diff --git a/.cspell.json b/.cspell.json index e81b4d931..07bb50187 100644 --- a/.cspell.json +++ b/.cspell.json @@ -36,6 +36,7 @@ "Auslander", "Axiomatising", "axiomatization", + "Baer", "bijection", "bijections", "bijective", @@ -97,8 +98,11 @@ "cogenerating", "cogenerator", "cogenerators", + "cogroup", "Cohomology", + "coidentity", "coimage", + "coinverse", "cokernel", "cokernels", "colimit", @@ -108,6 +112,7 @@ "comonadicity", "compactification", "compactifications", + "comultiplication", "concretizability", "concretizable", "conormal", @@ -138,6 +143,7 @@ "cospans", "cosymmetric", "cosymmetry", + "cototal", "cotransitive", "cotransitivity", "counit", @@ -167,6 +173,8 @@ "extensivity", "extremal", "Faddeev", + "Farb", + "fibration", "fieldification", "filtrations", "finitary", @@ -192,6 +200,7 @@ "Hertweck", "Heunen", "Heyting", + "homeomorphic", "homotopic", "homotopy", "Hušek", @@ -201,6 +210,7 @@ "hypercollections", "idempotents", "Ieke", + "indeterminates", "infima", "infimum", "infinitary", @@ -254,6 +264,8 @@ "Noetherian", "Noncommutative", "objectwise", + "opfibration", + "Perrone", "pointwise", "Pontryagin", "poset", @@ -289,6 +301,7 @@ "saft", "Schapira", "Schepler", + "Schreier", "semigroup", "semigroups", "semisimple", @@ -326,12 +339,14 @@ "Szamuely", "Tarski", "tensoring", + "Tholen", "Tietze", "topoi", "tripleability", "Turso", "Tychonoff", "Ulmer", + "ultrafilters", "uncountably", "unital", "unitalization", diff --git a/content/Grp_total_explicit_proof.md b/content/Grp_total_explicit_proof.md new file mode 100644 index 000000000..3e5a6bf55 --- /dev/null +++ b/content/Grp_total_explicit_proof.md @@ -0,0 +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 +--- + +# Explicit Proof that the Category of Groups is Total + +The definition of a total category is very abstract; furthermore, it is not immediately clear how it is possible for _any_ category which is not essentially small to satisfy the definition, much less a wide variety of the algebraic and topological categories which are considered in practice. Thus, to illustrate the definition, we give an explicit construction of the functor +$$L : [\Grp^{\op},\Set] \to \Grp$$ +that is left adjoint to the Yoneda embedding $y : \Grp \hookrightarrow [\Grp^{\op},\Set]$. + +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 + +$$ +\begin{CD} +T(H) @> = >> T(H)\\ +@V T(hh') VV @VVV\\ +T(\IZ * \IZ') @>>> T(\IZ) +\end{CD} +$$ + +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 $H$. + +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 satisfied. + +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 + +$$ +\begin{CD} +TH @> \alpha_H >> \Hom(H, G) \\ +@V Th VV @VV {-} \circ h V \\ +T\IZ @> \alpha_{\IZ} >> \Hom(\IZ, G). +\end{CD} +$$ + +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 da80c70a8..ac3b7675d 100644 --- a/content/missing_cogenerator.md +++ b/content/missing_cogenerator.md @@ -12,9 +12,11 @@ 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 $X$. + +Now assume that $\C$ is 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 (namely the implication (i) $\Rightarrow$ (iii)), $\C$ 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. ::: diff --git a/database/data/categories/Alg(R).yaml b/database/data/categories/Alg(R).yaml index d352aa820..935935a1b 100644 --- a/database/data/categories/Alg(R).yaml +++ b/database/data/categories/Alg(R).yaml @@ -41,8 +41,10 @@ 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. + references: + - calg_not_cototal - 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/database/data/categories/CAlg(R).yaml b/database/data/categories/CAlg(R).yaml index 936a1a145..840932f19 100644 --- a/database/data/categories/CAlg(R).yaml +++ b/database/data/categories/CAlg(R).yaml @@ -18,7 +18,7 @@ satisfied_properties: proof: There is a forgetful functor $\CAlg(R) \to \Set$ and $\Set$ is locally small. - property: one-sorted finitary algebraic - proof: Take the one-sorted algebraic theory of a commutative algebra. + proof: Take the one-sorted algebraic theory of a commutative $R$-algebra. - property: strict terminal object proof: 'If $f : 0 \to A$ is a homomorphism of $R$-algebras, then $A$ satisfies $1=f(1)=f(0)=0$, so that $A=0$.' @@ -55,9 +55,6 @@ 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: 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 $(X^n)_{n} \in A^{\IN}$.' @@ -84,6 +81,13 @@ unsatisfied_properties: \end{CD}$$ In the limit, it induces the inclusion $K[X] \hookrightarrow K[[X]]$, where $K[[X]]$ is the algebra of formal power series over $K$. It is clearly not surjective, but this is not sufficient, we need to argue that it is not an epimorphism in $\CAlg(R)$, or equivalently, in $\CRing$. For a proof, see MSE/2391187. + - property: cototal + proof: >- + This proof will be similar to the proof of this lemma. To start, for each infinite cardinal $\kappa \ge \card(U(R))$, choose a field $F_\kappa$ of cardinality $\kappa$ including a quotient field of $R$. (For example, fix some such quotient field $R / \m$ and then let $F_\kappa$ be the extension of $R / \m$ by $\kappa$ many indeterminates.) We first claim that for any commutative $R$-algebra $A$ with $\kappa > \card(U(A))$, there is exactly one $R$-algebra homomorphism $\varphi : R \times F_\kappa \to A$, which can be described as the projection $R \times F_\kappa \to R$ followed by the unique morphism $R \to A$. To see this, note that for any other such morphism $\varphi$, we must have $\varphi(0, 1) \ne 0$. It follows that the rng homomorphism $F_\kappa \to A$ formed by composing $F_\kappa \to R \times F_\kappa$, $x \mapsto (0, x)$, with $\varphi$ is non-zero, and it is therefore injective since $F_\kappa$ was chosen to be a field. This gives a contradiction. + + It follows that for any commutative $R$-algebra $A$, the set of cocones $R \times F_\kappa \to A$ from the discrete diagram on $R \times F_\kappa$ is bijective to a set. On the other hand, we claim that $\CAlg(R)$ has no coproduct of all $R \times F_\kappa$. This will imply that $\CAlg(R)$ is not cototal, by G. M. Kelly, A survey of totality for enriched and ordinary categories, Thm. 5.6 (namely the contrapositive of the implication (i) $\Rightarrow$ (iii)). To see this, suppose we had such a coproduct $A$, and choose a cardinal $\kappa$ greater than $\max(\card(U(A)), \card(U(R)), \aleph_0)$. Then the coprojection $R \times F_\kappa \to A$ would be split injective, where we can construct the splitting $A \to R\times F_\kappa$ such that the $\kappa$ component is the identity, whereas for $\lambda \ne \kappa$, the $\lambda$ component is given by the projection $R \times F_\lambda \to R$ composed with the unique morphism $R \to R \times F_\kappa$. However, from this injection, we would get $\card(A) \ge \card(R \times F_\kappa) = \kappa$, giving a contradiction. + label: calg_not_cototal + special_objects: initial object: description: $R$ diff --git a/database/data/categories/Cat.yaml b/database/data/categories/Cat.yaml index 11fd7d2ce..2552d19e7 100644 --- a/database/data/categories/Cat.yaml +++ b/database/data/categories/Cat.yaml @@ -44,9 +44,6 @@ 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. @@ -93,6 +90,13 @@ 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 $G_\kappa$ of cardinality $\kappa$ (such as the alternating group on $\kappa$). Now consider the large span diagram $1 \dottedrightrightarrows B G_\kappa$. Then for any small category $\C$, the collection of cocones $1 \dottedrightrightarrows B G_\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 $G_\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 \dottedrightrightarrows B G_\kappa$ does not have a pushout in $\Cat$; by G. M. Kelly, A survey of totality for enriched and ordinary categories, Thm. 5.6 (namely the contrapositive of the implication (i) $\Rightarrow$ (iii)), this will imply that $\Cat$ is not cototal. To see this, suppose we have a pushout $\C$ of $1 \dottedrightrightarrows B G_\kappa$, and choose a cardinal $\lambda > \card(\Mor(\C))$ (which is possible since $\C$ is a small category). Then the coprojection $i_\lambda : B G_\lambda \to \C$ must be split monic, since we can construct a cocone $1 \dottedrightrightarrows B G_\kappa \to B G_\lambda$ in which $B G_\kappa \to B G_\lambda$ corresponds to the zero map for $\kappa \ne \lambda$, and in which $B G_\lambda \to B G_\lambda$ is the identity. It follows that if $X$ is the image in $\C$ of the object of $B G_\lambda$ under $i_\lambda$, then $i_\lambda$ induces an injective map $G_\lambda \to \End_{\C}(X)$. This gives a contradiction since $\lambda > \card(\End_{\C}(X))$ and $G_\lambda$ has cardinality $\lambda$. + label: cat_not_cototal + special_objects: initial object: description: empty category diff --git a/database/data/categories/Grp.yaml b/database/data/categories/Grp.yaml index a117d21d0..b989a4744 100644 --- a/database/data/categories/Grp.yaml +++ b/database/data/categories/Grp.yaml @@ -42,6 +42,10 @@ 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 + - property: extremal generator proof: The group $\IZ$ is an extremal generator since it represents the forgetful functor $\Grp \to \Set$, which is faithful and conservative. check_redundancy: false @@ -54,7 +58,7 @@ 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).' label: grp_no_cogenerator diff --git a/database/data/categories/Haus.yaml b/database/data/categories/Haus.yaml index 328fb0321..8ff38d0e5 100644 --- a/database/data/categories/Haus.yaml +++ b/database/data/categories/Haus.yaml @@ -27,9 +27,11 @@ 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. @@ -76,9 +78,6 @@ unsatisfied_properties: references: - met_no_filtered_colimit_stable_monos - - property: cogenerator - 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).' @@ -89,6 +88,13 @@ 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 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. (Note that the result there applies for general $T_1$ spaces $Q$, not just for $Q_\kappa$ as constructed above. In the reference, 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 a large span diagram $1 \dottedrightrightarrows X_\kappa$ where each morphism $1 \to X_\kappa$ corresponds to $x_\kappa$. Then for any Hausdorff space $Y$, the collection of cocones $1 \dottedrightrightarrows X_\kappa \to Y$ is bijective to a set: 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 \dottedrightrightarrows 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 (namely the contrapositive of the implication (i) $\Rightarrow$ (iii)), this will imply that $\Haus$ 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 \dottedrightrightarrows 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. references: diff --git a/database/data/categories/LRS_R.yaml b/database/data/categories/LRS_R.yaml index 83387ceec..03d2971fe 100644 --- a/database/data/categories/LRS_R.yaml +++ b/database/data/categories/LRS_R.yaml @@ -78,11 +78,8 @@ unsatisfied_properties: references: - top_not_co-malcev - - 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 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/Meas.yaml b/database/data/categories/Meas.yaml index af5336c0e..c324ddc85 100644 --- a/database/data/categories/Meas.yaml +++ b/database/data/categories/Meas.yaml @@ -34,6 +34,7 @@ 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. diff --git a/database/data/categories/Mon.yaml b/database/data/categories/Mon.yaml index f34a72888..4ca823ac7 100644 --- a/database/data/categories/Mon.yaml +++ b/database/data/categories/Mon.yaml @@ -40,7 +40,7 @@ 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/database/data/categories/On.yaml b/database/data/categories/On.yaml index 45fe331bb..495785fab 100644 --- a/database/data/categories/On.yaml +++ b/database/data/categories/On.yaml @@ -42,9 +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. - - property: inverse proof: Consider the strictly increasing sequence $0 < 1 < 2 < \cdots$. diff --git a/database/data/categories/Rng.yaml b/database/data/categories/Rng.yaml index dea398ba9..0d76d48db 100644 --- a/database/data/categories/Rng.yaml +++ b/database/data/categories/Rng.yaml @@ -40,7 +40,7 @@ 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/database/data/categories/SemiGrp.yaml b/database/data/categories/SemiGrp.yaml index d9899349c..f1c698fc9 100644 --- a/database/data/categories/SemiGrp.yaml +++ b/database/data/categories/SemiGrp.yaml @@ -55,16 +55,6 @@ 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: cogenerator - # 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 cogenerator $Q$. There is an infinite simple group $G$ larger than $Q$ (such as an alternating group). Since $\id_G, 1 : G \rightrightarrows G$ are different, there is a homomorphism of semigroups $f : G \to Q$ with $f \neq f \circ 1$. Then - $$N \coloneqq \{g \in G : f(g) = f(1)\}$$ - is a normal subgroup of $G$. It is proper, and hence trivial. But then $f$ is injective, which is a contradiction. - references: - - grp_no_cogenerator - - property: cofiltered-limit-stable epimorphisms proof: We already know that $\Set$ does not have this property (by this result). Now apply the contrapositive of the dual of Lemma 2 here to the functor $\Set \to \SemiGrp$ that equips a set with the multiplication $a \cdot b \coloneqq a$. @@ -76,6 +66,16 @@ unsatisfied_properties: references: - mon_no_effective_cocongruences + - property: cototal + proof: >- + The proof is similar to the proof for $\Cat$. For each infinite cardinal $\kappa$, let $G_\kappa$ be a simple group of cardinality $\kappa$ (such as the alternating group on $\kappa$). We can then form the large span diagram $1 \dottedrightrightarrows G_\kappa$ in $\SemiGrp$. For every semigroup $A$, the collection of cocones $1 \dottedrightrightarrows G_\kappa \to A$ is bijective to a set: for every such cocone, we must first choose an idempotent $e$ of $A$ corresponding to the map $1 \to A$. Then, whenever $\kappa > \card(U(A))$, where $U(-)$ denotes the underlying set, for $f_\kappa : G_\kappa \to A$ in the cocone, we see + $$N \coloneqq \{g \in G_\kappa : f_\kappa(g) = e\}$$ + is a normal subgroup of $G_\kappa$. It must be non-trivial since otherwise $f_\kappa$ would induce an injective group homomorphism from $G_\kappa$ to a group contained in $A$. Therefore, $N$ is all of $G_\kappa$, so $f_\kappa$ is the constant map with image $e$. + + We now claim that $1 \dottedrightrightarrows G_\kappa$ does not have a pushout in $\SemiGrp$; by G. M. Kelly, A survey of totality for enriched and ordinary categories, Thm. 5.6 (namely the contrapositive of the implication (i) $\Rightarrow$ (iii)), 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 $G_\lambda\to A$ must be split monic, since we can construct a cocone $1 \dottedrightrightarrows G_\kappa \to G_\lambda$ such that the map $G_\kappa \to G_\lambda$ is the trivial (semi-)group homomorphism if $\kappa \ne \lambda$, while the map $G_\lambda \to G_\lambda$ is the identity. But this contradicts the choice of $\lambda$. + references: + - cat_not_cototal + - property: parametrized natural numbers object proof: >- Assume that a parametrized natural numbers object exists. Then by Lemma 4 here, for every semigroup $A$ the natural homomorphism diff --git a/database/data/categories/Top.yaml b/database/data/categories/Top.yaml index 488ce0d56..060551b75 100644 --- a/database/data/categories/Top.yaml +++ b/database/data/categories/Top.yaml @@ -24,6 +24,7 @@ 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/database/data/categories/Unif.yaml b/database/data/categories/Unif.yaml index 5e3c9d61c..afecfd969 100644 --- a/database/data/categories/Unif.yaml +++ b/database/data/categories/Unif.yaml @@ -19,6 +19,7 @@ satisfied_properties: - property: complete proof: 'Take the limit of the underlying sets and endow it with the coarsest uniform structure making all projections uniform; cf. Bourbaki, General Topology (Part 1), Chapter II, § 3, no. 3 on initial uniformities. More concretely, products are described below on this page, and the equalizer of two uniform maps $f,g : (X,\Phi) \rightrightarrows (Y,\Psi)$ is the subset $E \coloneqq \{x \in X : f(x) = g(x)\}$ equipped with the uniform structure $\{U \cap (E \times E) : U \in \Phi\}$.' + check_redundancy: false - property: cocomplete proof: 'Take the colimit of the underlying sets and endow it with the finest uniform structure making all inclusions uniform. More concretely, coproducts are described below on this page, and the coequalizer of two uniform maps $f,g : (X,\Phi) \rightrightarrows (Y,\Psi)$ is the $\Set$-based coequalizer $Q = Y / (f(x) \sim g(x))$ equipped with the following uniform structure, which makes the projection $p : Y \to Q$ uniform. Let $\Theta$ be the set of all subsets $U \subseteq Q \times Q$ such that $(p \times p)^*(U) \in \Psi$. It satisfies all the axioms of a uniform structure except one, namely the composition axiom. To fix this (and this construction works in complete generality), let $\Sigma \subseteq \Theta$ be the set of all $U \in \Theta$ for which there exists a sequence $U_1,U_2,\dotsc$ in $\Theta$ such that $U_1 \subseteq U$ and $U_{k+1} \circ U_{k+1} \subseteq U_k$ for all $k$. It is then straightforward to check that $\Sigma$ is a uniform structure on $Q$. Moreover, by construction, the map $p : (Y,\Psi) \to (Q,\Sigma)$ is uniform, and one verifies that it satisfies the required universal property.' diff --git a/database/data/category-implications/total.yaml b/database/data/category-implications/total.yaml new file mode 100644 index 000000000..a65030f82 --- /dev/null +++ b/database/data/category-implications/total.yaml @@ -0,0 +1,33 @@ +# results on total and cototal categories + +- id: total_loc_ess_small + assumptions: + - total + conclusions: + - locally essentially small + proof: This is trivial. + +- id: total_cocomplete + assumptions: + - total + conclusions: + - cocomplete + proof: 'If the category $\C$ is locally small and total, then the Yoneda embedding $y : \C^{\op} \to [\C, \Set]$ makes $\C^{\op}$ into a reflective subcategory of the presheaf category $[\C, \Set]$, where the latter is complete. For a general total category $\C$, use the equivalence to a locally small, total category.' + +- id: total_complete + assumptions: + - total + conclusions: + - complete + proof: This is proven in G. M. Kelly, A survey of totality for enriched and ordinary categories, Thm. 5.6. + +# 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 + 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, Thm. 1. diff --git a/database/data/category-properties/cototal.yaml b/database/data/category-properties/cototal.yaml new file mode 100644 index 000000000..a9e58b504 --- /dev/null +++ b/database/data/category-properties/cototal.yaml @@ -0,0 +1,27 @@ +id: cototal +relation: is +description: >- + A locally small category $\C$ is called cototal if it satisfies one of the following equivalent conditions: +
    +
  1. The contravariant Yoneda embedding + $$y : \C^{\op} \to [\C, \Set]$$ + has a left adjoint. +
  2. 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$.)
    3. +
  3. 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$.
    4. +
+ The equivalence is proven as Thm. 5.2 and 5.5 in G. M. Kelly, A survey of totality for enriched and ordinary categories. + + A general category is called cototal if it is equivalent to a locally small category which is cototal. +nlab_link: https://ncatlab.org/nlab/show/cototal+category +dual: total +invariant_under_equivalences: true + +related: + - cocomplete + - complete + - locally copresentable + +tags: + - limits + - colimits + - size diff --git a/database/data/category-properties/locally copresentable.yaml b/database/data/category-properties/locally copresentable.yaml index 60978091f..7a1298247 100644 --- a/database/data/category-properties/locally copresentable.yaml +++ b/database/data/category-properties/locally copresentable.yaml @@ -8,6 +8,7 @@ invariant_under_equivalences: true related: - coaccessible - complete + - cototal tags: - accessibility diff --git a/database/data/category-properties/locally presentable.yaml b/database/data/category-properties/locally presentable.yaml index fca891a41..ef2d78df0 100644 --- a/database/data/category-properties/locally presentable.yaml +++ b/database/data/category-properties/locally presentable.yaml @@ -17,6 +17,7 @@ invariant_under_equivalences: true related: - accessible - cocomplete + - total - locally finitely presentable - locally multi-presentable - locally poly-presentable diff --git a/database/data/category-properties/total.yaml b/database/data/category-properties/total.yaml new file mode 100644 index 000000000..5023fd42f --- /dev/null +++ b/database/data/category-properties/total.yaml @@ -0,0 +1,27 @@ +id: total +relation: is +description: >- + A locally small category $\C$ is called total if it satisfies one of the following equivalent conditions: +
    +
  1. The covariant Yoneda embedding + $$y : \C \to [\C^{\op}, \Set]$$ + has a left adjoint. For a concrete example of how such a left adjoint could look, see here. +
  2. Every discrete fibration $X : \I \to \C$ (with $\I$ not necessarily essentially small) whose fibers are bijective to sets has a colimit in $\C$. (Recall that $X$ is a discrete fibration if for every morphism $f : Y \to X_i$ in $\C$ there exists a unique $\alpha : j \to i$ in $\I$ such that $X_\alpha = f$.)
    3. +
  3. Let $X : \I \to \C$ be a diagram (with $\I$ not necessarily essentially small) such that for each object $Y$ of $\C$, the collection of connected components of the comma category $Y \downarrow X$ is bijective to a set. Then $X$ has a colimit in $\C$.
    4. +
+ The equivalence is proven as Thm. 5.2 and 5.5 in G. M. Kelly, A survey of totality for enriched and ordinary categories. + + A general category is called total if it is equivalent to a locally small category which is total. +nlab_link: https://ncatlab.org/nlab/show/total+category +dual: cototal +invariant_under_equivalences: true + +related: + - cocomplete + - complete + - locally presentable + +tags: + - limits + - colimits + - size diff --git a/database/data/functors/brauer_group.yaml b/database/data/functors/brauer_group.yaml index 9888bc89c..f33d40bf4 100644 --- a/database/data/functors/brauer_group.yaml +++ b/database/data/functors/brauer_group.yaml @@ -69,7 +69,7 @@ unsatisfied_properties: proof: >- A functor preserving binary products also preserves subterminal objects. In $\Fld$, the field $\IQ$ is subterminal. However, $\Br(\IQ)$ is not subterminal in $\Ab$, since otherwise it would be trivial, which is not the case. In fact, the rational quaternion algebra $$H(\IQ) = \langle i,j : i^2=j^2=-1,\, ij=-ji \rangle_{\IQ}$$ - is a non-trivial central finite-dimensional division algebra over $\mathbb{Q}$, and more precisely, $\Br(\IQ)$ is an infinite abelian group. See for example Exercise 4 in Section 12.5 in Associative Algebras by Pierce. + is a non-trivial central finite-dimensional division algebra over $\IQ$, and more precisely, $\Br(\IQ)$ is an infinite abelian group. See for example Exercise 4 in Section 12.5 in Associative Algebras by Pierce. - property: preserves binary coproducts proof: If $K$ is any field of characteristic $0$, then $K \sqcup \IQ = K$ in $\Fld$. More precisely, the cospan $\IQ \rightarrow K \leftarrow K$ is a coproduct, where the map on the right is the identity. However, the cospan $\Br(\IQ) \rightarrow \Br(K) \leftarrow \Br(K)$ is not a coproduct in $\Ab$, since otherwise the group $\Br(\IQ)$ would be trivial. diff --git a/database/data/macros.yaml b/database/data/macros.yaml index 89a01c15b..6eaf48082 100644 --- a/database/data/macros.yaml +++ b/database/data/macros.yaml @@ -73,6 +73,7 @@ \Id: \operatorname{Id} \Br: \operatorname{Br} \rightrightrightarrows: \mathrel{\substack{\rightarrow\\[-0.6ex]\rightarrow\\[-0.6ex]\rightarrow}} +\dottedrightrightarrows: \mathrel{\substack{\rightarrow\\[-0.6ex]\cdots\\[-0.6ex]\rightarrow}} # categories \Set: \mathbf{Set} diff --git a/database/scripts/expected-data/Ab.json b/database/scripts/expected-data/Ab.json index 54cd258d6..0f86509e8 100644 --- a/database/scripts/expected-data/Ab.json +++ b/database/scripts/expected-data/Ab.json @@ -126,6 +126,8 @@ "equalizers of cokernel pairs": true, "coequalizers of kernel pairs": true, "concretizable": true, + "total": true, + "cototal": true, "cartesian closed": false, "locally cartesian closed": false, diff --git a/database/scripts/expected-data/Set.json b/database/scripts/expected-data/Set.json index 6a2502ea6..fd4593e11 100644 --- a/database/scripts/expected-data/Set.json +++ b/database/scripts/expected-data/Set.json @@ -124,6 +124,8 @@ "equalizers of cokernel pairs": true, "coequalizers of kernel pairs": true, "concretizable": true, + "total": true, + "cototal": true, "Grothendieck abelian": false, "Malcev": false, diff --git a/database/scripts/expected-data/Top.json b/database/scripts/expected-data/Top.json index 50fc8175a..6a8437fc1 100644 --- a/database/scripts/expected-data/Top.json +++ b/database/scripts/expected-data/Top.json @@ -89,6 +89,8 @@ "equalizers of cokernel pairs": true, "coequalizers of kernel pairs": true, "concretizable": true, + "total": true, + "cototal": true, "abelian": false, "additive": false,