From 81a81b5982fc3cdf1331cf3c3e4be7325cccecd5 Mon Sep 17 00:00:00 2001 From: Script Raccoon Date: Wed, 19 Aug 2026 22:16:17 +0200 Subject: [PATCH 1/3] add results on free cocompletions --- .cspell.json | 1 + content/free-cocompletion.md | 157 +++++++++++++++++++++++++++++++ database/data/categories/Sp.yaml | 2 +- database/data/macros.yaml | 1 + 4 files changed, 160 insertions(+), 1 deletion(-) create mode 100644 content/free-cocompletion.md diff --git a/.cspell.json b/.cspell.json index f8e7b4fc..2e5aac2d 100644 --- a/.cspell.json +++ b/.cspell.json @@ -212,6 +212,7 @@ "Kerodon", "Kolmogorov", "Kunen", + "Kuratowski", "Lawvere", "libsql", "Lindelöf", diff --git a/content/free-cocompletion.md b/content/free-cocompletion.md new file mode 100644 index 00000000..eab55e7c --- /dev/null +++ b/content/free-cocompletion.md @@ -0,0 +1,157 @@ +--- +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 Kan Extensions are Partial Colimits 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 Kashiwara-Schapira. + +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 object-wise. +::: + +::: Proof +This follows from cocompleteness of $[\C^{\op},\Set]$ with object-wise 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 object-wise 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 object-wise. 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 object-wise. +::: + +::: 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. +::: diff --git a/database/data/categories/Sp.yaml b/database/data/categories/Sp.yaml index b43eeec4..7187a858 100644 --- a/database/data/categories/Sp.yaml +++ b/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 here; 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.' diff --git a/database/data/macros.yaml b/database/data/macros.yaml index 3f7829fd..f1f32856 100644 --- a/database/data/macros.yaml +++ b/database/data/macros.yaml @@ -48,6 +48,7 @@ \Bilin: \operatorname{Bilin} \Ob: \operatorname{Ob} \id: \operatorname{id} +\ev: \operatorname{ev} \card: \operatorname{card} \colim: \operatorname{colim} \im: \operatorname{im} From 6b47611c66a9b25541d5f45bf1313efe3a81724a Mon Sep 17 00:00:00 2001 From: Script Raccoon Date: Wed, 19 Aug 2026 22:17:32 +0200 Subject: [PATCH 2/3] add example of a cocomplete category without equalizers; decide its properties --- .../example-cocomplete-no-equalizers.yaml | 161 ++++++++++++++++++ 1 file changed, 161 insertions(+) create mode 100644 database/data/categories/example-cocomplete-no-equalizers.yaml diff --git a/database/data/categories/example-cocomplete-no-equalizers.yaml b/database/data/categories/example-cocomplete-no-equalizers.yaml new file mode 100644 index 00000000..ac236c41 --- /dev/null +++ b/database/data/categories/example-cocomplete-no-equalizers.yaml @@ -0,0 +1,161 @@ +id: example-cocomplete-no-equalizers +name: example of a cocomplete category without equalizers +notation: $\widehat{\C}$ +objects: small presheaves on a suitable category extending $\Set$ (see below) +morphisms: natural transformations +description: >- + This rather technical and artificial category has solely been added as an example of a cocomplete category that does not have equalizers. + + To construct it, we start with the category $\C$ that has two objects $A,B$ and every set $X$ as an object. (Instead of the collection of sets, we can take every other non-small collection.) There are the identities, two morphisms $f,g : A \rightrightarrows B$, a unique morphism $u_X : X \to A$ for every set $X$, and a unique morphism $v_X : X \to B$ for every set $X$. The composition is defined by $f \circ u_X = g \circ u_X = v_X$. There are no morphisms between different sets. + $$\begin{array}{c} + X \\[0.25ex] + {\raisebox{1ex}{$\scriptstyle u_X$}} \!\! \swarrow + \qquad + \searrow \!\! {\raisebox{1ex}{$\scriptstyle v_X$}} \\ + A \;\; + \begin{array}{c} + \xrightarrow{\quad f \quad }\\[-1.25ex] + \xrightarrow[\quad g \quad ]{} + \end{array} + \;\; B + \end{array}$$ + The category in this entry is the free cocompletion $\widehat{\C}$. It consists of small presheaves $F : \C^{\op} \to \Set$, i.e. those presheaves that can be written as a small colimit of representable functors. Equivalently, the category of elements $\int F$ is finally small. +nlab_link: null +tags: + - category theory + +related: [] + +comments: + - This category has been suggested by Simon Henry in MO/509754 and further explained in MSE/5137415. + +satisfied_properties: + - property: locally essentially small + proof: See Lemma 1 here. + + - property: cocomplete + check_redundancy: false + proof: Colimits can be constructed object-wise. See Lemma 3 here. + + - property: terminal object + label: terminal_presheaf_small + proof: >- + We need to prove that the constant presheaf $1 : \C^{\op} \to \Set$ is small. We prove that it is actually the coequalizer of $f_*,g_* : \Hom(-,A) \rightrightarrows \Hom(-,B)$. This can be checked object-wise. At the object $A \in \C$, the two maps + $$f_*(A),g_*(A) : \{\id_A\} = \Hom(A,A) \rightrightarrows \Hom(A,B) = \{f,g\}$$ + are given by $\id_A \mapsto f$ resp. $\id_A \mapsto g$. Thus, their coequalizer is $1$. At the object $B \in \C$ the two maps + $$f_*(B),g_*(B) : \varnothing = \Hom(B,A) \rightrightarrows \Hom(B,B) = \{\id_B\}$$ + also have coequalizer $1$. At the object $X \in \Set$ the maps + $$f_*(X),g_*(X) : \{u_X\} = \Hom(X,A) \rightrightarrows \Hom(X,B) = \{v_X\}$$ + are equal and have coequalizer $1$. + + - property: mono-regular + proof: See Lemma 6 here. + + - property: epi-regular + proof: See Proposition 10 here. + + - property: filtered-colimit-stable monomorphisms + proof: See Lemma 6 here. + + - property: infinitary extensive + proof: See Lemma 7 here. + + - property: co-Malcev + proof: See Lemma 9 here. + +unsatisfied_properties: + - property: skeletal + proof: This is trivial. + + - property: locally small + proof: Since $\C$ is not small, this follows from Lemma 2 here; but this result is not really relevant and what only matters is that $\widehat{\C}$ is locally essentially small. + + - property: semi-strongly connected + proof: Pick two different sets $X$ and $Y$. There is no morphism $\Hom(-,X) \to \Hom(-,Y)$, since the image of $\id_X$ would be a morphism $X \to Y$, which does not exist. Likewise, there is no morphism in the other direction. + + - property: well-powered + references: + - terminal_presheaf_small + proof: We already know that the terminal presheaf $1$ is small. For every set $X$ the unique morphism $\Hom(-,X) \to 1$ is a monomorphism since $X$ is subterminal in $\C$ (actually, any morphism with codomain $X$ is the identity). For different sets $X,X'$ we have $\Hom(-,X) \not\cong \Hom(-,X')$ since there is not even a morphism $X \to X'$ in $\C$. This shows that $\Sub(1)$ is not small. + + - property: well-copowered + proof: >- + For every set $X$ we use $u_X : X \to A$ to construct the pushout + $$P_X \coloneqq \Hom(-,A) \sqcup_{\Hom(-,X)} \Hom(-,A).$$ + It is a quotient of $\Hom(-,A) \sqcup \Hom(-,A)$. For $X \neq Y$ we have $P_X \not\cong P_Y$ because $P_X(Y)$ has two elements, the two copies of $u_Y$, while $P_Y(Y)$ has exactly one element, the image of $u_Y$. + + - property: equalizers + proof: >- + Assume that $f_*,g_* : \Hom(-,A) \rightrightarrows \Hom(-,B)$ have an equalizer $E : \C^{\op} \to \Set$ in $\widehat{\C}$. By Lemma 4 here, this equalizer is object-wise, so that for every object $O \in \C$, the set $E(O)$ identifies with the equalizer of $f_*(O),g_*(O) : \Hom(O,A) \rightrightarrows \Hom(O,B)$. Therefore, $E(X) = \Hom(A,X) = \{u_X\}$ for every set $X$, $E(A) = \varnothing$, and $E(B) = \varnothing$. But then its category of elements $\int E$ is a discrete category indexed by all sets, which thus is not finally small. Therefore, $E$ is not small. + + - property: binary powers + proof: >- + We will show that the product of $\Hom(-,B)$ with itself does not exist. By Lemma 4 here, this must be the limit in the category of all presheaves. Therefore, we need to show that the product presheaf $\Hom(-,B) \times \Hom(-,B)$ is not small. Equivalently, its category of elements is not finally small. It identifies with the category of spans over the pair $(B,B)$ and has these objects: + $$\begin{align*} + X_{vv} & ~ \coloneqq ~ B \xleftarrow{v_X} X \xrightarrow{v_X} B \\ + A_{ff} & ~ \coloneqq ~ B \xleftarrow{f} A \xrightarrow{f} B \\ + A_{fg} & ~ \coloneqq ~ B \xleftarrow{f} A \xrightarrow{g} B \\ + A_{gf} & ~ \coloneqq ~ B \xleftarrow{g} A \xrightarrow{f} B \\ + A_{gg} & ~ \coloneqq ~ B \xleftarrow{g} A \xrightarrow{g} B \\ + B_{11} & ~ \coloneqq ~ B \xleftarrow{\id_B} B \xrightarrow{\id_B} B + \end{align*}$$ + For the span $X_{vv}$, $X$ can be any set. Between these spans, there are only the following non-identity morphisms: + $$\begin{align*} + u_X &: X_{vv} \to A_{ff} \\ + u_X &: X_{vv} \to A_{fg} \\ + u_X &: X_{vv} \to A_{gf} \\ + u_X &: X_{vv} \to A_{gg} \\ + f &: A_{ff} \to B_{11} \\ + g &: A_{gg} \to B_{11} \\ + v_X &: X_{vv} \to B_{11} + \end{align*}$$ + It follows that the category of spans is thin. Therefore, we can imagine the above morphisms as $\leq$-relations in a partially ordered collection. + $$\begin{array}{c} + X_{vv} \\[0.5ex] + \swarrow ~~ \downarrow ~~~ \downarrow ~~ \searrow \\[0.5ex] + A_{fg} ~~ A_{ff} ~~ A_{gg} ~~ A_{gf} \\[0.5ex] + \downarrow ~~~ \downarrow \\[0.5ex] + B_{11} + \end{array}$$ + Assume that that there is small final collection of spans $\F$. This means that for every span $S$ the subset $S \downarrow \F = \{T \in \F : S \leq T\}$ is connected (and in particular, non-empty). Since $A_{fg} \downarrow \F$ is non-empty and $A_{fg}$ is maximal, we see that $A_{fg} \in \F$. Likewise, we have $A_{gf} \in \F$. Now choose a set $X$ such that $X_{vv} \notin \F$; it surely exists since $\F$ is small. The two elements $A_{fg}$, $A_{gf}$ of $X_{vv} \downarrow \F$ are connected by a zig-zag path in $X_{vv} \downarrow \F$. By the structure of the partial order, one of the connecting elements must be $Y_{vv}$ for some set $Y$. Then $X_{vv} \leq Y_{vv}$ implies $X_{vv} = Y_{vv} \in \F$, a contradiction. + + - property: cofiltered-limit-stable epimorphisms + references: + - terminal_presheaf_small + proof: We already know that the terminal presheaf $1$ is small. Now consider for $n \in \IN$ the copower $\IN_{\geq n} \otimes 1$ and for $n < m$ the morphism $\IN_{\geq m} \otimes 1 \to \IN_{\geq n} \otimes 1$ induced by the inclusion $\IN_{\geq m} \subseteq \IN_{\geq n}$. This sequence has a limit, the empty presheaf; this can easily be deduced from $\bigcap_{n \in \IN} \IN_{\geq n} = \varnothing$ in $\IN$. Therefore, the unique morphisms $\IN_{\geq n} \otimes 1 \to 1$ are epimorphisms, but their limit $0 \to 1$ is not. + + - property: generating set + proof: Assume that a generating set exists. By writing each presheaf in that generating set as a small colimit of representable functors, we find a set of sets $\S$ such that the representable functors $\Hom(-,A)$, $\Hom(-,B)$ and the $\Hom(-,X)$ for $X \in \S$ provide a generating set. (It is possible that $\Hom(-,A)$ and/or $\Hom(-,B)$ are not required, but it does not hurt to add them.) Let $Y$ be any set that is not contained in $\S$; it surely exists. There are two evident endomorphisms of the small presheaf $\Hom(-,Y) + \Hom(-,Y)$, the identity and the flip. They are not equal since $\Hom(Y,Y) = \{\id_Y\}$ is not empty. Thus, there is a morphism from one of the mentioned representable functors that distinguishes them. But actually, there is no morphism $\Hom(A,-) \to \Hom(-,Y) + \Hom(-,Y)$ at all since there is no morphism $A \to Y$, and likewise there is no morphism from $\Hom(B,-)$, and also not from $\Hom(-,X)$ for $X \in \S$ since $X \neq Y$. + + - property: cogenerator + proof: >- + Assume that a cogenerator $Q$ exists. For every set $X$, consider the equivalence relation $\ker(u_X^* : Q(A) \to Q(X))$ on $Q(A)$; here, we write $u_X^*$ instead of $Q(u_X)$. Since $Q(A)$ is a set, while there is a proper collection of sets, there are distinct sets $X,Y$ that induce the same equivalence relation: + $$\ker(u_X^*) = \ker(u_Y^*).$$ + Using the morphism $u_X : X \to A$, consider the small presheaf + $$P \coloneqq \Hom(-,A) \sqcup_{\Hom(-,X)} \Hom(-,A).$$ + Denote the two inclusions $\Hom(-,A) \to P$ by $i_1$ and $i_2$. For almost every object $O \in \C$, the canonical map $\Hom(O,A) \sqcup \Hom(O,A) \to P(O)$ is bijective; the only exception is $O = X$, for which $P(X)$ consists of the single element $i_1(u_X) = i_2(u_X)$. In particular, $P(A)$ consists of the two elements $i_1(\id_A)$ and $i_2(\id_A)$, while $P(Y)$ consists of the two elements $i_1(u_Y)$ and $i_2(u_Y)$. By the Yoneda Lemma, these correspond to two morphisms + $$\alpha_1,\alpha_2 : \Hom(-,Y) \rightrightarrows P.$$ + We claim that $\gamma \circ \alpha_1 = \gamma \circ \alpha_2$ for every morphism $\gamma : P \to Q$, showing that $Q$ is not a cogenerator. + + Consider the two elements $q_1 \coloneqq \gamma_A(i_1(\id_A))$ and $q_2 \coloneqq \gamma_A(i_2(\id_A))$ of $Q(A)$. They have the same image under $u_X^* : Q(A) \to Q(X)$ because, by construction of $P$, the elements $i_1(\id_A)$ and $i_2(\id_A)$ of $P(A)$ have the same image under $u_X^* : P(A) \to P(X)$. Hence, $\ker(u_X^*) = \ker(u_Y^*)$ implies that $q_1$ and $q_2$ have the same image under $u_Y^*$. Thus, + $$\gamma_Y(i_1(u_Y)) = u_Y^*(q_1) = u_Y^*(q_2) = \gamma_Y(i_2(u_Y)).$$ + By the Yoneda Lemma, this is equivalent to $\gamma \circ \alpha_1 = \gamma \circ \alpha_2$. + +special_objects: + initial object: + description: constant presheaf with value $0$ + terminal object: + description: constant presheaf with value $1$ + coproducts: + description: object-wise defined disjoint union of presheaves + +special_morphisms: + isomorphisms: + description: natural isomorphisms + proof: This is trivial. + monomorphisms: + description: natural transformations that are injective at every object + proof: See Lemma 5 here. + epimorphisms: + description: natural transformations that are surjective at every object + proof: See Lemma 5 here. From 311968b6f2c6d4987c7c6d0f60b6b646a1fe1009 Mon Sep 17 00:00:00 2001 From: Script Raccoon Date: Thu, 20 Aug 2026 09:36:05 +0200 Subject: [PATCH 3/3] the example category does not have sequential limits --- .../example-cocomplete-no-equalizers.yaml | 32 ++++++++++++++++--- 1 file changed, 27 insertions(+), 5 deletions(-) diff --git a/database/data/categories/example-cocomplete-no-equalizers.yaml b/database/data/categories/example-cocomplete-no-equalizers.yaml index ac236c41..e2b60e46 100644 --- a/database/data/categories/example-cocomplete-no-equalizers.yaml +++ b/database/data/categories/example-cocomplete-no-equalizers.yaml @@ -1,7 +1,7 @@ id: example-cocomplete-no-equalizers name: example of a cocomplete category without equalizers notation: $\widehat{\C}$ -objects: small presheaves on a suitable category extending $\Set$ (see below) +objects: small presheaves on a suitable large category (see below) morphisms: natural transformations description: >- This rather technical and artificial category has solely been added as an example of a cocomplete category that does not have equalizers. @@ -119,10 +119,32 @@ unsatisfied_properties: \end{array}$$ Assume that that there is small final collection of spans $\F$. This means that for every span $S$ the subset $S \downarrow \F = \{T \in \F : S \leq T\}$ is connected (and in particular, non-empty). Since $A_{fg} \downarrow \F$ is non-empty and $A_{fg}$ is maximal, we see that $A_{fg} \in \F$. Likewise, we have $A_{gf} \in \F$. Now choose a set $X$ such that $X_{vv} \notin \F$; it surely exists since $\F$ is small. The two elements $A_{fg}$, $A_{gf}$ of $X_{vv} \downarrow \F$ are connected by a zig-zag path in $X_{vv} \downarrow \F$. By the structure of the partial order, one of the connecting elements must be $Y_{vv}$ for some set $Y$. Then $X_{vv} \leq Y_{vv}$ implies $X_{vv} = Y_{vv} \in \F$, a contradiction. - - property: cofiltered-limit-stable epimorphisms - references: - - terminal_presheaf_small - proof: We already know that the terminal presheaf $1$ is small. Now consider for $n \in \IN$ the copower $\IN_{\geq n} \otimes 1$ and for $n < m$ the morphism $\IN_{\geq m} \otimes 1 \to \IN_{\geq n} \otimes 1$ induced by the inclusion $\IN_{\geq m} \subseteq \IN_{\geq n}$. This sequence has a limit, the empty presheaf; this can easily be deduced from $\bigcap_{n \in \IN} \IN_{\geq n} = \varnothing$ in $\IN$. Therefore, the unique morphisms $\IN_{\geq n} \otimes 1 \to 1$ are epimorphisms, but their limit $0 \to 1$ is not. + - property: sequential limits + proof: >- + By Lemma 4 here, it suffices to find a sequence of small presheaves on $\C$ whose limit in the category of all presheaves is not small. + + For $n \in \IN$ define the presheaf $F_n$ on objects by $F_n(A) = F_n(B) = \IN_{\geq n}$ and $F_n(X)=\{\ast\}$ for every set $X$. The maps $u_X^* : F_n(A) \to F_n(X)$ and $v_X^* : F_n(B) \to F_n(X)$ are uniquely determined. The map $f^* : F_n(B) \to F_n(A)$ is the identity map, while the map $g^* : F_n(B) \to F_n(A)$ is defined by $k \mapsto k + 1$. This is indeed a presheaf since $F_n(X)$ is a singleton. + + Let us check that $F_n$ is small. We do this by proving that $\int F_n$ is finally small. Let $K_n$ be the full subcategory of $\int F_n$ consisting of $(A,k)$ and $(B,k)$ for $k \geq n$. Clearly, $K_n$ is small, and we claim that for every object $T \in \int F_n$ the slice category $T \downarrow K_n$ is connected. This is trivial when $T \in K_n$ since then the slice category has an initial object. Otherwise, we have $T = (X,*)$ for some set $X$. The objects of the slice category are then + $$\begin{align*} + u_X^{[k]} &: (X,*) \to (A,k) \\ + v_X^{[k]} &: (X,*) \to (B,k) + \end{align*}$$ + for $k \geq n$. The morphism $f : A \to B$ provides a morphism $(A,k) \to (B,k)$ in $\int F_n$. It is a morphism + $$u_X^{[k]} \to v_X^{[k]}$$ + in the slice category since $f u_X = v_X$. Next, the morphism $g : A \to B$ induces a morphism $(A,k+1) \to (B,k)$ in $\int F_n$ since $g^*(k) = k+1$. It is a morphism + $$u_X^{[k+1]} \to v_X^{[k]}$$ + in the slice category since $g u_X = v_X$. This proves that all objects in the slice category are connected to each other. + + Define the morphism $F_{n+1} \to F_n$ as the inclusion at $A$ and $B$ and as the identity at $X$. Naturality is easy to check. Let $F_\infty$ denote the limit presheaf of this sequence. We compute $F_\infty(A) = \bigcap_{n \in \IN} \IN_{\geq n} = \varnothing$. Likewise, we have $F_\infty(B) = \varnothing$. But we have $F_\infty(X) = \{\ast\}$. This presheaf is not small since $\int F_\infty$ is a discrete category indexed by all sets, which thus is not finally small. + + # We can keep this proof here because at some point we will drop + # the assumption of cofiltered limits for this property to hold, + # and then this proof will not be redundant anymore. + # - property: cofiltered-limit-stable epimorphisms + # references: + # - terminal_presheaf_small + # proof: We already know that the terminal presheaf $1$ is small. Now consider for $n \in \IN$ the copower $\IN_{\geq n} \otimes 1$ and for $n < m$ the morphism $\IN_{\geq m} \otimes 1 \to \IN_{\geq n} \otimes 1$ induced by the inclusion $\IN_{\geq m} \subseteq \IN_{\geq n}$. This sequence has a limit, the empty presheaf; this can easily be deduced from $\bigcap_{n \in \IN} \IN_{\geq n} = \varnothing$ in $\IN$. Therefore, the unique morphisms $\IN_{\geq n} \otimes 1 \to 1$ are epimorphisms, but their limit $0 \to 1$ is not. - property: generating set proof: Assume that a generating set exists. By writing each presheaf in that generating set as a small colimit of representable functors, we find a set of sets $\S$ such that the representable functors $\Hom(-,A)$, $\Hom(-,B)$ and the $\Hom(-,X)$ for $X \in \S$ provide a generating set. (It is possible that $\Hom(-,A)$ and/or $\Hom(-,B)$ are not required, but it does not hurt to add them.) Let $Y$ be any set that is not contained in $\S$; it surely exists. There are two evident endomorphisms of the small presheaf $\Hom(-,Y) + \Hom(-,Y)$, the identity and the flip. They are not equal since $\Hom(Y,Y) = \{\id_Y\}$ is not empty. Thus, there is a morphism from one of the mentioned representable functors that distinguishes them. But actually, there is no morphism $\Hom(A,-) \to \Hom(-,Y) + \Hom(-,Y)$ at all since there is no morphism $A \to Y$, and likewise there is no morphism from $\Hom(B,-)$, and also not from $\Hom(-,X)$ for $X \in \S$ since $X \neq Y$.