with $u_X^*(a) = x$. Then $u_X : (X,x) \to (A,a)$ is a morphism in $\int F$, showing that $(X,x) \downarrow \D$ is non-empty.
+
+There are two types of objects in $(X,x) \downarrow \D$. The first type consists of morphisms
+$$u_X : (X,x) \to (A,a),$$
+where $a \in F(A)$ satisfies $u_X^*(a)=x$. The second type consists of morphisms
+$$v_X : (X,x) \to (B,b),$$
+where $b \in F(B)$ satisfies $v_X^*(b)=x$. Every object of the second type is connected to an object of the first type, since $f : (A,f^*(b)) \to (B,b)$ is a morphism with $f \circ u_X = v_X$.
+
+It remains to show that every two objects
+$$(X,x) \to (A,a), \quad (X,x) \to (A,a')$$
+are connected, where $a,a' \in F(A)$ satisfy $u_X^*(a)=u_X^*(a')=x$. Since the diagram $(1)$ is a coequalizer diagram, there is a finite sequence of elements $a_0,\dotsc,a_n$ in $F(A)$, where $a_0=a$ and $a_n=a'$, and a finite sequence of elements $b_0,\dotsc,b_{n-1} \in F(B)$ such that, for every $0 \leq i < n$, either
+$$a_i=f^*(b_i), \quad a_{i+1}=g^*(b_i),$$
+or
+$$a_i=g^*(b_i), \quad a_{i+1}=f^*(b_i).$$
+It suffices to show that $(X,x) \to (A,a_i)$ and $(X,x) \to (A,a_{i+1})$ are connected. We may assume without loss of generality that
+$$a_i=f^*(b_i), \quad a_{i+1}=g^*(b_i).$$
+But then both are connected to $(X,x) \to (B,b_i)$ via the morphisms
+
+$$f : (A,a_i) \to (B,b_i), \quad g : (A,a_{i+1}) \to (B,b_i),$$
+
+respectively.
+:::

database/data/categories/Sp.yaml

@@ -40,7 +40,7 @@ unsatisfied_properties:
     proof: 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: locally small
-    proof: '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.'
+    proof: Since $\FinSet$ is not small, this follows exactly like Lemma 2 <a href="/content/free-cocompletion">here</a>; but this result is not really relevant and what only matters is that $\Sp$ is locally essentially small.
 
   - property: essentially countable
     proof: 'Any function $f : \IN \to \IN$ can be regarded as a combinatorial species with trivial actions, and distinct functions yield non-isomorphic species.'