From ea4d3dad6b9c32f88728276fd21f3e20de920406 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Fri, 26 Jun 2026 19:54:03 -0400 Subject: [PATCH 01/13] Add total/cototal category properties --- content/Grp_total_explicit_proof.md | 37 +++++++++++++++++++ content/missing_cogenerator.md | 14 ++++++- databases/catdat/data/categories/Alg(R).yaml | 3 ++ databases/catdat/data/categories/CAlg(R).yaml | 5 ++- databases/catdat/data/categories/CRing.yaml | 3 ++ databases/catdat/data/categories/Cat.yaml | 13 +++++-- databases/catdat/data/categories/Grp.yaml | 6 ++- databases/catdat/data/categories/Haus.yaml | 2 + databases/catdat/data/categories/LRS_R.yaml | 3 ++ databases/catdat/data/categories/Meas.yaml | 2 + databases/catdat/data/categories/Mon.yaml | 2 +- databases/catdat/data/categories/On.yaml | 1 + databases/catdat/data/categories/Ring.yaml | 3 ++ databases/catdat/data/categories/Rng.yaml | 2 +- databases/catdat/data/categories/Top.yaml | 1 + .../category-implications/accessible.yaml | 8 ++++ .../data/category-implications/total.yaml | 28 ++++++++++++++ .../data/category-properties/cototal.yaml | 11 ++++++ .../locally copresentable.yaml | 1 + .../locally presentable.yaml | 1 + .../data/category-properties/total.yaml | 11 ++++++ .../catdat/scripts/expected-data/Ab.json | 2 + .../catdat/scripts/expected-data/Set.json | 2 + .../catdat/scripts/expected-data/Top.json | 2 + 24 files changed, 154 insertions(+), 9 deletions(-) create mode 100644 content/Grp_total_explicit_proof.md create mode 100644 databases/catdat/data/category-implications/total.yaml create mode 100644 databases/catdat/data/category-properties/cototal.yaml create mode 100644 databases/catdat/data/category-properties/total.yaml diff --git a/content/Grp_total_explicit_proof.md b/content/Grp_total_explicit_proof.md new file mode 100644 index 000000000..2149442f0 --- /dev/null +++ b/content/Grp_total_explicit_proof.md @@ -0,0 +1,37 @@ +--- +title: Explicit Proof that the Category of Groups is Total +description: An explicit construction of the left adjoint to the covariant Yoneda embedding on the category of groups +author: Daniel Schepler +--- + +## Explicit Proof that the Category of Groups is Total + +We will fix a functor $T : \Grp^{\op} \to \Set$. Based on this, we need to construct a group $L(T)$. The construction will work as follows: we start with a set of generators $e_x$, one for each element $x \in T\IZ$. Now let $\mu : \IZ \to \IZ * \IZ'$ be the comultiplication homomorphism, $1 \mapsto 1 \cdot 1'$, and $i_1, i_2 : \IZ \rightrightarrows \IZ * \IZ'$ the two coprojections. We now add a relation $e_{T\mu(x)} = e_{Ti_1(x)} \cdot e_{Ti_2(x)}$ for each element $x \in T(\IZ * \IZ')$. Similarly, letting $\iota : \IZ \to \IZ$ be the coinverse homomorphism, we add a relation $e_{T\iota(x)} = e_x^{-1}$ for each element $x \in T\IZ$; and letting $\varepsilon : \IZ \to 0$ be the coidentity homomorphism, we add a relation $e_{T\varepsilon(x)} = 1$ for each element $x \in T0$. + +We first need to define the natural transformation $\eta_T : T \to \Hom({-}, L(T))$. To start, for each group $H$ we need a function $TH \to \Hom(H, L(T))$. We will define this function to send $x \in TH$ to $h \mapsto e_{Th(x)}$, where we abuse notation to identify $h \in H$ with the corresponding morphism $\IZ \to H$, so that $Th : TH \to T\IZ$. To see that this defines a group homomorphism from $H$ to $L(T)$, note that for $h, h' \in H$ we have three commutative diagrams of the form +$$\begin{CD} +T(H) @> = >> T(H)\\ +@V T(h * h') VV @VVV\\ +T(\IZ * \IZ') @>>> T(\IZ) +\end{CD}$$ +where on the bottom we use $T\mu, Ti_1, Ti_2$. Applying this to $x\in TH$, we get $Th(x)$, $Th'(x)$, and $T(h h')(x)$, respectively. Thus, the relation $e_{T\mu(y)} = e_{Ti_1(y)} \cdot e_{Ti_2(y)}$ with $y \coloneqq T(h * h')(x)$ implies +$$e_{T(hh')(x)} = e_{Th(x)} e_{Th'(x)},$$ +as required. Similar proofs show that the map $H \to G$ respects inverses and the identity. We leave it as an exercise for the reader to show this is natural in $H$. + +We now need to show that for each group $G$ and natural transformation $\alpha : T \to y_G$, there exists a unique group homomorphism $\varphi : L(T) \to G$ such that $\alpha = y_{\varphi} \circ \eta_T : T \to \Hom({-}, L(T)) \to \Hom({-}, G)$. We start with uniqueness: suppose $x \in T\IZ$. Then by hypothesis, $\alpha_{\IZ} = y_{\varphi} \circ (\eta_T)_{\alpha} : T\IZ \to \Hom(\IZ, L(T)) \to \Hom(\IZ, G)$. For each $x \in T\IZ$, the first step on the right hand side maps $x \mapsto (1 \mapsto e_x)$, and the second step then maps this to $1 \mapsto \varphi(e_x)$. Therefore, $\varphi(e_x) = \alpha_{\IZ}(x)(1)$ for each $x$, which establishes the uniqueness of $\varphi$. + +For the existence part, the first step is to show there is a gruop homomorphism $L(T) \to G$ with the images of $e_x$ required by the previous part, i.e. $e_x \mapsto \alpha_{\IZ}(x)(1)$. To prove this, we need to check that the relations in $L(T)$ are collapsed in $G$. Now, for each $x \in T(\IZ * \IZ')$, we have three commutative diagrams of the form +$$\begin{CD} +T(\IZ * \IZ') @> \alpha_{\IZ * \IZ'} >> \Hom(\IZ * \IZ', G) @> \simeq >> UG \times UG\\ +@VVV @VVV @VVV\\ +T(\IZ) @> \alpha_{\IZ} >> \Hom(\IZ, G) @> \simeq >> UG +\end{CD}$$ +applying naturality to $\mu, i_1, i_2 : \IZ \to \IZ * \IZ'$. On the right hand side, we get multiplication, first projection, and second projection respectively. From this, we conclude that the images of $e_{T\mu(x)}$ and $e_{Ti_1(x)} \cdot e_{Ti_2(x)}$ in $UG$ agree for any element $x \in T(\IZ * \IZ')$. Similar proofs show that the other relations are also collapsed. + +Finally, we need to show $\alpha = y_{\varphi} \circ \eta_T$. It suffices to show $\alpha_H = (y_{\varphi})_H \circ (\eta_T)_H : TH \to \Hom(H, L(T)) \to \Hom(H, G)$ for each group $H$. By definition, for each $x \in TH$, the first step gives the homomorphism $h \mapsto e_{Th(x)}$; then the second step is formed by composition with $\varphi$. By the specification of $\varphi$, this gives the homorphism $h \mapsto \alpha_{\IZ}(Th(x))(1)$. However, by the assumption that $\alpha$ is a natural transformation, for each $h \in H$ we have a commutative diagram +$$\begin{CD} +TH @> \alpha_H >> \Hom(H, G) \\ +@V Th VV @VV {-} \circ h V \\ +T\IZ @> \alpha_{\IZ} >> \Hom(\IZ, G). +\end{CD}$$ +Applying this to $x \in TH$ gives exactly that $\alpha_{\IZ}(Th(x))(1) = \alpha_H(x)(h)$. $\square$ diff --git a/content/missing_cogenerator.md b/content/missing_cogenerator.md index a59d81f79..a397ba6bb 100644 --- a/content/missing_cogenerator.md +++ b/content/missing_cogenerator.md @@ -13,8 +13,18 @@ Let $\C$ be a pointed category with a faithful functor $U: \C \to \Set$. Assume 1. For any $X \in \F$ and any $Y \in \C$, every non-zero morphism $f: X \to Y$ is injective on underlying sets. 2. For every $Y \in \C$ there is some object $X \in \F$ such that $\card(U(X)) > \card(U(Y))$. -Then $\C$ does not have a cogenerator. +Then $\C$ does not have a cogenerator. Moreover, $\C$ is not cototal. ::: _Proof._ -Assume that there is a cogenerator $Y$. By assumption (2) there is an object $X \in \F$ such that $U(X)$ is larger than $U(Y)$ (w.r.t. cardinalities). Since $0,\id_X : X \rightrightarrows X$ are distinct, there is a morphism $f : X \to Y$ with $f \neq 0$. But then $U(f) : U(X) \to U(Y)$ is injective by assumption (1), which contradicts our choice of $X$. $\square$ +Assume that there is a cogenerator $Y$. By assumption (2) there is an object $X \in \F$ such that $U(X)$ is larger than $U(Y)$ (w.r.t. cardinalities). Since $0,\id_X : X \rightrightarrows X$ are distinct, there is a morphism $f : X \to Y$ with $f \neq 0$. But then $U(f) : U(X) \to U(Y)$ is injective by assumption (1), which contradicts our choice of $X$. + +Now assume that $\C$ is cocotal. Using the axiom of choice, we may assume that for each small cardinal $\kappa$, there is at most one element $X \in \F$ such that $\card(U(X)) = \kappa$. We define a functor $T : \C \to \Set$ which is morally defined by $T(Y) \coloneqq \prod_{X \in \F} \Hom(X, Y)$. In this product, we know that $\Hom(X, Y) = \{ 0 \}$ whenever $\card(U(X)) > \card(U(Y))$, so every term except for a small number is a singleton and does not change the product. Thus, we may define +$$T(Y) \coloneqq \prod_{X \in \F : \card(U(X)) \le \card(U(Y))} \Hom(X, Y),$$ +and define the functor structure by extending tuples with zero morphisms as necessary. + +Now, if $L : [\C, \Set] \to \C^{\op}$ is the left adjoint of the contravariant Yoneda embedding $y : \C^{op} \to [\C, \Set]$, then we have a bijection +$$\Hom_{\C}(X, L(T)) \simeq \Hom(T, \Hom(X, {-}))$$ +for each object $X$ of $\C$. Now let $X_0$ be an element of $\F$ such that $\card(U(X_0)) > \card(U(L(T)))$. Then we can define two natural transformations $T \to \Hom({-}, X_0)$: one is the constant zero map; and the other is selection of the $X$ component, i.e. given an element of $TY = \prod_{X\in F : \card(U(X)) \le \card(U(Y))} \Hom(X, Y)$ we select the $X$ component if $\card(U(X)) \le \card(U(Y))$, or set the result to the zero morphism if $\card(U(X)) > \card(U(Y))$. Note these two natural transformations are unequal, since for example we can define an element of $T(X_0)$ which is $\id_{X_0}$ on the $X_0$ component and zero on the other components, and the two transformations map this element to $\id_{X_0} \ne 0$. + +Therefore, by the above bijection, $\Hom_{\C}(X_0, L(T))$ has at least two elements, out of which at least one must be non-zero and therefore injective on underlying sets. This contradicts the assumption that $\card(U(X_0)) > \card(U(L(T))$. $\square$ diff --git a/databases/catdat/data/categories/Alg(R).yaml b/databases/catdat/data/categories/Alg(R).yaml index 96fc09ade..2cf709753 100644 --- a/databases/catdat/data/categories/Alg(R).yaml +++ b/databases/catdat/data/categories/Alg(R).yaml @@ -43,6 +43,9 @@ unsatisfied_properties: - property: cogenerating set proof: '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: cototal + proof: Essentially the same proof as for $\CAlg(R)$ works here. + - property: codistributive proof: '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 $\langle X,E : E^2=E \rangle$.' diff --git a/databases/catdat/data/categories/CAlg(R).yaml b/databases/catdat/data/categories/CAlg(R).yaml index 0244eb450..9a8fad0d7 100644 --- a/databases/catdat/data/categories/CAlg(R).yaml +++ b/databases/catdat/data/categories/CAlg(R).yaml @@ -19,7 +19,7 @@ satisfied_properties: proof: There is a forgetful functor $\CAlg(R) \to \Set$ and $\Set$ is locally small. - property: finitary algebraic - proof: Take the algebraic theory of a commutative ring. + proof: Take the algebraic theory of a commutative $R$-algebra. - property: strict terminal object proof: 'If $f : 0 \to R$ is a homomorphism, then $R$ satisfies $1=f(1)=f(0)=0$, so that $R=0$.' @@ -41,6 +41,9 @@ unsatisfied_properties: - property: cogenerating set proof: '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: cototal + proof: 'Let $\F$ be the family of commutative $R$-algebras of the form $R \times k$ where $k$ is an infinite field including a quotient field of $R$. Then for any commutative $R$-algebra $A$, we have a distinguished morphism $R \times k \to A$ consisting of the projection to $R$ followed by the unique morphism $R \to A$. Moreover, if we have any morphism $\varphi : R \times k \to A$ which is not equal to the distinguished morphism, that implies that $\varphi(0, 1) \ne 0$, so the rng homomorphism $k \to R \times k \to A$ is injective, implying $\card(U(A)) \ge \card(U(k))$. From here, an argument similar to the one here gives a contradiction, using the distinguished morphisms in place of zero morphisms.' + - property: countably codistributive proof: '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 $A \coloneqq K[X]$ and consider the sequence $(X^n)_{n} \in A^{\IN}$.' diff --git a/databases/catdat/data/categories/CRing.yaml b/databases/catdat/data/categories/CRing.yaml index d95061e94..b90c6bc4e 100644 --- a/databases/catdat/data/categories/CRing.yaml +++ b/databases/catdat/data/categories/CRing.yaml @@ -47,6 +47,9 @@ unsatisfied_properties: - property: cogenerating set proof: '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: cototal + proof: 'This is a special case of the proof for $\CAlg(R)$ with $R = \IZ$.' + - property: countably codistributive proof: '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.' diff --git a/databases/catdat/data/categories/Cat.yaml b/databases/catdat/data/categories/Cat.yaml index 80e08f454..702677bbd 100644 --- a/databases/catdat/data/categories/Cat.yaml +++ b/databases/catdat/data/categories/Cat.yaml @@ -40,9 +40,6 @@ unsatisfied_properties: - property: balanced proof: 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: cogenerating set - proof: '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: regular proof: See Example 3.14 at the nLab. @@ -85,6 +82,16 @@ unsatisfied_properties: $$\Sub_{\reg}(\{ 0 \to 1 \to 2 \}) \to \Sub_{\reg}(\{ 0 \to 1 \}) \times_{\Sub_{\reg}(\{1\})} \Sub_{\reg}(\{ 1 \to 2 \})$$ is not injective. Therefore, $\Sub_{\reg} : \Cat^{\op} \to \Set^+$ does not preserve pullbacks, so it cannot be representable. + - property: cototal + proof: >- + For each infinite cardinal $\kappa$, choose a simple group $S_\kappa$ of cardinality $\kappa$ with at least $\kappa$ distinct automorphisms (for example the group of permutations of $\kappa$ of finite support which are even). We will define a functor $T : \Cat \to \Set$ which morally will send a category $\C$ to the collection-sized wide pullback of $[B S_\kappa, \C] \to [1, \C]$. Namely, if $\kappa$ is greater than the cardinality of any endomorphism monoid of an object of $\C$, then the image in $[1, \C]$ uniquely determines a functor in $[B S_\kappa, \C]$. Thus, we can restrict to the corresponding pullback with $\kappa$ restricted to be at most $\max(\aleph_0, \lambda)$ where $\lambda$ is the supremum of the cardinalities of endomorphism monoids; and this also forms a pullback of the full collection-sized diagram. + + Now assume $\Cat$ is cototal with $L : [\Cat, \Set] \to \Cat^{\op}$ a left adjoint to the contravariant Yoneda embedding. Then for each infinite cardinal $\kappa$, we would have a bijection + $$[B S_\kappa, L(T)] \simeq \Hom(T, [B S_\kappa, -]).$$ + On the other hand, we have at least $\kappa$ distinct morphisms $T \to [B S_\kappa, -]$: for each automorphism $\pi$ of $S_\kappa$, we can take the natural transformation which selects the $\kappa$ component of the limit, and then composes with $B \pi$. To see these are distinct, apply them to the element of $T(B S_\kappa)$ which is the identity on the $\kappa$ component and $B(0)$ on the $\lambda$ component for $\lambda \ne \kappa$. + + However, if $\kappa$ is greater than the cardinality of any endomorphism monoid of $L(T)$, then any functor $B S_\kappa \to L(T)$ must act on the morphisms of $B S_\kappa$ by sending each to the identity morphism on the image object. Therefore, $\card([B S_\kappa, L(T)]) \le \card(\Ob(L(T)))$. Thus, if we also choose $\kappa > \card(\Ob(L(T)))$, then we get a contradiction. + special_objects: initial object: description: empty category diff --git a/databases/catdat/data/categories/Grp.yaml b/databases/catdat/data/categories/Grp.yaml index 1b541e609..190d37b79 100644 --- a/databases/catdat/data/categories/Grp.yaml +++ b/databases/catdat/data/categories/Grp.yaml @@ -39,6 +39,10 @@ satisfied_properties: - property: effective cocongruences proof: A proof can be found here. + - property: total + proof: This follows formally from the fact that $\Grp$ is finitary algebraic and therefore locally presentable. For a more explicit proof, see here. + check_redundancy: false + unsatisfied_properties: - property: skeletal proof: This is trivial. @@ -46,7 +50,7 @@ unsatisfied_properties: - property: normal proof: Every non-normal subgroup (such as $C_2 \hookrightarrow S_3$) provides a counterexample. - - property: cogenerator + - property: cototal proof: '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: coregular diff --git a/databases/catdat/data/categories/Haus.yaml b/databases/catdat/data/categories/Haus.yaml index 81bcd23e4..6ed887832 100644 --- a/databases/catdat/data/categories/Haus.yaml +++ b/databases/catdat/data/categories/Haus.yaml @@ -26,9 +26,11 @@ satisfied_properties: - property: equalizers proof: This follows from the corresponding fact for $\Top$ since subspaces of Hausdorff spaces are again Hausdorff. + check_redundancy: false - property: products proof: This follows from the corresponding fact for $\Top$ since products of Hausdorff spaces are again Hausdorff. + check_redundancy: false - property: cocomplete proof: 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. diff --git a/databases/catdat/data/categories/LRS_R.yaml b/databases/catdat/data/categories/LRS_R.yaml index 9bb3f5768..769554621 100644 --- a/databases/catdat/data/categories/LRS_R.yaml +++ b/databases/catdat/data/categories/LRS_R.yaml @@ -53,6 +53,9 @@ unsatisfied_properties: Alternatively, using the usual adjunction between affine schemes and locally ringed spaces (EGA I (1971), Ch. 1, Prop. 1.6.3), a generating set in $\LRS_R$ would induce a generating set in the category of affine $R$-schemes, which contradicts the fact that $\CAlg(R)$ does not have a cogenerating set. + - property: total + proof: 'The adjunction between the global sections functor and the $\Spec$ functor (EGA I (1971), Ch. 1, Prop. 1.6.3) makes $\CAlg(R)^{\op}$ into a reflective subcategory of $\LRS_R$. Therefore, if $LRS_R$ were total, then \CAlg(R) would be cototal, which we know is not the case.' + - property: cartesian closed proof: This is Corollary 4(a) here. check_redundancy: false diff --git a/databases/catdat/data/categories/Meas.yaml b/databases/catdat/data/categories/Meas.yaml index d7ce5a657..e5aafc471 100644 --- a/databases/catdat/data/categories/Meas.yaml +++ b/databases/catdat/data/categories/Meas.yaml @@ -36,9 +36,11 @@ satisfied_properties: - property: complete proof: Take the limit of the underlying sets and take the smallest $\sigma$-algebra making all projections measurable. + check_redundancy: false - property: cocomplete proof: 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. + check_redundancy: false - property: infinitary extensive proof: '[Sketch] Since $\Set$ is infinitary extensive, a map $f : Y \to \coprod_i X_i \eqqcolon X$ 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.' diff --git a/databases/catdat/data/categories/Mon.yaml b/databases/catdat/data/categories/Mon.yaml index 10474de50..2d1981158 100644 --- a/databases/catdat/data/categories/Mon.yaml +++ b/databases/catdat/data/categories/Mon.yaml @@ -39,7 +39,7 @@ unsatisfied_properties: - property: Malcev proof: 'Consider the submonoid $\{(a,b) : a \leq b \}$ of $\IN^2$.' - - property: cogenerator + - property: cototal proof: '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: counital diff --git a/databases/catdat/data/categories/On.yaml b/databases/catdat/data/categories/On.yaml index 5f67174eb..1e16de13a 100644 --- a/databases/catdat/data/categories/On.yaml +++ b/databases/catdat/data/categories/On.yaml @@ -45,6 +45,7 @@ unsatisfied_properties: - property: well-copowered proof: The "quotients" of $0$ are all ordinals. + check_redundancy: false - property: inverse proof: Consider the strictly increasing sequence $0 < 1 < 2 < \cdots$. diff --git a/databases/catdat/data/categories/Ring.yaml b/databases/catdat/data/categories/Ring.yaml index 0e2a63428..6167ca973 100644 --- a/databases/catdat/data/categories/Ring.yaml +++ b/databases/catdat/data/categories/Ring.yaml @@ -46,6 +46,9 @@ unsatisfied_properties: - property: cogenerating set proof: '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: cototal + proof: 'This is a special case of the proof for $\Alg(R)$ with $R = \IZ$.' + - property: codistributive proof: '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 $\langle X,E : E^2=E \rangle$.' diff --git a/databases/catdat/data/categories/Rng.yaml b/databases/catdat/data/categories/Rng.yaml index d491880cf..75ba1d641 100644 --- a/databases/catdat/data/categories/Rng.yaml +++ b/databases/catdat/data/categories/Rng.yaml @@ -36,7 +36,7 @@ unsatisfied_properties: - property: balanced proof: The inclusion $\IZ \hookrightarrow \IQ$ is a counterexample. (The proof can be reduced to the unital case.) - - property: cogenerator + - property: cototal proof: '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: counital diff --git a/databases/catdat/data/categories/Top.yaml b/databases/catdat/data/categories/Top.yaml index c6a2b2741..b739aaa37 100644 --- a/databases/catdat/data/categories/Top.yaml +++ b/databases/catdat/data/categories/Top.yaml @@ -21,6 +21,7 @@ satisfied_properties: - property: complete proof: Take the limit of the underlying sets and endow it with the coarsest topology making all projections continuous. + check_redundancy: false - property: cocomplete proof: Take the colimit of the underlying sets and endow it with the finest topology making all inclusions continuous. diff --git a/databases/catdat/data/category-implications/accessible.yaml b/databases/catdat/data/category-implications/accessible.yaml index 94033586c..c26fdb79e 100644 --- a/databases/catdat/data/category-implications/accessible.yaml +++ b/databases/catdat/data/category-implications/accessible.yaml @@ -28,6 +28,14 @@ proof: This follows from one of equivalent formulations of locally presentable categories. is_equivalence: true +- id: locally_presentable_consequence + assumptions: + - locally presentable + conclusions: + - total + proof: ... + is_equivalence: false + - id: locally_finitely_presentable_consequence assumptions: - locally finitely presentable diff --git a/databases/catdat/data/category-implications/total.yaml b/databases/catdat/data/category-implications/total.yaml new file mode 100644 index 000000000..3eafb7b0f --- /dev/null +++ b/databases/catdat/data/category-implications/total.yaml @@ -0,0 +1,28 @@ +# results on total and cototal categories + +- id: total_definition + assumptions: + - total + conclusions: + - locally essentially small + proof: This is true by definition. + is_equivalence: false + +- id: total_consequences + assumptions: + - total + conclusions: + - complete + - cocomplete + proof: 'If a category $\C$ is total, then $\C$ is a reflective subcategory of the presheaf category $[\C^{\op}, \Set]$ where the latter is cocomplete. As for completeness: suppose $(X_i)_{i\in \I}$ is a limit diagram in $\C$, and consider the functor $T$ which sends an object $U$ to the set of cones $U \to X_i$. Then $L(T)$ is a limit of $X_i$. To see this, first of all for each $i$ we have a morphism $T \to \Hom({-}, X_i)$ which selects the $i$ component of the cone; this corresponds to a morphism $L(T) \to X_i$. Similarly, for each $f : i \to j$ in $\I$ we have a morphism from the $i$ component functor to the $j$ component functor, so naturality of the adjunction implies that we get the required compatibility conditions for a cone $L(T) \to X_i$. Conversely, the identity morphism $\id : L(T) \to L(T)$ corresponds to a morphism $T \to \Hom({-}, L(T))$, so for any cone $U \to X_i$ we have a corresponding element of $T(U)$, which maps to a morphism in $\Hom(U, L(T))$.' + is_equivalence: false + +- id: cocomplete_well-copowered_generator_implies_total + assumptions: + - cocomplete + - well-copowered + - generator + conclusions: + - total + proof: ... + is_equivalence: false diff --git a/databases/catdat/data/category-properties/cototal.yaml b/databases/catdat/data/category-properties/cototal.yaml new file mode 100644 index 000000000..445c5d383 --- /dev/null +++ b/databases/catdat/data/category-properties/cototal.yaml @@ -0,0 +1,11 @@ +id: cototal +relation: is +description: 'A category $\C$ is cototal when its dual category is total, i.e. it is locally essentially small and the contravariant Yoneda embedding $y : \C \to [\C^{\op}, \Set]$ has a left adjoint.' +nlab_link: https://ncatlab.org/nlab/show/total+category +dual_property: total +invariant_under_equivalences: true + +related_properties: + - cocomplete + - complete + - locally copresentable diff --git a/databases/catdat/data/category-properties/locally copresentable.yaml b/databases/catdat/data/category-properties/locally copresentable.yaml index 12563b17d..27865e350 100644 --- a/databases/catdat/data/category-properties/locally copresentable.yaml +++ b/databases/catdat/data/category-properties/locally copresentable.yaml @@ -7,3 +7,4 @@ invariant_under_equivalences: true related_properties: - coaccessible - complete + - cototal diff --git a/databases/catdat/data/category-properties/locally presentable.yaml b/databases/catdat/data/category-properties/locally presentable.yaml index 53c681d2b..37683d540 100644 --- a/databases/catdat/data/category-properties/locally presentable.yaml +++ b/databases/catdat/data/category-properties/locally presentable.yaml @@ -17,6 +17,7 @@ invariant_under_equivalences: true related_properties: - accessible - cocomplete + - total - locally finitely presentable - locally multi-presentable - locally poly-presentable diff --git a/databases/catdat/data/category-properties/total.yaml b/databases/catdat/data/category-properties/total.yaml new file mode 100644 index 000000000..9d3266c1f --- /dev/null +++ b/databases/catdat/data/category-properties/total.yaml @@ -0,0 +1,11 @@ +id: total +relation: is +description: 'A category $\C$ is total when it is locally essentially small and the covariant Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ has a left adjoint. (For a concrete example of how this left adjoint might look, see here.)' +nlab_link: https://ncatlab.org/nlab/show/total+category +dual_property: cototal +invariant_under_equivalences: true + +related_properties: + - cocomplete + - complete + - locally presentable diff --git a/databases/catdat/scripts/expected-data/Ab.json b/databases/catdat/scripts/expected-data/Ab.json index 8e9060677..1900d9837 100644 --- a/databases/catdat/scripts/expected-data/Ab.json +++ b/databases/catdat/scripts/expected-data/Ab.json @@ -116,6 +116,8 @@ "ℵ₂-small coproducts": true, "ℵ₂-small powers": true, "ℵ₂-small copowers": true, + "total": true, + "cototal": true, "cartesian closed": false, "locally cartesian closed": false, diff --git a/databases/catdat/scripts/expected-data/Set.json b/databases/catdat/scripts/expected-data/Set.json index 5596e6796..763b547dd 100644 --- a/databases/catdat/scripts/expected-data/Set.json +++ b/databases/catdat/scripts/expected-data/Set.json @@ -112,6 +112,8 @@ "ℵ₂-small powers": true, "ℵ₂-small copowers": true, "pretopos": true, + "total": true, + "cototal": true, "Grothendieck abelian": false, "Malcev": false, diff --git a/databases/catdat/scripts/expected-data/Top.json b/databases/catdat/scripts/expected-data/Top.json index 8da72ea9c..91d7b7253 100644 --- a/databases/catdat/scripts/expected-data/Top.json +++ b/databases/catdat/scripts/expected-data/Top.json @@ -80,6 +80,8 @@ "ℵ₂-small coproducts": true, "ℵ₂-small powers": true, "ℵ₂-small copowers": true, + "total": true, + "cototal": true, "abelian": false, "additive": false, From dce21a3233e3ba2ab025da828c782394d3bae31f Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Tue, 30 Jun 2026 09:30:58 -0400 Subject: [PATCH 02/13] Fix notation in definitions --- databases/catdat/data/category-properties/cototal.yaml | 2 +- databases/catdat/data/category-properties/total.yaml | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/databases/catdat/data/category-properties/cototal.yaml b/databases/catdat/data/category-properties/cototal.yaml index 445c5d383..6a11e0a7a 100644 --- a/databases/catdat/data/category-properties/cototal.yaml +++ b/databases/catdat/data/category-properties/cototal.yaml @@ -1,6 +1,6 @@ id: cototal relation: is -description: 'A category $\C$ is cototal when its dual category is total, i.e. it is locally essentially small and the contravariant Yoneda embedding $y : \C \to [\C^{\op}, \Set]$ has a left adjoint.' +description: 'A category $\C$ is cototal when its dual category is total, i.e. it is locally essentially small and the contravariant Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ has a left adjoint.' nlab_link: https://ncatlab.org/nlab/show/total+category dual_property: total invariant_under_equivalences: true diff --git a/databases/catdat/data/category-properties/total.yaml b/databases/catdat/data/category-properties/total.yaml index 9d3266c1f..97145f3e0 100644 --- a/databases/catdat/data/category-properties/total.yaml +++ b/databases/catdat/data/category-properties/total.yaml @@ -1,6 +1,6 @@ id: total relation: is -description: 'A category $\C$ is total when it is locally essentially small and the covariant Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ has a left adjoint. (For a concrete example of how this left adjoint might look, see here.)' +description: 'A category $\C$ is total when it is locally essentially small and the covariant Yoneda embedding $y : \C \to [\C^{\op}, \Set]$ has a left adjoint. (For a concrete example of how this left adjoint might look, see here.)' nlab_link: https://ncatlab.org/nlab/show/total+category dual_property: cototal invariant_under_equivalences: true From 4e10c808ab44cd0a010574a48ba998865c52f64c Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Sat, 25 Jul 2026 10:29:03 -0400 Subject: [PATCH 03/13] Apply suggestions from code review Co-authored-by: Script Raccoon --- databases/catdat/data/category-properties/cototal.yaml | 4 ++-- databases/catdat/data/category-properties/total.yaml | 4 ++-- 2 files changed, 4 insertions(+), 4 deletions(-) diff --git a/databases/catdat/data/category-properties/cototal.yaml b/databases/catdat/data/category-properties/cototal.yaml index 6a11e0a7a..997e3f1d8 100644 --- a/databases/catdat/data/category-properties/cototal.yaml +++ b/databases/catdat/data/category-properties/cototal.yaml @@ -2,10 +2,10 @@ id: cototal relation: is description: 'A category $\C$ is cototal when its dual category is total, i.e. it is locally essentially small and the contravariant Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ has a left adjoint.' nlab_link: https://ncatlab.org/nlab/show/total+category -dual_property: total +dual: total invariant_under_equivalences: true -related_properties: +related: - cocomplete - complete - locally copresentable diff --git a/databases/catdat/data/category-properties/total.yaml b/databases/catdat/data/category-properties/total.yaml index 97145f3e0..20372a388 100644 --- a/databases/catdat/data/category-properties/total.yaml +++ b/databases/catdat/data/category-properties/total.yaml @@ -2,10 +2,10 @@ id: total relation: is description: 'A category $\C$ is total when it is locally essentially small and the covariant Yoneda embedding $y : \C \to [\C^{\op}, \Set]$ has a left adjoint. (For a concrete example of how this left adjoint might look, see here.)' nlab_link: https://ncatlab.org/nlab/show/total+category -dual_property: cototal +dual: cototal invariant_under_equivalences: true -related_properties: +related: - cocomplete - complete - locally presentable From a4d0184bf0c866fe57ecbfd28a0ae6ada00bbc50 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Sat, 25 Jul 2026 12:17:34 -0400 Subject: [PATCH 04/13] Fill in missing references Also update definitions to versions that make sense even for non locally small categories --- database/data/categories/Alg(R).yaml | 3 --- database/data/categories/CAlg(R).yaml | 3 --- database/data/categories/CRing.yaml | 3 --- database/data/categories/Grp.yaml | 2 +- database/data/categories/LRS_R.yaml | 6 ------ database/data/categories/Ring.yaml | 3 --- .../data/category-implications/accessible.yaml | 8 -------- database/data/category-implications/total.yaml | 18 ++++++++++-------- database/data/category-properties/cototal.yaml | 16 +++++++++++++++- database/data/category-properties/total.yaml | 16 +++++++++++++++- 10 files changed, 41 insertions(+), 37 deletions(-) diff --git a/database/data/categories/Alg(R).yaml b/database/data/categories/Alg(R).yaml index 23180ad7e..7e1887578 100644 --- a/database/data/categories/Alg(R).yaml +++ b/database/data/categories/Alg(R).yaml @@ -40,9 +40,6 @@ unsatisfied_properties: - property: semi-strongly connected proof: This is because already the full subcategory $\CAlg(R)$ of commutative algebras is not semi-strongly connected. - - property: cogenerating set - proof: '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: cototal proof: Essentially the same proof as for $\CAlg(R)$ works here. diff --git a/database/data/categories/CAlg(R).yaml b/database/data/categories/CAlg(R).yaml index 093a6401a..782b31d4d 100644 --- a/database/data/categories/CAlg(R).yaml +++ b/database/data/categories/CAlg(R).yaml @@ -38,9 +38,6 @@ unsatisfied_properties: - property: balanced proof: Take a prime ideal $P \subseteq R$ and consider the commutative $R$-algebra $A \coloneqq R/P$ (which is an integral domain). Then the inclusion $A \hookrightarrow Q(A)$ is a counterexample. - - property: cogenerating set - proof: '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: cototal proof: 'Let $\F$ be the family of commutative $R$-algebras of the form $R \times k$ where $k$ is an infinite field including a quotient field of $R$. Then for any commutative $R$-algebra $A$, we have a distinguished morphism $R \times k \to A$ consisting of the projection to $R$ followed by the unique morphism $R \to A$. Moreover, if we have any morphism $\varphi : R \times k \to A$ which is not equal to the distinguished morphism, that implies that $\varphi(0, 1) \ne 0$, so the rng homomorphism $k \to R \times k \to A$ is injective, implying $\card(U(A)) \ge \card(U(k))$. From here, an argument similar to the one here gives a contradiction, using the distinguished morphisms in place of zero morphisms.' diff --git a/database/data/categories/CRing.yaml b/database/data/categories/CRing.yaml index f25e4def7..7e53ba6d4 100644 --- a/database/data/categories/CRing.yaml +++ b/database/data/categories/CRing.yaml @@ -44,9 +44,6 @@ unsatisfied_properties: - property: balanced proof: The inclusion $\IZ \hookrightarrow \IQ$ is a counterexample. - - property: cogenerating set - proof: '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: cototal proof: 'This is a special case of the proof for $\CAlg(R)$ with $R = \IZ$.' diff --git a/database/data/categories/Grp.yaml b/database/data/categories/Grp.yaml index 427e5fd0e..fccde1f9a 100644 --- a/database/data/categories/Grp.yaml +++ b/database/data/categories/Grp.yaml @@ -40,7 +40,7 @@ satisfied_properties: proof: A proof can be found here. - property: total - proof: This follows formally from the fact that $\Grp$ is finitary algebraic and therefore locally presentable. For a more explicit proof, see here. + proof: This follows formally from the fact that $\Grp$ is finitary algebraic and therefore locally presentable. For a more explicit proof, see here. check_redundancy: false - property: extremal generator diff --git a/database/data/categories/LRS_R.yaml b/database/data/categories/LRS_R.yaml index eea6e8ea0..f052d9351 100644 --- a/database/data/categories/LRS_R.yaml +++ b/database/data/categories/LRS_R.yaml @@ -47,12 +47,6 @@ unsatisfied_properties: - property: co-Malcev proof: 'We can adjust the proof for $\Top$ (see MO/509548) as follows: Let $K$ be a residue field of $R$, let $X$ 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.' - - property: generating set - proof: >- - Out of any small set $S$ of locally ringed spaces, there is only a small set of residue fields at their points. Therefore, if $K$ is a field over $R$ with a strictly larger cardinality than any of these residue fields, then the only possible morphism from an element of $S$ to $\Spec K(X,Y)$ is one with an empty domain. However, that makes it impossible for $S$ to distinguish the two canonical automorphisms of $\Spec K(X,Y)$. - - Alternatively, using the usual adjunction between affine schemes and locally ringed spaces (EGA I (1971), Ch. 1, Prop. 1.6.3), a generating set in $\LRS_R$ would induce a generating set in the category of affine $R$-schemes, which contradicts the fact that $\CAlg(R)$ does not have a cogenerating set. - - property: total proof: 'The adjunction between the global sections functor and the $\Spec$ functor (EGA I (1971), Ch. 1, Prop. 1.6.3) makes $\CAlg(R)^{\op}$ into a reflective subcategory of $\LRS_R$. Therefore, if $LRS_R$ were total, then \CAlg(R) would be cototal, which we know is not the case.' diff --git a/database/data/categories/Ring.yaml b/database/data/categories/Ring.yaml index fb79b9d2e..3435a83e8 100644 --- a/database/data/categories/Ring.yaml +++ b/database/data/categories/Ring.yaml @@ -43,9 +43,6 @@ unsatisfied_properties: - property: semi-strongly connected proof: This is because already the full subcategory $\CRing$ does not have this property. - - property: cogenerating set - proof: '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: cototal proof: 'This is a special case of the proof for $\Alg(R)$ with $R = \IZ$.' diff --git a/database/data/category-implications/accessible.yaml b/database/data/category-implications/accessible.yaml index b65ed3ce3..b5870acc3 100644 --- a/database/data/category-implications/accessible.yaml +++ b/database/data/category-implications/accessible.yaml @@ -28,14 +28,6 @@ proof: This follows from one of equivalent formulations of locally presentable categories. is_equivalence: true -- id: locally_presentable_consequence - assumptions: - - locally presentable - conclusions: - - total - proof: ... - is_equivalence: false - - id: locally_finitely_presentable_consequence assumptions: - locally finitely presentable diff --git a/database/data/category-implications/total.yaml b/database/data/category-implications/total.yaml index 3eafb7b0f..c22c85dbc 100644 --- a/database/data/category-implications/total.yaml +++ b/database/data/category-implications/total.yaml @@ -1,28 +1,30 @@ # results on total and cototal categories -- id: total_definition +- id: total_cocomplete assumptions: - total conclusions: - - locally essentially small - proof: This is true by definition. + - cocomplete + proof: 'A diagram $X : I \to \C$ with $I$ small trivially satisfies the hypotheses of condition (2) in the definition of a total category.' is_equivalence: false -- id: total_consequences +- id: total_complete assumptions: - total + - locally essentially small conclusions: - complete - - cocomplete - proof: 'If a category $\C$ is total, then $\C$ is a reflective subcategory of the presheaf category $[\C^{\op}, \Set]$ where the latter is cocomplete. As for completeness: suppose $(X_i)_{i\in \I}$ is a limit diagram in $\C$, and consider the functor $T$ which sends an object $U$ to the set of cones $U \to X_i$. Then $L(T)$ is a limit of $X_i$. To see this, first of all for each $i$ we have a morphism $T \to \Hom({-}, X_i)$ which selects the $i$ component of the cone; this corresponds to a morphism $L(T) \to X_i$. Similarly, for each $f : i \to j$ in $\I$ we have a morphism from the $i$ component functor to the $j$ component functor, so naturality of the adjunction implies that we get the required compatibility conditions for a cone $L(T) \to X_i$. Conversely, the identity morphism $\id : L(T) \to L(T)$ corresponds to a morphism $T \to \Hom({-}, L(T))$, so for any cone $U \to X_i$ we have a corresponding element of $T(U)$, which maps to a morphism in $\Hom(U, L(T))$.' + proof: 'This is proven in Kelly, Thm. 5.6.' is_equivalence: false +# TODO: replace "well-copowered" with "epi-cocomplete" if adding the latter property - id: cocomplete_well-copowered_generator_implies_total assumptions: - cocomplete + - locally essentially small - well-copowered - - generator + - generating set conclusions: - total - proof: ... + proof: 'Since the category is cocomplete and well-copowered, it is epi-cocomplete (meaning that it has wide pushouts of epimorphisms, even of non-small families of epimorphisms). The result then follows from Day, Thm. 1.' is_equivalence: false diff --git a/database/data/category-properties/cototal.yaml b/database/data/category-properties/cototal.yaml index 997e3f1d8..0838c635e 100644 --- a/database/data/category-properties/cototal.yaml +++ b/database/data/category-properties/cototal.yaml @@ -1,6 +1,15 @@ id: cototal relation: is -description: 'A category $\C$ is cototal when its dual category is total, i.e. it is locally essentially small and the contravariant Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ has a left adjoint.' +description: >- + Let $\C$ be a category. Then the following are equivalent (Kelly, Thm. 5.5): + + 1. A diagram $X : \I \to \C$ (with $\I$ not necessarily essentially small) is called a discrete op-fibration if for every $i \in \I$ and morphism $f : X_i \to Y$ in $\C$, there exists a unique $\alpha : i\to j$ in $\I$ such that $X_\alpha = f$. This condition is that every discrete op-fibration whose fibers are bijective to sets has a limit in $\C$. + + 2. Let $X : \I \to \C$ be a diagram (with $\I$ not necessarily essentially small) such that for each object $Y$ of $\C$, the collection of connected components of the comma category $X \downarrow Y$ is bijective to a set. Then $X$ has a limit in $\C$. + + A general category is cototal if it satisfies one of the above conditions. + + Note that if $\C$ is locally small, $\C$ is total if and only if the contravariant Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ has a left adjoint (Kelly, Thm. 5.2). nlab_link: https://ncatlab.org/nlab/show/total+category dual: total invariant_under_equivalences: true @@ -9,3 +18,8 @@ related: - cocomplete - complete - locally copresentable + +tags: + - limits + - colimits + - size diff --git a/database/data/category-properties/total.yaml b/database/data/category-properties/total.yaml index 20372a388..85487fc4d 100644 --- a/database/data/category-properties/total.yaml +++ b/database/data/category-properties/total.yaml @@ -1,6 +1,15 @@ id: total relation: is -description: 'A category $\C$ is total when it is locally essentially small and the covariant Yoneda embedding $y : \C \to [\C^{\op}, \Set]$ has a left adjoint. (For a concrete example of how this left adjoint might look, see here.)' +description: >- + Let $\C$ be a category. Then the following are equivalent (Kelly, Thm. 5.5): + + 1. A diagram $X : \I \to \C$ (with $\I$ not necessarily essentially small) is called a discrete fibration if for every $i \in \I$ and morphism $f : Y \to X_i$ in $\C$, there exists a unique $\alpha : j \to i$ in $\I$ such that $X_\alpha = f$. This condition is that every discrete fibration whose fibers are bijective to sets has a colimit in $\C$. + + 2. Let $X : \I \to \C$ be a diagram (with $\I$ not necessarily essentially small) such that for each object $Y$ of $\C$, the collection of connected components of the comma category $Y \downarrow X$ is bijective to a set. Then $X$ has a colimit in $\C$. + + A general category is total if it satisfies one of the above conditions. + + Note that if $\C$ is locally small, $\C$ is total if and only if the covariant Yoneda embedding $y : \C \to [\C^{\op}, \Set]$ has a left adjoint (Kelly, Thm. 5.2). For a concrete example of how such a left adjoint could look, see here. nlab_link: https://ncatlab.org/nlab/show/total+category dual: cototal invariant_under_equivalences: true @@ -9,3 +18,8 @@ related: - cocomplete - complete - locally presentable + +tags: + - limits + - colimits + - size From b45356557c00c0c10f21ab172414c3f4eb966ee3 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Sat, 25 Jul 2026 13:18:39 -0400 Subject: [PATCH 05/13] Simplify a couple proofs using the fact that a cototal category is "hyper-cocomplete" to adapt the terminology in Kelly --- content/missing_cogenerator.md | 12 ++---------- database/data/categories/Cat.yaml | 8 ++------ 2 files changed, 4 insertions(+), 16 deletions(-) diff --git a/content/missing_cogenerator.md b/content/missing_cogenerator.md index a397ba6bb..a475180a1 100644 --- a/content/missing_cogenerator.md +++ b/content/missing_cogenerator.md @@ -13,18 +13,10 @@ Let $\C$ be a pointed category with a faithful functor $U: \C \to \Set$. Assume 1. For any $X \in \F$ and any $Y \in \C$, every non-zero morphism $f: X \to Y$ is injective on underlying sets. 2. For every $Y \in \C$ there is some object $X \in \F$ such that $\card(U(X)) > \card(U(Y))$. -Then $\C$ does not have a cogenerator. Moreover, $\C$ is not cototal. +Then $\C$ does not have a cogenerator. Moreover, if $\C$ is locally essentially small, then $\C$ is not cototal. ::: _Proof._ Assume that there is a cogenerator $Y$. By assumption (2) there is an object $X \in \F$ such that $U(X)$ is larger than $U(Y)$ (w.r.t. cardinalities). Since $0,\id_X : X \rightrightarrows X$ are distinct, there is a morphism $f : X \to Y$ with $f \neq 0$. But then $U(f) : U(X) \to U(Y)$ is injective by assumption (1), which contradicts our choice of $X$. -Now assume that $\C$ is cocotal. Using the axiom of choice, we may assume that for each small cardinal $\kappa$, there is at most one element $X \in \F$ such that $\card(U(X)) = \kappa$. We define a functor $T : \C \to \Set$ which is morally defined by $T(Y) \coloneqq \prod_{X \in \F} \Hom(X, Y)$. In this product, we know that $\Hom(X, Y) = \{ 0 \}$ whenever $\card(U(X)) > \card(U(Y))$, so every term except for a small number is a singleton and does not change the product. Thus, we may define -$$T(Y) \coloneqq \prod_{X \in \F : \card(U(X)) \le \card(U(Y))} \Hom(X, Y),$$ -and define the functor structure by extending tuples with zero morphisms as necessary. - -Now, if $L : [\C, \Set] \to \C^{\op}$ is the left adjoint of the contravariant Yoneda embedding $y : \C^{op} \to [\C, \Set]$, then we have a bijection -$$\Hom_{\C}(X, L(T)) \simeq \Hom(T, \Hom(X, {-}))$$ -for each object $X$ of $\C$. Now let $X_0$ be an element of $\F$ such that $\card(U(X_0)) > \card(U(L(T)))$. Then we can define two natural transformations $T \to \Hom({-}, X_0)$: one is the constant zero map; and the other is selection of the $X$ component, i.e. given an element of $TY = \prod_{X\in F : \card(U(X)) \le \card(U(Y))} \Hom(X, Y)$ we select the $X$ component if $\card(U(X)) \le \card(U(Y))$, or set the result to the zero morphism if $\card(U(X)) > \card(U(Y))$. Note these two natural transformations are unequal, since for example we can define an element of $T(X_0)$ which is $\id_{X_0}$ on the $X_0$ component and zero on the other components, and the two transformations map this element to $\id_{X_0} \ne 0$. - -Therefore, by the above bijection, $\Hom_{\C}(X_0, L(T))$ has at least two elements, out of which at least one must be non-zero and therefore injective on underlying sets. This contradicts the assumption that $\card(U(X_0)) > \card(U(L(T))$. $\square$ +Now assume that $\C$ is locally essentially small and cocotal. Using the axiom of choice, we may assume that for each small cardinal $\kappa$, there is at most one element $X \in \F$ such that $\card(U(X)) = \kappa$. Treating $\F$ as a discrete diagram in $\C$, assumption (1) implies that for any object $Y$ of $\C$, the collection of cocones $\F \to Y$ is bijective with a set, since the maps $X \to Y$ with $\card(U(X)) > \card(U(Y))$ must all be zero in such a cocone. Therefore, by Kelly, Thm. 5.6, $\C$ must have a coproduct $Y$ of all elements of $\F$. But then by assumption (2), there exists $X \in \F$ such that $\card(U(X)) > \card(U(Y))$; and since $\C$ is pointed, the coprojection $X \to Y$ must be split monic and therefore non-zero. Using assumption (1), we get a contradiction. $\square$ diff --git a/database/data/categories/Cat.yaml b/database/data/categories/Cat.yaml index b4ad80277..35a3926b0 100644 --- a/database/data/categories/Cat.yaml +++ b/database/data/categories/Cat.yaml @@ -87,13 +87,9 @@ unsatisfied_properties: - property: cototal proof: >- - For each infinite cardinal $\kappa$, choose a simple group $S_\kappa$ of cardinality $\kappa$ with at least $\kappa$ distinct automorphisms (for example the group of permutations of $\kappa$ of finite support which are even). We will define a functor $T : \Cat \to \Set$ which morally will send a category $\C$ to the collection-sized wide pullback of $[B S_\kappa, \C] \to [1, \C]$. Namely, if $\kappa$ is greater than the cardinality of any endomorphism monoid of an object of $\C$, then the image in $[1, \C]$ uniquely determines a functor in $[B S_\kappa, \C]$. Thus, we can restrict to the corresponding pullback with $\kappa$ restricted to be at most $\max(\aleph_0, \lambda)$ where $\lambda$ is the supremum of the cardinalities of endomorphism monoids; and this also forms a pullback of the full collection-sized diagram. + For each infinite cardinal $\kappa$, choose a simple group $S_\kappa$ of cardinality $\kappa$ (for example the group of permutations of $\kappa$ of finite support which are even). Now consider the ultra-wide pushout diagram $1 \rightrightarrows B S_\kappa$. Then for any small category $\C$, the collection of cocones $1 \rightrightarrows B S_\kappa \to \C$ is bijective with a set: to form any such cocone, we must first choose the object $X$ of $\C$ which is the image of the object of $1$. Then, we must choose the morphisms $S_\kappa \to \End_{\C}(X)$; but for $\kappa > \card(\End_{\C}(X))$, the corresponding morphism must be zero. - Now assume $\Cat$ is cototal with $L : [\Cat, \Set] \to \Cat^{\op}$ a left adjoint to the contravariant Yoneda embedding. Then for each infinite cardinal $\kappa$, we would have a bijection - $$[B S_\kappa, L(T)] \simeq \Hom(T, [B S_\kappa, -]).$$ - On the other hand, we have at least $\kappa$ distinct morphisms $T \to [B S_\kappa, -]$: for each automorphism $\pi$ of $S_\kappa$, we can take the natural transformation which selects the $\kappa$ component of the limit, and then composes with $B \pi$. To see these are distinct, apply them to the element of $T(B S_\kappa)$ which is the identity on the $\kappa$ component and $B(0)$ on the $\lambda$ component for $\lambda \ne \kappa$. - - However, if $\kappa$ is greater than the cardinality of any endomorphism monoid of $L(T)$, then any functor $B S_\kappa \to L(T)$ must act on the morphisms of $B S_\kappa$ by sending each to the identity morphism on the image object. Therefore, $\card([B S_\kappa, L(T)]) \le \card(\Ob(L(T)))$. Thus, if we also choose $\kappa > \card(\Ob(L(T)))$, then we get a contradiction. + On the other hand, we claim that $1 \rightrightarrows B S_\kappa$ does not have a pushout in $\Cat$; by Kelly, Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we have a pushout $\C$ of $1 \rightrightarrows B S_\kappa$, and choose a cardinal $\lambda > \card(\Mor(\C))$. Then the coprojection $i_\lambda : B S_\lambda \to \C$ must be split monic, since we can construct a cocone $1 \rightrightarrows B S_\kappa \to B S_\lambda$ in which $B S_\kappa \to B S_\lambda$ corresponds to the zero map for $\kappa \ne \lambda$, and in which $B S_\lambda \to B S_\lambda$ is the identity. It follows that if $X$ is the image in $\C$ of the object of $B S_\lambda$ under $i_\lambda$, then $i_\lambda$ induces an injective map $S_\lambda \to \End_{\C}(X)$. This gives a contradiction since $\lambda > \card(\End_{\C}(X))$ and $S_\lambda$ is a simple group. special_objects: initial object: From 0f2395a76ed8031154767df9b473ce50d0192414 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Sat, 25 Jul 2026 16:20:58 -0400 Subject: [PATCH 06/13] Add proof that Haus is not cototal --- database/data/categories/Haus.yaml | 11 +++++++---- 1 file changed, 7 insertions(+), 4 deletions(-) diff --git a/database/data/categories/Haus.yaml b/database/data/categories/Haus.yaml index 5d307c7e3..94805cc1f 100644 --- a/database/data/categories/Haus.yaml +++ b/database/data/categories/Haus.yaml @@ -73,10 +73,6 @@ unsatisfied_properties: $$(-\infty, -1/n] \cup [1/n, \infty) \hookrightarrow \IR$$ with itself. That is, $X_n$ is the union of two lines $\IR \times \{1\}$ and $\IR \times \{2\}$ where we identify $(x,1) \equiv (x,2)$ when $|x| \geq 1/n$. Then $X_n$ is Hausdorff, and there is a canonical surjective continuous map $X_n \to X_{n+1}$. The colimit in $\Top$ is the union of two lines where we identify $(x,1) \equiv (x,2)$ when $|x| \geq 1/n$ for some $n$, i.e. when $x \neq 0$. This is the line with the double origin, which is not Hausdorff. Its Hausdorff reflection is the line $\IR$ where all points of both lines are identified, and it provides the colimit in $\Haus$. Now, the injective continuous maps $\{1,2\} \to X_n$, $i \mapsto (0,i)$ (where $\{1,2\}$ is discrete) become the constant map $0 : \{1,2\} \to \IR$ in the colimit, which is not a monomorphism. - - property: cogenerator - # cspell: disable-next-line - proof: 'Assume that $Q$ is a cogenerator. Since $Q$ is Hausdorff, $Q$ is $T_1$. By a theorem of Herrlich (Wann sind alle stetigen Abbildungen in Y konstant. Math. Z. 90 (1965): 152-154. EUMDL), there is a regular Hausdorff space $X$ with $\geq 2$ points such that every continuous map $X \to Q$ is constant. (The author only states that $X$ is regular, but actually, $X$ is regular and $T_1$, hence Hausdorff.) But since $Q$ is a cogenerator, this implies that all maps $1 \rightrightarrows X$ are equal, i.e. that $X$ has just one point. This is a contradiction.' - - property: regular proof: 'The regular epimorphisms are precisely the surjective quotient maps of Hausdorff spaces (see below). In a regular category, for every regular epimorphism $X \to Y$ and every object $Z$, the induced morphism $X \times Z \to Y \times Z$ is again a regular epimorphism. This is not the case in $\Haus$ (or $\Top$, for that matter). The standard example is the quotient map $\IR \to \IR / \IZ^+$, for which the induced map $\IR \times \IQ \to \IR/\IZ^+ \times \IQ$ is not a quotient map (MSE/1907972).' @@ -87,6 +83,13 @@ unsatisfied_properties: Let $C \coloneqq \{1,2\}$ be the discrete two-point space. The map $f : A \to C$ defined by $f(a)=1$ for $a \in A_1$ and $f(a)=2$ for $a \in A_2$ is continuous, since $A$ is discrete. The pushout $C \sqcup_A \Gamma$ in $\Haus$ is the Hausdorff reflection of the pushout $Q$ in $\Top$. Notice that $Q$ is the quotient space of $\Gamma$ in which $A_1$ and $A_2$ are each collapsed to a point, denoted by $[A_1]$ and $[A_2]$. The canonical map $C \to Q$ is given by $i \mapsto [A_i]$. Now, $[A_1]$ and $[A_2]$ cannot be separated by disjoint open neighborhoods in $Q$, since such neighborhoods would pull back to disjoint open neighborhoods of $A_1$ and $A_2$ in $\Gamma$. Thus, they are identified in the Hausdorff reflection. This shows that the canonical map $C \to C \sqcup_A \Gamma$ is not injective and hence not a regular monomorphism. + - property: cototal + # cspell: disable-next-line + proof: >- + For each small cardinal $\kappa$, let $Q_\kappa$ be the product of all Hausdorff topological spaces whose underlying set is a non-empty subset of $\kappa$. By a theorem of Herrlich (Wann sind alle stetigen Abbildungen in Y konstant. Math. Z. 90 (1965): 152-154. EUMDL), there is a regular Hausdorff space $X_\kappa$ with at least two points such that every continuous map $X_\kappa \to Q_\kappa$ is constant. (The author only states that $X_\kappa$ is regular, but actually, $X_\kappa$ is regular and $T_1$, hence Hausdorff.) Choose a base point $x_\kappa \in X_\kappa$ for each $\kappa$. We can form an ultra-wide pushout diagram $1 \rightrightarrows X_\kappa$ where each morphism $1 \to X_\kappa$ corresponds to $x_\kappa$. Then for any Hausdorff space $Y$, the collection of cocones $1 \rightrightarrows X_\kappa \to Y$ is bijective to a set: if $Y$ is empty, then the collection of cocones is obviously empty. Otherwise, in order to form a cocone, we must first choose $y \in Y$ corresponding to the morphism $1 \to Y$. Then for each $\kappa \ge \card(U(Y))$, $Y$ is homeomorphic to one of the spaces in the product forming $Q_\kappa$. Therefore, there is a morphism $Y \to Q_\kappa$ splitting the projection map $Q_\kappa \to Y$. It follows that the map $X_\kappa \to Y$ is constant, and in fact it must be the constant map with image $y$. + + On the other hand, we claim that $1 \rightrightarrows X_\kappa$ does not have a pushout in $\Haus$; by Kelly, Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we had a pushout $Y$, and let $\lambda \coloneqq \card(U(Y))$. Then the coprojection $X_\lambda \to Y$ is split monic, since we can construct a cocone $1 \rightrightarrows X_\kappa \to X_\lambda$ where the map $X_\kappa \to X_\lambda$ is the constant map with image $x_\lambda$ when $\kappa \ne \lambda$, and the map $X_\lambda \to X_\lambda$ is the identity. But similarly to the previous paragraph, we can show any morphism $X_\lambda \to Y$ must be constant, giving a contradiction since $X_\lambda$ has at least two points. + - property: extremal generating set proof: The proof is the same as the one for $\Top$; there the test spaces we use are of the form $\kappa \sqcup \{ \kappa \}$ and $\kappa + 1$, which are both Hausdorff spaces. From 08f682287c4db24e8d58832806184883e021439a Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Sat, 25 Jul 2026 17:27:20 -0400 Subject: [PATCH 07/13] Add missing target="_blank" to some external references --- content/missing_cogenerator.md | 2 +- database/data/categories/Cat.yaml | 2 +- database/data/categories/Haus.yaml | 2 +- database/data/categories/SemiGrp.yaml | 16 ++++++++-------- database/data/category-implications/total.yaml | 4 ++-- database/data/category-properties/cototal.yaml | 6 +++--- database/data/category-properties/total.yaml | 6 +++--- 7 files changed, 19 insertions(+), 19 deletions(-) diff --git a/content/missing_cogenerator.md b/content/missing_cogenerator.md index a475180a1..ffcf9dfc2 100644 --- a/content/missing_cogenerator.md +++ b/content/missing_cogenerator.md @@ -19,4 +19,4 @@ Then $\C$ does not have a cogenerator. Moreover, if $\C$ is locally essentially _Proof._ Assume that there is a cogenerator $Y$. By assumption (2) there is an object $X \in \F$ such that $U(X)$ is larger than $U(Y)$ (w.r.t. cardinalities). Since $0,\id_X : X \rightrightarrows X$ are distinct, there is a morphism $f : X \to Y$ with $f \neq 0$. But then $U(f) : U(X) \to U(Y)$ is injective by assumption (1), which contradicts our choice of $X$. -Now assume that $\C$ is locally essentially small and cocotal. Using the axiom of choice, we may assume that for each small cardinal $\kappa$, there is at most one element $X \in \F$ such that $\card(U(X)) = \kappa$. Treating $\F$ as a discrete diagram in $\C$, assumption (1) implies that for any object $Y$ of $\C$, the collection of cocones $\F \to Y$ is bijective with a set, since the maps $X \to Y$ with $\card(U(X)) > \card(U(Y))$ must all be zero in such a cocone. Therefore, by Kelly, Thm. 5.6, $\C$ must have a coproduct $Y$ of all elements of $\F$. But then by assumption (2), there exists $X \in \F$ such that $\card(U(X)) > \card(U(Y))$; and since $\C$ is pointed, the coprojection $X \to Y$ must be split monic and therefore non-zero. Using assumption (1), we get a contradiction. $\square$ +Now assume that $\C$ is locally essentially small and cocotal. Using the axiom of choice, we may assume that for each small cardinal $\kappa$, there is at most one element $X \in \F$ such that $\card(U(X)) = \kappa$. Treating $\F$ as a discrete diagram in $\C$, assumption (1) implies that for any object $Y$ of $\C$, the collection of cocones $\F \to Y$ is bijective with a set, since the maps $X \to Y$ with $\card(U(X)) > \card(U(Y))$ must all be zero in such a cocone. Therefore, by Kelly, Thm. 5.6, $\C$ must have a coproduct $Y$ of all elements of $\F$. But then by assumption (2), there exists $X \in \F$ such that $\card(U(X)) > \card(U(Y))$; and since $\C$ is pointed, the coprojection $X \to Y$ must be split monic and therefore non-zero. Using assumption (1), we get a contradiction. $\square$ diff --git a/database/data/categories/Cat.yaml b/database/data/categories/Cat.yaml index 35a3926b0..6495ff237 100644 --- a/database/data/categories/Cat.yaml +++ b/database/data/categories/Cat.yaml @@ -89,7 +89,7 @@ unsatisfied_properties: proof: >- For each infinite cardinal $\kappa$, choose a simple group $S_\kappa$ of cardinality $\kappa$ (for example the group of permutations of $\kappa$ of finite support which are even). Now consider the ultra-wide pushout diagram $1 \rightrightarrows B S_\kappa$. Then for any small category $\C$, the collection of cocones $1 \rightrightarrows B S_\kappa \to \C$ is bijective with a set: to form any such cocone, we must first choose the object $X$ of $\C$ which is the image of the object of $1$. Then, we must choose the morphisms $S_\kappa \to \End_{\C}(X)$; but for $\kappa > \card(\End_{\C}(X))$, the corresponding morphism must be zero. - On the other hand, we claim that $1 \rightrightarrows B S_\kappa$ does not have a pushout in $\Cat$; by Kelly, Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we have a pushout $\C$ of $1 \rightrightarrows B S_\kappa$, and choose a cardinal $\lambda > \card(\Mor(\C))$. Then the coprojection $i_\lambda : B S_\lambda \to \C$ must be split monic, since we can construct a cocone $1 \rightrightarrows B S_\kappa \to B S_\lambda$ in which $B S_\kappa \to B S_\lambda$ corresponds to the zero map for $\kappa \ne \lambda$, and in which $B S_\lambda \to B S_\lambda$ is the identity. It follows that if $X$ is the image in $\C$ of the object of $B S_\lambda$ under $i_\lambda$, then $i_\lambda$ induces an injective map $S_\lambda \to \End_{\C}(X)$. This gives a contradiction since $\lambda > \card(\End_{\C}(X))$ and $S_\lambda$ is a simple group. + On the other hand, we claim that $1 \rightrightarrows B S_\kappa$ does not have a pushout in $\Cat$; by Kelly, Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we have a pushout $\C$ of $1 \rightrightarrows B S_\kappa$, and choose a cardinal $\lambda > \card(\Mor(\C))$. Then the coprojection $i_\lambda : B S_\lambda \to \C$ must be split monic, since we can construct a cocone $1 \rightrightarrows B S_\kappa \to B S_\lambda$ in which $B S_\kappa \to B S_\lambda$ corresponds to the zero map for $\kappa \ne \lambda$, and in which $B S_\lambda \to B S_\lambda$ is the identity. It follows that if $X$ is the image in $\C$ of the object of $B S_\lambda$ under $i_\lambda$, then $i_\lambda$ induces an injective map $S_\lambda \to \End_{\C}(X)$. This gives a contradiction since $\lambda > \card(\End_{\C}(X))$ and $S_\lambda$ is a simple group. special_objects: initial object: diff --git a/database/data/categories/Haus.yaml b/database/data/categories/Haus.yaml index 94805cc1f..ddbd4541c 100644 --- a/database/data/categories/Haus.yaml +++ b/database/data/categories/Haus.yaml @@ -88,7 +88,7 @@ unsatisfied_properties: proof: >- For each small cardinal $\kappa$, let $Q_\kappa$ be the product of all Hausdorff topological spaces whose underlying set is a non-empty subset of $\kappa$. By a theorem of Herrlich (Wann sind alle stetigen Abbildungen in Y konstant. Math. Z. 90 (1965): 152-154. EUMDL), there is a regular Hausdorff space $X_\kappa$ with at least two points such that every continuous map $X_\kappa \to Q_\kappa$ is constant. (The author only states that $X_\kappa$ is regular, but actually, $X_\kappa$ is regular and $T_1$, hence Hausdorff.) Choose a base point $x_\kappa \in X_\kappa$ for each $\kappa$. We can form an ultra-wide pushout diagram $1 \rightrightarrows X_\kappa$ where each morphism $1 \to X_\kappa$ corresponds to $x_\kappa$. Then for any Hausdorff space $Y$, the collection of cocones $1 \rightrightarrows X_\kappa \to Y$ is bijective to a set: if $Y$ is empty, then the collection of cocones is obviously empty. Otherwise, in order to form a cocone, we must first choose $y \in Y$ corresponding to the morphism $1 \to Y$. Then for each $\kappa \ge \card(U(Y))$, $Y$ is homeomorphic to one of the spaces in the product forming $Q_\kappa$. Therefore, there is a morphism $Y \to Q_\kappa$ splitting the projection map $Q_\kappa \to Y$. It follows that the map $X_\kappa \to Y$ is constant, and in fact it must be the constant map with image $y$. - On the other hand, we claim that $1 \rightrightarrows X_\kappa$ does not have a pushout in $\Haus$; by Kelly, Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we had a pushout $Y$, and let $\lambda \coloneqq \card(U(Y))$. Then the coprojection $X_\lambda \to Y$ is split monic, since we can construct a cocone $1 \rightrightarrows X_\kappa \to X_\lambda$ where the map $X_\kappa \to X_\lambda$ is the constant map with image $x_\lambda$ when $\kappa \ne \lambda$, and the map $X_\lambda \to X_\lambda$ is the identity. But similarly to the previous paragraph, we can show any morphism $X_\lambda \to Y$ must be constant, giving a contradiction since $X_\lambda$ has at least two points. + On the other hand, we claim that $1 \rightrightarrows X_\kappa$ does not have a pushout in $\Haus$; by Kelly, Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we had a pushout $Y$, and let $\lambda \coloneqq \card(U(Y))$. Then the coprojection $X_\lambda \to Y$ is split monic, since we can construct a cocone $1 \rightrightarrows X_\kappa \to X_\lambda$ where the map $X_\kappa \to X_\lambda$ is the constant map with image $x_\lambda$ when $\kappa \ne \lambda$, and the map $X_\lambda \to X_\lambda$ is the identity. But similarly to the previous paragraph, we can show any morphism $X_\lambda \to Y$ must be constant, giving a contradiction since $X_\lambda$ has at least two points. - property: extremal generating set proof: The proof is the same as the one for $\Top$; there the test spaces we use are of the form $\kappa \sqcup \{ \kappa \}$ and $\kappa + 1$, which are both Hausdorff spaces. diff --git a/database/data/categories/SemiGrp.yaml b/database/data/categories/SemiGrp.yaml index d7c904688..9d008990d 100644 --- a/database/data/categories/SemiGrp.yaml +++ b/database/data/categories/SemiGrp.yaml @@ -54,14 +54,6 @@ unsatisfied_properties: 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: cogenerating set - # TODO: find a variant of the lemma missing_cogenerating_sets - # (or missing_cogenerator) which handles this. - proof: >- - The proof is similar to the proof for $\Grp$. Assume that there is a cogenerating set $S$. There is an infinite simple group $G$ larger than all the semigroups in $S$ (such as an alternating group). Since $\id_G, 1 : G \rightrightarrows G$ are different, there is a semigroup $H \in S$ and a homomorphism of semigroups $f : G \to H$ with $f \neq f \circ 1$. Then - $$N \coloneqq \{g \in G : f(g) = f(1)\}$$ - is a normal subgroup of $G$. It is proper, and hence trivial. But then $f$ is injective, which is a contradiction. - - property: cofiltered-limit-stable epimorphisms proof: We already know that $\Set$ does not have this property (by this result). Now apply the contrapositive of the dual of Lemma 2 here to the functor $\Set \to \SemiGrp$ that equips a set with the multiplication $a \cdot b \coloneqq a$. @@ -71,6 +63,14 @@ unsatisfied_properties: $$\begin{align*} X & \coloneqq \langle p \mid p^2 = p \rangle,\\ E & \coloneqq \langle p, q \mid p^2 = p,\, q^2 = q,\, pq = q,\, qp = p \rangle, \end{align*}$$ 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: cototal + proof: >- + The proof is similar to the proof for $\Grp$. For each infinite cardinal $\kappa$, let $S_\kappa$ be a simple group of cardinality $\kappa$ (such as the alternating group on $\kappa$). We can then form the ultra-wide pushout diagram $1 \rightrightarrows S_\kappa$ in $\SemiGrp$. For every semigroup $A$, the collection of cocones $1 \rightrightarrows S_\kappa \to A$ is bijective to a set: for every such cocone, we must first choose an idempotent $e$ of $A$ corresponding to the map $1 \to A$. Then, whenever $\kappa > \card(U(A))$, then for $f_\kappa : S_\kappa \to A$ in the cocone, we see + $$N \coloneqq \{g \in S_\kappa : f_\kappa(g) = e\}$$ + is a normal subgroup of $G$. It must be non-trivial since otherwise $f_\kappa$ would induce an injective group homomorphism from $G$ to a group contained in $A$. Therefore, $N$ is all of $G$, so $f_\kappa$ is the constant map with image $a$. + + We now claim that $1 \rightrightarrows S_\kappa$ does not have a pushout in $\SemiGrp$; by Kelly, Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we had a pushout $A$, and let $\lambda$ be a cardinal strictly greater than $\card(U(A))$. Then the coprojection $S_\lambda\to A$ must be split monic, since we can construct a cocone $1 \rightrightarrows S_\kappa \to S_\lambda$ such that the map $S_\kappa \to S_\lambda$ is the constant map with image 1 if $\kappa \ne \lambda$, while the map $S_\lambda \to S_\lambda$ is the identity. But this contradicts the choice of $\lambda$. + - property: natural numbers object proof: >- Assume that a natural numbers object exists. Then by this result, for every semigroup $A$ the natural homomorphism diff --git a/database/data/category-implications/total.yaml b/database/data/category-implications/total.yaml index c22c85dbc..aa0333e66 100644 --- a/database/data/category-implications/total.yaml +++ b/database/data/category-implications/total.yaml @@ -14,7 +14,7 @@ - locally essentially small conclusions: - complete - proof: 'This is proven in Kelly, Thm. 5.6.' + proof: 'This is proven in Kelly, Thm. 5.6.' is_equivalence: false # TODO: replace "well-copowered" with "epi-cocomplete" if adding the latter property @@ -26,5 +26,5 @@ - generating set conclusions: - total - proof: 'Since the category is cocomplete and well-copowered, it is epi-cocomplete (meaning that it has wide pushouts of epimorphisms, even of non-small families of epimorphisms). The result then follows from Day, Thm. 1.' + proof: 'Since the category is cocomplete and well-copowered, it is epi-cocomplete (meaning that it has wide pushouts of epimorphisms, even of non-small families of epimorphisms). The result then follows from Day, Thm. 1.' is_equivalence: false diff --git a/database/data/category-properties/cototal.yaml b/database/data/category-properties/cototal.yaml index 0838c635e..dd948d968 100644 --- a/database/data/category-properties/cototal.yaml +++ b/database/data/category-properties/cototal.yaml @@ -1,15 +1,15 @@ id: cototal relation: is description: >- - Let $\C$ be a category. Then the following are equivalent (Kelly, Thm. 5.5): + Let $\C$ be a category. Then the following are equivalent (Kelly, Thm. 5.5): - 1. A diagram $X : \I \to \C$ (with $\I$ not necessarily essentially small) is called a discrete op-fibration if for every $i \in \I$ and morphism $f : X_i \to Y$ in $\C$, there exists a unique $\alpha : i\to j$ in $\I$ such that $X_\alpha = f$. This condition is that every discrete op-fibration whose fibers are bijective to sets has a limit in $\C$. + 1. A diagram $X : \I \to \C$ (with $\I$ not necessarily essentially small) is called a discrete op-fibration if for every $i \in \I$ and morphism $f : X_i \to Y$ in $\C$, there exists a unique $\alpha : i\to j$ in $\I$ such that $X_\alpha = f$. This condition is that every discrete op-fibration whose fibers are bijective to sets has a limit in $\C$. 2. Let $X : \I \to \C$ be a diagram (with $\I$ not necessarily essentially small) such that for each object $Y$ of $\C$, the collection of connected components of the comma category $X \downarrow Y$ is bijective to a set. Then $X$ has a limit in $\C$. A general category is cototal if it satisfies one of the above conditions. - Note that if $\C$ is locally small, $\C$ is total if and only if the contravariant Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ has a left adjoint (Kelly, Thm. 5.2). + Note that if $\C$ is locally small, $\C$ is total if and only if the contravariant Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ has a left adjoint (Kelly, Thm. 5.2). nlab_link: https://ncatlab.org/nlab/show/total+category dual: total invariant_under_equivalences: true diff --git a/database/data/category-properties/total.yaml b/database/data/category-properties/total.yaml index 85487fc4d..62fb22bc4 100644 --- a/database/data/category-properties/total.yaml +++ b/database/data/category-properties/total.yaml @@ -1,15 +1,15 @@ id: total relation: is description: >- - Let $\C$ be a category. Then the following are equivalent (Kelly, Thm. 5.5): + Let $\C$ be a category. Then the following are equivalent (Kelly, Thm. 5.5): - 1. A diagram $X : \I \to \C$ (with $\I$ not necessarily essentially small) is called a discrete fibration if for every $i \in \I$ and morphism $f : Y \to X_i$ in $\C$, there exists a unique $\alpha : j \to i$ in $\I$ such that $X_\alpha = f$. This condition is that every discrete fibration whose fibers are bijective to sets has a colimit in $\C$. + 1. A diagram $X : \I \to \C$ (with $\I$ not necessarily essentially small) is called a discrete fibration if for every $i \in \I$ and morphism $f : Y \to X_i$ in $\C$, there exists a unique $\alpha : j \to i$ in $\I$ such that $X_\alpha = f$. This condition is that every discrete fibration whose fibers are bijective to sets has a colimit in $\C$. 2. Let $X : \I \to \C$ be a diagram (with $\I$ not necessarily essentially small) such that for each object $Y$ of $\C$, the collection of connected components of the comma category $Y \downarrow X$ is bijective to a set. Then $X$ has a colimit in $\C$. A general category is total if it satisfies one of the above conditions. - Note that if $\C$ is locally small, $\C$ is total if and only if the covariant Yoneda embedding $y : \C \to [\C^{\op}, \Set]$ has a left adjoint (Kelly, Thm. 5.2). For a concrete example of how such a left adjoint could look, see here. + Note that if $\C$ is locally small, $\C$ is total if and only if the covariant Yoneda embedding $y : \C \to [\C^{\op}, \Set]$ has a left adjoint (Kelly, Thm. 5.2). For a concrete example of how such a left adjoint could look, see here. nlab_link: https://ncatlab.org/nlab/show/total+category dual: cototal invariant_under_equivalences: true From 008c153df4dadfd4a2e67b1c535a277b12574beb Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Wed, 29 Jul 2026 22:45:53 -0400 Subject: [PATCH 08/13] Fix implications for non locally essentially small categories --- database/data/category-implications/total.yaml | 12 +++++++++++- 1 file changed, 11 insertions(+), 1 deletion(-) diff --git a/database/data/category-implications/total.yaml b/database/data/category-implications/total.yaml index aa0333e66..057d6e058 100644 --- a/database/data/category-implications/total.yaml +++ b/database/data/category-implications/total.yaml @@ -3,9 +3,19 @@ - id: total_cocomplete assumptions: - total + - locally essentially small conclusions: - cocomplete - proof: 'A diagram $X : I \to \C$ with $I$ small trivially satisfies the hypotheses of condition (2) in the definition of a total category.' + proof: 'A diagram $X : \I \to \C$ with $\I$ small and $\C$ locally essentially small is easily seen to satisfy the hypotheses of condition (2) in the definition of a total category.' + is_equivalence: false + +# TODO: If adding a property "has a weak terminal object", that can be added to the conclusions here: The identity functor $\C \to \C$ is also a discrete fibration, and its colimit must certainly at least be a weakly terminal object. It's unclear to me at this time whether it must necessarily be a terminal object in case $\C$ is not locally essentially small. +- id: total_initial + assumptions: + - total + conclusions: + - initial object + proof: 'The empty diagram is vacuously a discrete fibration; thus, it has a colimit, which must be an initial object.' is_equivalence: false - id: total_complete From e90a4a360e3f9a40b82f68c90fc6839b344cbaa6 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Sat, 1 Aug 2026 10:21:14 -0400 Subject: [PATCH 09/13] Update Kelly and Day references with article titles --- content/missing_cogenerator.md | 2 +- database/data/categories/Cat.yaml | 2 +- database/data/categories/Haus.yaml | 2 +- database/data/categories/SemiGrp.yaml | 2 +- database/data/category-implications/total.yaml | 4 ++-- database/data/category-properties/cototal.yaml | 4 ++-- database/data/category-properties/total.yaml | 4 ++-- 7 files changed, 10 insertions(+), 10 deletions(-) diff --git a/content/missing_cogenerator.md b/content/missing_cogenerator.md index ffcf9dfc2..6649dea1f 100644 --- a/content/missing_cogenerator.md +++ b/content/missing_cogenerator.md @@ -19,4 +19,4 @@ Then $\C$ does not have a cogenerator. Moreover, if $\C$ is locally essentially _Proof._ Assume that there is a cogenerator $Y$. By assumption (2) there is an object $X \in \F$ such that $U(X)$ is larger than $U(Y)$ (w.r.t. cardinalities). Since $0,\id_X : X \rightrightarrows X$ are distinct, there is a morphism $f : X \to Y$ with $f \neq 0$. But then $U(f) : U(X) \to U(Y)$ is injective by assumption (1), which contradicts our choice of $X$. -Now assume that $\C$ is locally essentially small and cocotal. Using the axiom of choice, we may assume that for each small cardinal $\kappa$, there is at most one element $X \in \F$ such that $\card(U(X)) = \kappa$. Treating $\F$ as a discrete diagram in $\C$, assumption (1) implies that for any object $Y$ of $\C$, the collection of cocones $\F \to Y$ is bijective with a set, since the maps $X \to Y$ with $\card(U(X)) > \card(U(Y))$ must all be zero in such a cocone. Therefore, by Kelly, Thm. 5.6, $\C$ must have a coproduct $Y$ of all elements of $\F$. But then by assumption (2), there exists $X \in \F$ such that $\card(U(X)) > \card(U(Y))$; and since $\C$ is pointed, the coprojection $X \to Y$ must be split monic and therefore non-zero. Using assumption (1), we get a contradiction. $\square$ +Now assume that $\C$ is locally essentially small and cocotal. Using the axiom of choice, we may assume that for each small cardinal $\kappa$, there is at most one element $X \in \F$ such that $\card(U(X)) = \kappa$. Treating $\F$ as a discrete diagram in $\C$, assumption (1) implies that for any object $Y$ of $\C$, the collection of cocones $\F \to Y$ is bijective with a set, since the maps $X \to Y$ with $\card(U(X)) > \card(U(Y))$ must all be zero in such a cocone. Therefore, by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.6, $\C$ must have a coproduct $Y$ of all elements of $\F$. But then by assumption (2), there exists $X \in \F$ such that $\card(U(X)) > \card(U(Y))$; and since $\C$ is pointed, the coprojection $X \to Y$ must be split monic and therefore non-zero. Using assumption (1), we get a contradiction. $\square$ diff --git a/database/data/categories/Cat.yaml b/database/data/categories/Cat.yaml index 6495ff237..7c20e7d83 100644 --- a/database/data/categories/Cat.yaml +++ b/database/data/categories/Cat.yaml @@ -89,7 +89,7 @@ unsatisfied_properties: proof: >- For each infinite cardinal $\kappa$, choose a simple group $S_\kappa$ of cardinality $\kappa$ (for example the group of permutations of $\kappa$ of finite support which are even). Now consider the ultra-wide pushout diagram $1 \rightrightarrows B S_\kappa$. Then for any small category $\C$, the collection of cocones $1 \rightrightarrows B S_\kappa \to \C$ is bijective with a set: to form any such cocone, we must first choose the object $X$ of $\C$ which is the image of the object of $1$. Then, we must choose the morphisms $S_\kappa \to \End_{\C}(X)$; but for $\kappa > \card(\End_{\C}(X))$, the corresponding morphism must be zero. - On the other hand, we claim that $1 \rightrightarrows B S_\kappa$ does not have a pushout in $\Cat$; by Kelly, Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we have a pushout $\C$ of $1 \rightrightarrows B S_\kappa$, and choose a cardinal $\lambda > \card(\Mor(\C))$. Then the coprojection $i_\lambda : B S_\lambda \to \C$ must be split monic, since we can construct a cocone $1 \rightrightarrows B S_\kappa \to B S_\lambda$ in which $B S_\kappa \to B S_\lambda$ corresponds to the zero map for $\kappa \ne \lambda$, and in which $B S_\lambda \to B S_\lambda$ is the identity. It follows that if $X$ is the image in $\C$ of the object of $B S_\lambda$ under $i_\lambda$, then $i_\lambda$ induces an injective map $S_\lambda \to \End_{\C}(X)$. This gives a contradiction since $\lambda > \card(\End_{\C}(X))$ and $S_\lambda$ is a simple group. + On the other hand, we claim that $1 \rightrightarrows B S_\kappa$ does not have a pushout in $\Cat$; by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we have a pushout $\C$ of $1 \rightrightarrows B S_\kappa$, and choose a cardinal $\lambda > \card(\Mor(\C))$. Then the coprojection $i_\lambda : B S_\lambda \to \C$ must be split monic, since we can construct a cocone $1 \rightrightarrows B S_\kappa \to B S_\lambda$ in which $B S_\kappa \to B S_\lambda$ corresponds to the zero map for $\kappa \ne \lambda$, and in which $B S_\lambda \to B S_\lambda$ is the identity. It follows that if $X$ is the image in $\C$ of the object of $B S_\lambda$ under $i_\lambda$, then $i_\lambda$ induces an injective map $S_\lambda \to \End_{\C}(X)$. This gives a contradiction since $\lambda > \card(\End_{\C}(X))$ and $S_\lambda$ is a simple group. special_objects: initial object: diff --git a/database/data/categories/Haus.yaml b/database/data/categories/Haus.yaml index ddbd4541c..c9cd24a18 100644 --- a/database/data/categories/Haus.yaml +++ b/database/data/categories/Haus.yaml @@ -88,7 +88,7 @@ unsatisfied_properties: proof: >- For each small cardinal $\kappa$, let $Q_\kappa$ be the product of all Hausdorff topological spaces whose underlying set is a non-empty subset of $\kappa$. By a theorem of Herrlich (Wann sind alle stetigen Abbildungen in Y konstant. Math. Z. 90 (1965): 152-154. EUMDL), there is a regular Hausdorff space $X_\kappa$ with at least two points such that every continuous map $X_\kappa \to Q_\kappa$ is constant. (The author only states that $X_\kappa$ is regular, but actually, $X_\kappa$ is regular and $T_1$, hence Hausdorff.) Choose a base point $x_\kappa \in X_\kappa$ for each $\kappa$. We can form an ultra-wide pushout diagram $1 \rightrightarrows X_\kappa$ where each morphism $1 \to X_\kappa$ corresponds to $x_\kappa$. Then for any Hausdorff space $Y$, the collection of cocones $1 \rightrightarrows X_\kappa \to Y$ is bijective to a set: if $Y$ is empty, then the collection of cocones is obviously empty. Otherwise, in order to form a cocone, we must first choose $y \in Y$ corresponding to the morphism $1 \to Y$. Then for each $\kappa \ge \card(U(Y))$, $Y$ is homeomorphic to one of the spaces in the product forming $Q_\kappa$. Therefore, there is a morphism $Y \to Q_\kappa$ splitting the projection map $Q_\kappa \to Y$. It follows that the map $X_\kappa \to Y$ is constant, and in fact it must be the constant map with image $y$. - On the other hand, we claim that $1 \rightrightarrows X_\kappa$ does not have a pushout in $\Haus$; by Kelly, Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we had a pushout $Y$, and let $\lambda \coloneqq \card(U(Y))$. Then the coprojection $X_\lambda \to Y$ is split monic, since we can construct a cocone $1 \rightrightarrows X_\kappa \to X_\lambda$ where the map $X_\kappa \to X_\lambda$ is the constant map with image $x_\lambda$ when $\kappa \ne \lambda$, and the map $X_\lambda \to X_\lambda$ is the identity. But similarly to the previous paragraph, we can show any morphism $X_\lambda \to Y$ must be constant, giving a contradiction since $X_\lambda$ has at least two points. + On the other hand, we claim that $1 \rightrightarrows X_\kappa$ does not have a pushout in $\Haus$; by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we had a pushout $Y$, and let $\lambda \coloneqq \card(U(Y))$. Then the coprojection $X_\lambda \to Y$ is split monic, since we can construct a cocone $1 \rightrightarrows X_\kappa \to X_\lambda$ where the map $X_\kappa \to X_\lambda$ is the constant map with image $x_\lambda$ when $\kappa \ne \lambda$, and the map $X_\lambda \to X_\lambda$ is the identity. But similarly to the previous paragraph, we can show any morphism $X_\lambda \to Y$ must be constant, giving a contradiction since $X_\lambda$ has at least two points. - property: extremal generating set proof: The proof is the same as the one for $\Top$; there the test spaces we use are of the form $\kappa \sqcup \{ \kappa \}$ and $\kappa + 1$, which are both Hausdorff spaces. diff --git a/database/data/categories/SemiGrp.yaml b/database/data/categories/SemiGrp.yaml index 9d008990d..b1a76d1f3 100644 --- a/database/data/categories/SemiGrp.yaml +++ b/database/data/categories/SemiGrp.yaml @@ -69,7 +69,7 @@ unsatisfied_properties: $$N \coloneqq \{g \in S_\kappa : f_\kappa(g) = e\}$$ is a normal subgroup of $G$. It must be non-trivial since otherwise $f_\kappa$ would induce an injective group homomorphism from $G$ to a group contained in $A$. Therefore, $N$ is all of $G$, so $f_\kappa$ is the constant map with image $a$. - We now claim that $1 \rightrightarrows S_\kappa$ does not have a pushout in $\SemiGrp$; by Kelly, Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we had a pushout $A$, and let $\lambda$ be a cardinal strictly greater than $\card(U(A))$. Then the coprojection $S_\lambda\to A$ must be split monic, since we can construct a cocone $1 \rightrightarrows S_\kappa \to S_\lambda$ such that the map $S_\kappa \to S_\lambda$ is the constant map with image 1 if $\kappa \ne \lambda$, while the map $S_\lambda \to S_\lambda$ is the identity. But this contradicts the choice of $\lambda$. + We now claim that $1 \rightrightarrows S_\kappa$ does not have a pushout in $\SemiGrp$; by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we had a pushout $A$, and let $\lambda$ be a cardinal strictly greater than $\card(U(A))$. Then the coprojection $S_\lambda\to A$ must be split monic, since we can construct a cocone $1 \rightrightarrows S_\kappa \to S_\lambda$ such that the map $S_\kappa \to S_\lambda$ is the constant map with image 1 if $\kappa \ne \lambda$, while the map $S_\lambda \to S_\lambda$ is the identity. But this contradicts the choice of $\lambda$. - property: natural numbers object proof: >- diff --git a/database/data/category-implications/total.yaml b/database/data/category-implications/total.yaml index 057d6e058..0e9f35ccf 100644 --- a/database/data/category-implications/total.yaml +++ b/database/data/category-implications/total.yaml @@ -24,7 +24,7 @@ - locally essentially small conclusions: - complete - proof: 'This is proven in Kelly, Thm. 5.6.' + proof: 'This is proven in G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.6.' is_equivalence: false # TODO: replace "well-copowered" with "epi-cocomplete" if adding the latter property @@ -36,5 +36,5 @@ - generating set conclusions: - total - proof: 'Since the category is cocomplete and well-copowered, it is epi-cocomplete (meaning that it has wide pushouts of epimorphisms, even of non-small families of epimorphisms). The result then follows from Day, Thm. 1.' + proof: 'Since the category is cocomplete and well-copowered, it is epi-cocomplete (meaning that it has wide pushouts of epimorphisms, even of non-small families of epimorphisms). The result then follows from B. J. Day, Further criteria for totality (NumDam), Thm. 1.' is_equivalence: false diff --git a/database/data/category-properties/cototal.yaml b/database/data/category-properties/cototal.yaml index dd948d968..fb4db82f1 100644 --- a/database/data/category-properties/cototal.yaml +++ b/database/data/category-properties/cototal.yaml @@ -1,7 +1,7 @@ id: cototal relation: is description: >- - Let $\C$ be a category. Then the following are equivalent (Kelly, Thm. 5.5): + Let $\C$ be a category. Then the following are equivalent by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.5: 1. A diagram $X : \I \to \C$ (with $\I$ not necessarily essentially small) is called a discrete op-fibration if for every $i \in \I$ and morphism $f : X_i \to Y$ in $\C$, there exists a unique $\alpha : i\to j$ in $\I$ such that $X_\alpha = f$. This condition is that every discrete op-fibration whose fibers are bijective to sets has a limit in $\C$. @@ -9,7 +9,7 @@ description: >- A general category is cototal if it satisfies one of the above conditions. - Note that if $\C$ is locally small, $\C$ is total if and only if the contravariant Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ has a left adjoint (Kelly, Thm. 5.2). + Note that if $\C$ is locally small, $\C$ is total if and only if the contravariant Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ has a left adjoint. This is by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.2. nlab_link: https://ncatlab.org/nlab/show/total+category dual: total invariant_under_equivalences: true diff --git a/database/data/category-properties/total.yaml b/database/data/category-properties/total.yaml index 62fb22bc4..bc6f9b7f4 100644 --- a/database/data/category-properties/total.yaml +++ b/database/data/category-properties/total.yaml @@ -1,7 +1,7 @@ id: total relation: is description: >- - Let $\C$ be a category. Then the following are equivalent (Kelly, Thm. 5.5): + Let $\C$ be a category. Then the following are equivalent by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.5: 1. A diagram $X : \I \to \C$ (with $\I$ not necessarily essentially small) is called a discrete fibration if for every $i \in \I$ and morphism $f : Y \to X_i$ in $\C$, there exists a unique $\alpha : j \to i$ in $\I$ such that $X_\alpha = f$. This condition is that every discrete fibration whose fibers are bijective to sets has a colimit in $\C$. @@ -9,7 +9,7 @@ description: >- A general category is total if it satisfies one of the above conditions. - Note that if $\C$ is locally small, $\C$ is total if and only if the covariant Yoneda embedding $y : \C \to [\C^{\op}, \Set]$ has a left adjoint (Kelly, Thm. 5.2). For a concrete example of how such a left adjoint could look, see here. + Note that if $\C$ is locally small, $\C$ is total if and only if the covariant Yoneda embedding $y : \C \to [\C^{\op}, \Set]$ has a left adjoint. This is by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.2). For a concrete example of how such a left adjoint could look, see here. nlab_link: https://ncatlab.org/nlab/show/total+category dual: cototal invariant_under_equivalences: true From d4536ecb01845a7a5c27e7a6e1f9c3eda64a78f2 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Sat, 1 Aug 2026 10:54:13 -0400 Subject: [PATCH 10/13] Add reference to result that reflective subcategory of a total category is total --- database/data/categories/LRS_R.yaml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/database/data/categories/LRS_R.yaml b/database/data/categories/LRS_R.yaml index 4f681fc04..368af023c 100644 --- a/database/data/categories/LRS_R.yaml +++ b/database/data/categories/LRS_R.yaml @@ -63,7 +63,7 @@ unsatisfied_properties: proof: 'We can adjust the proof for $\Top$ (see MO/509548) as follows: Let $K$ be a residue field of $R$, let $X$ 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.' - property: total - proof: 'The adjunction between the global sections functor and the $\Spec$ functor (EGA I (1971), Ch. 1, Prop. 1.6.3) makes $\CAlg(R)^{\op}$ into a reflective subcategory of $\LRS_R$. Therefore, if $LRS_R$ were total, then \CAlg(R) would be cototal, which we know is not the case.' + proof: 'The adjunction between the global sections functor and the $\Spec$ functor (EGA I (1971), Ch. 1, Prop. 1.6.3) makes $\CAlg(R)^{\op}$ into a reflective subcategory of $\LRS_R$. Therefore, if $LRS_R$ were total, then \CAlg(R) would be cototal by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Cor. 6.2, which we know is not the case.' - property: cartesian closed proof: This is Corollary 4(a) here. From 4fd8493f476e1f7aa378ecba0dea013549b4cb96 Mon Sep 17 00:00:00 2001 From: Script Raccoon Date: Sat, 1 Aug 2026 20:41:51 +0200 Subject: [PATCH 11/13] reformulate definitions of total and cototal categories --- .cspell.json | 3 +++ .../data/category-properties/cototal.yaml | 19 +++++++++---------- database/data/category-properties/total.yaml | 17 ++++++++--------- 3 files changed, 20 insertions(+), 19 deletions(-) diff --git a/.cspell.json b/.cspell.json index 8d092b139..8a21d5870 100644 --- a/.cspell.json +++ b/.cspell.json @@ -128,6 +128,7 @@ "cospans", "cosymmetric", "cosymmetry", + "cototal", "cotransitive", "cotransitivity", "counit", @@ -156,6 +157,7 @@ "extensivity", "extremal", "Faddeev", + "fibration", "fieldification", "filtrations", "finitary", @@ -238,6 +240,7 @@ "Noetherian", "Noncommutative", "objectwise", + "opfibration", "pointwise", "Pontryagin", "poset", diff --git a/database/data/category-properties/cototal.yaml b/database/data/category-properties/cototal.yaml index fb4db82f1..95203da5a 100644 --- a/database/data/category-properties/cototal.yaml +++ b/database/data/category-properties/cototal.yaml @@ -1,16 +1,15 @@ id: cototal relation: is description: >- - Let $\C$ be a category. Then the following are equivalent by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.5: - - 1. A diagram $X : \I \to \C$ (with $\I$ not necessarily essentially small) is called a discrete op-fibration if for every $i \in \I$ and morphism $f : X_i \to Y$ in $\C$, there exists a unique $\alpha : i\to j$ in $\I$ such that $X_\alpha = f$. This condition is that every discrete op-fibration whose fibers are bijective to sets has a limit in $\C$. - - 2. Let $X : \I \to \C$ be a diagram (with $\I$ not necessarily essentially small) such that for each object $Y$ of $\C$, the collection of connected components of the comma category $X \downarrow Y$ is bijective to a set. Then $X$ has a limit in $\C$. - - A general category is cototal if it satisfies one of the above conditions. - - Note that if $\C$ is locally small, $\C$ is total if and only if the contravariant Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ has a left adjoint. This is by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.2. -nlab_link: https://ncatlab.org/nlab/show/total+category + A category $\C$ is called cototal if it satisfies one of the following equivalent conditions: +
    +
  1. Every discrete opfibration $X : \I \to \C$ (with $\I$ not necessarily essentially small) whose fibers are bijective to sets has a limit in $\C$. (Recall that $X$ is a discrete opfibration if for every morphism $f : X_i \to Y$ in $\C$ there exists a unique $\alpha : i \to j$ in $\I$ such that $X_\alpha = f$.)
    2. +
  2. Let $X : \I \to \C$ be a diagram (with $\I$ not necessarily essentially small) such that for each object $Y$ of $\C$, the collection of connected components of the comma category $X \downarrow Y$ is bijective to a set. Then $X$ has a limit in $\C$.
    3. +
+ The equivalence is proven as Thm. 5.5 in G. M. Kelly, A survey of totality for enriched and ordinary categories. If $\C$ is locally small, $\C$ is cototal if and only if the contravariant Yoneda embedding + $$y : \C^{\op} \to [\C, \Set]$$ + has a left adjoint; see Thm. 5.2. in op. cit. +nlab_link: https://ncatlab.org/nlab/show/cototal+category dual: total invariant_under_equivalences: true diff --git a/database/data/category-properties/total.yaml b/database/data/category-properties/total.yaml index bc6f9b7f4..e47f5baec 100644 --- a/database/data/category-properties/total.yaml +++ b/database/data/category-properties/total.yaml @@ -1,15 +1,14 @@ id: total relation: is description: >- - Let $\C$ be a category. Then the following are equivalent by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.5: - - 1. A diagram $X : \I \to \C$ (with $\I$ not necessarily essentially small) is called a discrete fibration if for every $i \in \I$ and morphism $f : Y \to X_i$ in $\C$, there exists a unique $\alpha : j \to i$ in $\I$ such that $X_\alpha = f$. This condition is that every discrete fibration whose fibers are bijective to sets has a colimit in $\C$. - - 2. Let $X : \I \to \C$ be a diagram (with $\I$ not necessarily essentially small) such that for each object $Y$ of $\C$, the collection of connected components of the comma category $Y \downarrow X$ is bijective to a set. Then $X$ has a colimit in $\C$. - - A general category is total if it satisfies one of the above conditions. - - Note that if $\C$ is locally small, $\C$ is total if and only if the covariant Yoneda embedding $y : \C \to [\C^{\op}, \Set]$ has a left adjoint. This is by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.2). For a concrete example of how such a left adjoint could look, see here. + A category $\C$ is called total if it satisfies one of the following equivalent conditions: +
    +
  1. Every discrete fibration $X : \I \to \C$ (with $\I$ not necessarily essentially small) whose fibers are bijective to sets has a colimit in $\C$. (Recall that $X$ is a discrete fibration if for every morphism $f : Y \to X_i$ in $\C$ there exists a unique $\alpha : j \to i$ in $\I$ such that $X_\alpha = f$.)
    2. +
  2. Let $X : \I \to \C$ be a diagram (with $\I$ not necessarily essentially small) such that for each object $Y$ of $\C$, the collection of connected components of the comma category $Y \downarrow X$ is bijective to a set. Then $X$ has a colimit in $\C$.
    3. +
+ The equivalence is proven as Thm. 5.5 in G. M. Kelly, A survey of totality for enriched and ordinary categories. If $\C$ is locally small, $\C$ is total if and only if the covariant Yoneda embedding + $$y : \C \to [\C^{\op}, \Set]$$ + has a left adjoint; see Thm. 5.2. in op. cit. For a concrete example of how such a left adjoint could look, see here. nlab_link: https://ncatlab.org/nlab/show/total+category dual: cototal invariant_under_equivalences: true From 12689acfac006afbb2d3bd6d3aa1d804f0033a0c Mon Sep 17 00:00:00 2001 From: Script Raccoon Date: Sun, 2 Aug 2026 08:19:34 +0200 Subject: [PATCH 12/13] reformulate proof that Grp is total --- .cspell.json | 4 ++ content/Grp_total_explicit_proof.md | 65 +++++++++++++++++++++-------- 2 files changed, 52 insertions(+), 17 deletions(-) diff --git a/.cspell.json b/.cspell.json index 8a21d5870..5e33c5132 100644 --- a/.cspell.json +++ b/.cspell.json @@ -90,7 +90,10 @@ "cogenerating", "cogenerator", "cogenerators", + "cogroup", "Cohomology", + "coidentity", + "coinverse", "cokernel", "cokernels", "colimit", @@ -100,6 +103,7 @@ "comonadicity", "compactification", "compactifications", + "comultiplication", "conormal", "copower", "copowers", diff --git a/content/Grp_total_explicit_proof.md b/content/Grp_total_explicit_proof.md index 2149442f0..7a7f2d013 100644 --- a/content/Grp_total_explicit_proof.md +++ b/content/Grp_total_explicit_proof.md @@ -1,37 +1,68 @@ --- title: Explicit Proof that the Category of Groups is Total description: An explicit construction of the left adjoint to the covariant Yoneda embedding on the category of groups -author: Daniel Schepler --- ## Explicit Proof that the Category of Groups is Total -We will fix a functor $T : \Grp^{\op} \to \Set$. Based on this, we need to construct a group $L(T)$. The construction will work as follows: we start with a set of generators $e_x$, one for each element $x \in T\IZ$. Now let $\mu : \IZ \to \IZ * \IZ'$ be the comultiplication homomorphism, $1 \mapsto 1 \cdot 1'$, and $i_1, i_2 : \IZ \rightrightarrows \IZ * \IZ'$ the two coprojections. We now add a relation $e_{T\mu(x)} = e_{Ti_1(x)} \cdot e_{Ti_2(x)}$ for each element $x \in T(\IZ * \IZ')$. Similarly, letting $\iota : \IZ \to \IZ$ be the coinverse homomorphism, we add a relation $e_{T\iota(x)} = e_x^{-1}$ for each element $x \in T\IZ$; and letting $\varepsilon : \IZ \to 0$ be the coidentity homomorphism, we add a relation $e_{T\varepsilon(x)} = 1$ for each element $x \in T0$. +We give an explicit construction of the functor +$$L : [\Grp^{\op},\Set] \to \Grp$$ +that is left adjoint to the Yoneda embedding $y : \Grp \hookrightarrow [\Grp^{\op},\Set]$. -We first need to define the natural transformation $\eta_T : T \to \Hom({-}, L(T))$. To start, for each group $H$ we need a function $TH \to \Hom(H, L(T))$. We will define this function to send $x \in TH$ to $h \mapsto e_{Th(x)}$, where we abuse notation to identify $h \in H$ with the corresponding morphism $\IZ \to H$, so that $Th : TH \to T\IZ$. To see that this defines a group homomorphism from $H$ to $L(T)$, note that for $h, h' \in H$ we have three commutative diagrams of the form -$$\begin{CD} +Fix a functor $T : \Grp^{\op} \to \Set$. To construct the group $L(T)$, we will make use of the usual cogroup structure on $\IZ$ in $\Grp$, which includes + +- the comultiplication homomorphism $\mu : \IZ \to \IZ * \IZ'$, $1 \mapsto 1 \cdot 1'$ (where $\IZ'$ denotes a copy of $\IZ$), +- the coidentity homomorphism $\varepsilon : \IZ \to 0$, +- the coinverse homomorphism $\iota : \IZ \to \IZ$. + +Also, let $i_1,i_2 : \IZ \rightrightarrows \IZ * \IZ'$ denote the coprojections. We define the group $L(T)$ as the group generated by elements $e(x)$, one for each element $x \in T(\IZ)$, subject to the following relations: + +- $e(T\mu(x)) = e(Ti_1(x)) \cdot e(Ti_2(x))$ for each $x \in T(\IZ * \IZ')$, +- $e(T\varepsilon(x)) = 1$ for each $x \in T0$, +- $e(T\iota(x)) = e(x)^{-1}$ for each $x \in T\IZ$, + +We first need to define a natural transformation $\eta_T : T \to \Hom({-}, L(T))$. For each group $H$ we define the function $\eta_T(H) : TH \to \Hom(H, L(T))$ by sending $x \in TH$ to $h \mapsto e(Th(x))$, where we abuse notation to identify $h \in H$ with the corresponding morphism $\IZ \to H$ mapping $1 \mapsto h$, so that $Th : TH \to T\IZ$. To see that this defines a group homomorphism from $H$ to $L(T)$, note that for $h, h' \in H$ we have three commutative diagrams of the form + +$$ +\begin{CD} T(H) @> = >> T(H)\\ -@V T(h * h') VV @VVV\\ +@V T(hh') VV @VVV\\ T(\IZ * \IZ') @>>> T(\IZ) -\end{CD}$$ -where on the bottom we use $T\mu, Ti_1, Ti_2$. Applying this to $x\in TH$, we get $Th(x)$, $Th'(x)$, and $T(h h')(x)$, respectively. Thus, the relation $e_{T\mu(y)} = e_{Ti_1(y)} \cdot e_{Ti_2(y)}$ with $y \coloneqq T(h * h')(x)$ implies -$$e_{T(hh')(x)} = e_{Th(x)} e_{Th'(x)},$$ -as required. Similar proofs show that the map $H \to G$ respects inverses and the identity. We leave it as an exercise for the reader to show this is natural in $H$. +\end{CD} +$$ -We now need to show that for each group $G$ and natural transformation $\alpha : T \to y_G$, there exists a unique group homomorphism $\varphi : L(T) \to G$ such that $\alpha = y_{\varphi} \circ \eta_T : T \to \Hom({-}, L(T)) \to \Hom({-}, G)$. We start with uniqueness: suppose $x \in T\IZ$. Then by hypothesis, $\alpha_{\IZ} = y_{\varphi} \circ (\eta_T)_{\alpha} : T\IZ \to \Hom(\IZ, L(T)) \to \Hom(\IZ, G)$. For each $x \in T\IZ$, the first step on the right hand side maps $x \mapsto (1 \mapsto e_x)$, and the second step then maps this to $1 \mapsto \varphi(e_x)$. Therefore, $\varphi(e_x) = \alpha_{\IZ}(x)(1)$ for each $x$, which establishes the uniqueness of $\varphi$. +where on the bottom we use $T\mu, Ti_1, Ti_2$, and on the right we use $h h', h, h'$. Applying this to $x\in TH$, we get $Th(x)$, $Th'(x)$, and $T(h h')(x)$, respectively. Thus, the relation $e(T\mu(y)) = e(Ti_1(y)) \cdot e(Ti_2(y))$ with $y \coloneqq T(h h')(x)$ implies +$$e(T(hh')(x)) = e(Th(x)) \cdot e(Th'(x)),$$ +as required. Similar proofs show that the map $H \to L(T)$ respects inverses and the identity. We leave it as an exercise for the reader to show this is natural in $H$. -For the existence part, the first step is to show there is a gruop homomorphism $L(T) \to G$ with the images of $e_x$ required by the previous part, i.e. $e_x \mapsto \alpha_{\IZ}(x)(1)$. To prove this, we need to check that the relations in $L(T)$ are collapsed in $G$. Now, for each $x \in T(\IZ * \IZ')$, we have three commutative diagrams of the form -$$\begin{CD} +We now need to show that for each group $G$ and natural transformation $\alpha : T \to y_G$, there exists a unique group homomorphism $\varphi : L(T) \to G$ such that +$$\alpha = y_{\varphi} \circ \eta_T : T \to \Hom({-}, L(T)) \to \Hom({-}, G).$$ +We start with uniqueness: suppose $x \in T\IZ$. Then by hypothesis, +$$\alpha_{\IZ} = (y_{\varphi})_{\IZ} \circ (\eta_T)_{\IZ} : T\IZ \to \Hom(\IZ, L(T)) \to \Hom(\IZ, G).$$ +For each $x \in T\IZ$, the first step on the right hand side maps $x \mapsto (1 \mapsto e(x))$, and the second step then maps this to $1 \mapsto \varphi(e(x))$. Therefore, +$$\varphi(e(x)) = \alpha_{\IZ}(x)(1)$$ +for each $x$, which establishes the uniqueness of $\varphi$. + +For the existence part, the first step is to show there is a group homomorphism $L(T) \to G$ with the images of $e(x)$ required by the previous part, i.e. $e(x) \mapsto \alpha_{\IZ}(x)(1)$. To prove this, we need to check that the relations in $L(T)$ are satisfied in $G$. Now, for each $x \in T(\IZ * \IZ')$, we have three commutative diagrams of the form + +$$ +\begin{CD} T(\IZ * \IZ') @> \alpha_{\IZ * \IZ'} >> \Hom(\IZ * \IZ', G) @> \simeq >> UG \times UG\\ @VVV @VVV @VVV\\ T(\IZ) @> \alpha_{\IZ} >> \Hom(\IZ, G) @> \simeq >> UG -\end{CD}$$ -applying naturality to $\mu, i_1, i_2 : \IZ \to \IZ * \IZ'$. On the right hand side, we get multiplication, first projection, and second projection respectively. From this, we conclude that the images of $e_{T\mu(x)}$ and $e_{Ti_1(x)} \cdot e_{Ti_2(x)}$ in $UG$ agree for any element $x \in T(\IZ * \IZ')$. Similar proofs show that the other relations are also collapsed. +\end{CD} +$$ + +applying naturality to $\mu, i_1, i_2 : \IZ \to \IZ * \IZ'$. On the right hand side, we get multiplication, first projection, and second projection respectively. From this, we conclude that the images of $e(T\mu(x))$ and $e(Ti_1(x)) \cdot e(Ti_2(x))$ in $UG$ agree for any element $x \in T(\IZ * \IZ')$. Similar proofs show that the other relations are also satisfied. -Finally, we need to show $\alpha = y_{\varphi} \circ \eta_T$. It suffices to show $\alpha_H = (y_{\varphi})_H \circ (\eta_T)_H : TH \to \Hom(H, L(T)) \to \Hom(H, G)$ for each group $H$. By definition, for each $x \in TH$, the first step gives the homomorphism $h \mapsto e_{Th(x)}$; then the second step is formed by composition with $\varphi$. By the specification of $\varphi$, this gives the homorphism $h \mapsto \alpha_{\IZ}(Th(x))(1)$. However, by the assumption that $\alpha$ is a natural transformation, for each $h \in H$ we have a commutative diagram -$$\begin{CD} +Finally, we need to show $\alpha = y_{\varphi} \circ \eta_T$, i.e. $\alpha_H = (y_{\varphi})_H \circ (\eta_T)_H$ for each group $H$. By definition, for each $x \in TH$, the first step gives the homomorphism $h \mapsto e(Th(x))$; then the second step is formed by composition with $\varphi$. By the specification of $\varphi$, this gives the homomorphism $h \mapsto \alpha_{\IZ}(Th(x))(1)$. However, by the assumption that $\alpha$ is a natural transformation, for each $h \in H$ we have a commutative diagram + +$$ +\begin{CD} TH @> \alpha_H >> \Hom(H, G) \\ @V Th VV @VV {-} \circ h V \\ T\IZ @> \alpha_{\IZ} >> \Hom(\IZ, G). -\end{CD}$$ +\end{CD} +$$ + Applying this to $x \in TH$ gives exactly that $\alpha_{\IZ}(Th(x))(1) = \alpha_H(x)(h)$. $\square$ From 2d5d53e9db5ca666be1bc682f8a0f9b5bb39668d Mon Sep 17 00:00:00 2001 From: Script Raccoon Date: Sun, 2 Aug 2026 08:58:42 +0200 Subject: [PATCH 13/13] change citation style; small fixes --- .cspell.json | 1 + content/missing_cogenerator.md | 2 +- database/data/categories/Cat.yaml | 2 +- database/data/categories/Haus.yaml | 2 +- database/data/categories/LRS_R.yaml | 2 +- database/data/categories/On.yaml | 4 --- database/data/categories/SemiGrp.yaml | 4 +-- .../data/category-implications/total.yaml | 32 +++++++++---------- 8 files changed, 23 insertions(+), 26 deletions(-) diff --git a/.cspell.json b/.cspell.json index 5e33c5132..a07db56b5 100644 --- a/.cspell.json +++ b/.cspell.json @@ -184,6 +184,7 @@ "hausdorff", "Hertweck", "Heyting", + "homeomorphic", "homotopic", "homotopy", "Hušek", diff --git a/content/missing_cogenerator.md b/content/missing_cogenerator.md index 29ae0be60..36b52e8f1 100644 --- a/content/missing_cogenerator.md +++ b/content/missing_cogenerator.md @@ -18,4 +18,4 @@ Then $\C$ does not have a cogenerator. Moreover, if $\C$ is locally essentially _Proof._ Assume that there is a cogenerator $Y$. By assumption (2) there is an object $X \in \F$ such that $U(X)$ is larger than $U(Y)$ (w.r.t. cardinalities). Since $0,\id_X : X \rightrightarrows X$ are distinct, there is a morphism $f : X \to Y$ with $f \neq 0$. But then $U(f) : U(X) \to U(Y)$ is injective by assumption (1), which contradicts our choice of $X$. -Now assume that $\C$ is locally essentially small and cocotal. Using the axiom of choice, we may assume that for each small cardinal $\kappa$, there is at most one element $X \in \F$ such that $\card(U(X)) = \kappa$. Treating $\F$ as a discrete diagram in $\C$, assumption (1) implies that for any object $Y$ of $\C$, the collection of cocones $\F \to Y$ is bijective with a set, since the maps $X \to Y$ with $\card(U(X)) > \card(U(Y))$ must all be zero in such a cocone. Therefore, by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.6, $\C$ must have a coproduct $Y$ of all elements of $\F$. But then by assumption (2), there exists $X \in \F$ such that $\card(U(X)) > \card(U(Y))$; and since $\C$ is pointed, the coprojection $X \to Y$ must be split monic and therefore non-zero. Using assumption (1), we get a contradiction. $\square$ +Now assume that $\C$ is locally essentially small and cototal. Using the axiom of choice, we may assume that for each small cardinal $\kappa$, there is at most one element $X \in \F$ such that $\card(U(X)) = \kappa$. Treating $\F$ as a discrete diagram in $\C$, assumption (1) implies that for any object $Y$ of $\C$, the collection of cocones $\F \to Y$ is bijective with a set, since the maps $X \to Y$ with $\card(U(X)) > \card(U(Y))$ must all be zero in such a cocone. Therefore, by G. M. Kelly, A survey of totality for enriched and ordinary categories, Thm. 5.6, $\C$ must have a coproduct $Y$ of all elements of $\F$. But then by assumption (2), there exists $X \in \F$ such that $\card(U(X)) > \card(U(Y))$; and since $\C$ is pointed, the coprojection $X \to Y$ must be split monic and therefore non-zero. Using assumption (1), we get a contradiction. $\square$ diff --git a/database/data/categories/Cat.yaml b/database/data/categories/Cat.yaml index 705c834a7..ddb6a72eb 100644 --- a/database/data/categories/Cat.yaml +++ b/database/data/categories/Cat.yaml @@ -90,7 +90,7 @@ unsatisfied_properties: proof: >- For each infinite cardinal $\kappa$, choose a simple group $S_\kappa$ of cardinality $\kappa$ (for example the group of permutations of $\kappa$ of finite support which are even). Now consider the ultra-wide pushout diagram $1 \rightrightarrows B S_\kappa$. Then for any small category $\C$, the collection of cocones $1 \rightrightarrows B S_\kappa \to \C$ is bijective with a set: to form any such cocone, we must first choose the object $X$ of $\C$ which is the image of the object of $1$. Then, we must choose the morphisms $S_\kappa \to \End_{\C}(X)$; but for $\kappa > \card(\End_{\C}(X))$, the corresponding morphism must be zero. - On the other hand, we claim that $1 \rightrightarrows B S_\kappa$ does not have a pushout in $\Cat$; by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we have a pushout $\C$ of $1 \rightrightarrows B S_\kappa$, and choose a cardinal $\lambda > \card(\Mor(\C))$. Then the coprojection $i_\lambda : B S_\lambda \to \C$ must be split monic, since we can construct a cocone $1 \rightrightarrows B S_\kappa \to B S_\lambda$ in which $B S_\kappa \to B S_\lambda$ corresponds to the zero map for $\kappa \ne \lambda$, and in which $B S_\lambda \to B S_\lambda$ is the identity. It follows that if $X$ is the image in $\C$ of the object of $B S_\lambda$ under $i_\lambda$, then $i_\lambda$ induces an injective map $S_\lambda \to \End_{\C}(X)$. This gives a contradiction since $\lambda > \card(\End_{\C}(X))$ and $S_\lambda$ is a simple group. + On the other hand, we claim that $1 \rightrightarrows B S_\kappa$ does not have a pushout in $\Cat$; by G. M. Kelly, A survey of totality for enriched and ordinary categories, Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we have a pushout $\C$ of $1 \rightrightarrows B S_\kappa$, and choose a cardinal $\lambda > \card(\Mor(\C))$. Then the coprojection $i_\lambda : B S_\lambda \to \C$ must be split monic, since we can construct a cocone $1 \rightrightarrows B S_\kappa \to B S_\lambda$ in which $B S_\kappa \to B S_\lambda$ corresponds to the zero map for $\kappa \ne \lambda$, and in which $B S_\lambda \to B S_\lambda$ is the identity. It follows that if $X$ is the image in $\C$ of the object of $B S_\lambda$ under $i_\lambda$, then $i_\lambda$ induces an injective map $S_\lambda \to \End_{\C}(X)$. This gives a contradiction since $\lambda > \card(\End_{\C}(X))$ and $S_\lambda$ is a simple group. special_objects: initial object: diff --git a/database/data/categories/Haus.yaml b/database/data/categories/Haus.yaml index c663401f5..62c3abbcb 100644 --- a/database/data/categories/Haus.yaml +++ b/database/data/categories/Haus.yaml @@ -89,7 +89,7 @@ unsatisfied_properties: proof: >- For each small cardinal $\kappa$, let $Q_\kappa$ be the product of all Hausdorff topological spaces whose underlying set is a non-empty subset of $\kappa$. By a theorem of Herrlich (Wann sind alle stetigen Abbildungen in Y konstant. Math. Z. 90 (1965): 152-154. EUMDL), there is a regular Hausdorff space $X_\kappa$ with at least two points such that every continuous map $X_\kappa \to Q_\kappa$ is constant. (The author only states that $X_\kappa$ is regular, but actually, $X_\kappa$ is regular and $T_1$, hence Hausdorff.) Choose a base point $x_\kappa \in X_\kappa$ for each $\kappa$. We can form an ultra-wide pushout diagram $1 \rightrightarrows X_\kappa$ where each morphism $1 \to X_\kappa$ corresponds to $x_\kappa$. Then for any Hausdorff space $Y$, the collection of cocones $1 \rightrightarrows X_\kappa \to Y$ is bijective to a set: if $Y$ is empty, then the collection of cocones is obviously empty. Otherwise, in order to form a cocone, we must first choose $y \in Y$ corresponding to the morphism $1 \to Y$. Then for each $\kappa \ge \card(U(Y))$, $Y$ is homeomorphic to one of the spaces in the product forming $Q_\kappa$. Therefore, there is a morphism $Y \to Q_\kappa$ splitting the projection map $Q_\kappa \to Y$. It follows that the map $X_\kappa \to Y$ is constant, and in fact it must be the constant map with image $y$. - On the other hand, we claim that $1 \rightrightarrows X_\kappa$ does not have a pushout in $\Haus$; by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we had a pushout $Y$, and let $\lambda \coloneqq \card(U(Y))$. Then the coprojection $X_\lambda \to Y$ is split monic, since we can construct a cocone $1 \rightrightarrows X_\kappa \to X_\lambda$ where the map $X_\kappa \to X_\lambda$ is the constant map with image $x_\lambda$ when $\kappa \ne \lambda$, and the map $X_\lambda \to X_\lambda$ is the identity. But similarly to the previous paragraph, we can show any morphism $X_\lambda \to Y$ must be constant, giving a contradiction since $X_\lambda$ has at least two points. + On the other hand, we claim that $1 \rightrightarrows X_\kappa$ does not have a pushout in $\Haus$; by G. M. Kelly, A survey of totality for enriched and ordinary categories, Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we had a pushout $Y$, and let $\lambda \coloneqq \card(U(Y))$. Then the coprojection $X_\lambda \to Y$ is split monic, since we can construct a cocone $1 \rightrightarrows X_\kappa \to X_\lambda$ where the map $X_\kappa \to X_\lambda$ is the constant map with image $x_\lambda$ when $\kappa \ne \lambda$, and the map $X_\lambda \to X_\lambda$ is the identity. But similarly to the previous paragraph, we can show any morphism $X_\lambda \to Y$ must be constant, giving a contradiction since $X_\lambda$ has at least two points. - property: extremal generating set proof: The proof is the same as the one for $\Top$; there the test spaces we use are of the form $\kappa \sqcup \{ \kappa \}$ and $\kappa + 1$, which are both Hausdorff spaces. diff --git a/database/data/categories/LRS_R.yaml b/database/data/categories/LRS_R.yaml index 368af023c..5d09ef730 100644 --- a/database/data/categories/LRS_R.yaml +++ b/database/data/categories/LRS_R.yaml @@ -63,7 +63,7 @@ unsatisfied_properties: proof: 'We can adjust the proof for $\Top$ (see MO/509548) as follows: Let $K$ be a residue field of $R$, let $X$ 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.' - property: total - proof: 'The adjunction between the global sections functor and the $\Spec$ functor (EGA I (1971), Ch. 1, Prop. 1.6.3) makes $\CAlg(R)^{\op}$ into a reflective subcategory of $\LRS_R$. Therefore, if $LRS_R$ were total, then \CAlg(R) would be cototal by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Cor. 6.2, which we know is not the case.' + proof: 'The adjunction between the global sections functor and the $\Spec$ functor (EGA I (1971), Ch. 1, Prop. 1.6.3) makes $\CAlg(R)^{\op}$ into a reflective subcategory of $\LRS_R$. Therefore, if $\LRS_R$ were total, then $\CAlg(R)$ would be cototal by G. M. Kelly, A survey of totality for enriched and ordinary categories, Cor. 6.2, which we know is not the case.' - property: cartesian closed proof: This is Corollary 4(a) here. diff --git a/database/data/categories/On.yaml b/database/data/categories/On.yaml index 5724cac53..a843f6b6c 100644 --- a/database/data/categories/On.yaml +++ b/database/data/categories/On.yaml @@ -42,10 +42,6 @@ unsatisfied_properties: - property: terminal object proof: There is no largest ordinal $\alpha$ since $\alpha + 1$ will always be larger. - - property: well-copowered - proof: The "quotients" of $0$ are all ordinals. - check_redundancy: false - - property: inverse proof: Consider the strictly increasing sequence $0 < 1 < 2 < \cdots$. diff --git a/database/data/categories/SemiGrp.yaml b/database/data/categories/SemiGrp.yaml index b1a76d1f3..1e1cde3b0 100644 --- a/database/data/categories/SemiGrp.yaml +++ b/database/data/categories/SemiGrp.yaml @@ -65,11 +65,11 @@ unsatisfied_properties: - property: cototal proof: >- - The proof is similar to the proof for $\Grp$. For each infinite cardinal $\kappa$, let $S_\kappa$ be a simple group of cardinality $\kappa$ (such as the alternating group on $\kappa$). We can then form the ultra-wide pushout diagram $1 \rightrightarrows S_\kappa$ in $\SemiGrp$. For every semigroup $A$, the collection of cocones $1 \rightrightarrows S_\kappa \to A$ is bijective to a set: for every such cocone, we must first choose an idempotent $e$ of $A$ corresponding to the map $1 \to A$. Then, whenever $\kappa > \card(U(A))$, then for $f_\kappa : S_\kappa \to A$ in the cocone, we see + The proof is similar to the proof for $\Cat$. For each infinite cardinal $\kappa$, let $S_\kappa$ be a simple group of cardinality $\kappa$ (such as the alternating group on $\kappa$). We can then form the ultra-wide pushout diagram $1 \rightrightarrows S_\kappa$ in $\SemiGrp$. For every semigroup $A$, the collection of cocones $1 \rightrightarrows S_\kappa \to A$ is bijective to a set: for every such cocone, we must first choose an idempotent $e$ of $A$ corresponding to the map $1 \to A$. Then, whenever $\kappa > \card(U(A))$, then for $f_\kappa : S_\kappa \to A$ in the cocone, we see $$N \coloneqq \{g \in S_\kappa : f_\kappa(g) = e\}$$ is a normal subgroup of $G$. It must be non-trivial since otherwise $f_\kappa$ would induce an injective group homomorphism from $G$ to a group contained in $A$. Therefore, $N$ is all of $G$, so $f_\kappa$ is the constant map with image $a$. - We now claim that $1 \rightrightarrows S_\kappa$ does not have a pushout in $\SemiGrp$; by G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.6, this will imply that $\Cat$ is not cototal. To see this, suppose we had a pushout $A$, and let $\lambda$ be a cardinal strictly greater than $\card(U(A))$. Then the coprojection $S_\lambda\to A$ must be split monic, since we can construct a cocone $1 \rightrightarrows S_\kappa \to S_\lambda$ such that the map $S_\kappa \to S_\lambda$ is the constant map with image 1 if $\kappa \ne \lambda$, while the map $S_\lambda \to S_\lambda$ is the identity. But this contradicts the choice of $\lambda$. + We now claim that $1 \rightrightarrows S_\kappa$ does not have a pushout in $\SemiGrp$; by G. M. Kelly, A survey of totality for enriched and ordinary categories, Thm. 5.6, this will imply that $\SemiGrp$ is not cototal. To see this, suppose we had a pushout $A$, and let $\lambda$ be a cardinal strictly greater than $\card(U(A))$. Then the coprojection $S_\lambda\to A$ must be split monic, since we can construct a cocone $1 \rightrightarrows S_\kappa \to S_\lambda$ such that the map $S_\kappa \to S_\lambda$ is the constant map with image 1 if $\kappa \ne \lambda$, while the map $S_\lambda \to S_\lambda$ is the identity. But this contradicts the choice of $\lambda$. - property: natural numbers object proof: >- diff --git a/database/data/category-implications/total.yaml b/database/data/category-implications/total.yaml index 0e9f35ccf..8756b3c2a 100644 --- a/database/data/category-implications/total.yaml +++ b/database/data/category-implications/total.yaml @@ -2,39 +2,39 @@ - id: total_cocomplete assumptions: - - total - - locally essentially small + - total + - locally essentially small conclusions: - - cocomplete + - cocomplete proof: 'A diagram $X : \I \to \C$ with $\I$ small and $\C$ locally essentially small is easily seen to satisfy the hypotheses of condition (2) in the definition of a total category.' is_equivalence: false # TODO: If adding a property "has a weak terminal object", that can be added to the conclusions here: The identity functor $\C \to \C$ is also a discrete fibration, and its colimit must certainly at least be a weakly terminal object. It's unclear to me at this time whether it must necessarily be a terminal object in case $\C$ is not locally essentially small. - id: total_initial assumptions: - - total + - total conclusions: - - initial object - proof: 'The empty diagram is vacuously a discrete fibration; thus, it has a colimit, which must be an initial object.' + - initial object + proof: The empty diagram is vacuously a discrete fibration; thus, it has a colimit, which must be an initial object. is_equivalence: false - id: total_complete assumptions: - - total - - locally essentially small + - total + - locally essentially small conclusions: - - complete - proof: 'This is proven in G. M. Kelly, A survey of totality for enriched and ordinary categories (NumDam), Thm. 5.6.' + - complete + proof: This is proven in G. M. Kelly, A survey of totality for enriched and ordinary categories, Thm. 5.6. is_equivalence: false # TODO: replace "well-copowered" with "epi-cocomplete" if adding the latter property - id: cocomplete_well-copowered_generator_implies_total assumptions: - - cocomplete - - locally essentially small - - well-copowered - - generating set + - cocomplete + - locally essentially small + - well-copowered + - generating set conclusions: - - total - proof: 'Since the category is cocomplete and well-copowered, it is epi-cocomplete (meaning that it has wide pushouts of epimorphisms, even of non-small families of epimorphisms). The result then follows from B. J. Day, Further criteria for totality (NumDam), Thm. 1.' + - total + proof: Since the category is cocomplete and well-copowered, it is epi-cocomplete (meaning that it has wide pushouts of epimorphisms, even of non-small families of epimorphisms). The result then follows from B. J. Day, Further criteria for totality, Thm. 1. is_equivalence: false