Various community contributions 1

database/data/001_categories/200_comments.sql

@@ -38,7 +38,7 @@ VALUES
 ),
 (
     'Sh(X,Ab)',
-    'It is likely that neither of the currently remaining unknown properties (finitary algebraic, locally ℵ₁-presentable, split abelian, etc.) are satisfied for a <i>generic</i> space $X$, but we need to make this precise by adding additional requirements to $X$. Maybe we need to create separate entries for specific spaces $X$.'
+    'It is likely that neither of the currently remaining unknown properties (finitary algebraic, locally ℵ₁-presentable, etc.) are satisfied for a <i>generic</i> space $X$, but we need to make this precise by adding additional requirements to $X$. Maybe we need to create separate entries for specific spaces $X$.'
 ),
 (
     'M-Set',

database/data/004_property-assignments/Sh(X,Ab).sql

@@ -19,13 +19,14 @@ VALUES
 ),
 (
 	'Sh(X,Ab)',
-	'trivial',
+	'skeletal',
 	FALSE,
-	'Consider constant sheaves for non-isomorphic abelian groups.'
+	'Consider constant sheaves for isomorphic but non-equal abelian groups.'
 ),
 (
 	'Sh(X,Ab)',
-	'skeletal',
+	'split abelian',
 	FALSE,
-	'Consider constant sheaves for isomorphic but non-equal abelian groups.'
+	'Choose a point $x \in X$. The functor $x_* : \mathbf{Ab} \to \mathrm{Sh}(X,\mathbf{Ab})$ (skyscraper sheaf) is exact, and its left adjoint $x^* : \mathrm{Sh}(X,\mathbf{Ab}) \to \mathbf{Ab}$ (stalk) satisfies $x^* x_* \cong \mathrm{id}_{\mathbf{Ab}}$. Now, since $\mathbf{Ab}$ is not split abelian (see <a href="/category/Ab">here</a>), there is a short exact sequence of abelian groups $0 \to A \to B \to C \to 0$ that does not split. Then $0 \to x_* A \to x_* B \to x_* C \to 0$ is also exact, but it does not split: Otherwise it would also be split after applying $x^*$, which however gives the original sequence in $\mathbf{Ab}$.'
 );
+

database/data/004_property-assignments/Z.sql

@@ -88,4 +88,10 @@ VALUES
 	'cartesian closed',
 	FALSE,
 	'There are functors $F,G : \mathbf{CRing} \to \mathbf{Set}$ such that $\mathrm{Hom}(F,G)$ is not essentially small, see <a href="https://mathoverflow.net/questions/390611" target="_blank">MO/390611</a> for example. Now if the exponential $[F,G] : \mathbf{CRing} \to \mathbf{Set}$ exists, we get $[F,G](\mathbb{Z}) \cong \mathrm{Hom}(\mathrm{Hom}(\mathbb{Z},-),[F,G])$ by Yoneda, which simplifies to $\mathrm{Hom}(1,[F,G]) \cong \mathrm{Hom}(1 \times F,G) \cong \mathrm{Hom}(F,G)$, a contradiction.'
-);
+),
+(
+	'Z',
+	'well-powered',
+	FALSE,
+	'Consider the functor $F$ from <a href="https://mathoverflow.net/questions/390611" target="_blank">MO/390611</a> for example. The collection of subobjects of $F$ is not isomorphic to a set: for each infinite cardinal $\kappa$, simply cut off the construction of $F$ at $\kappa$. This yields a different subobject for each $\kappa$.'
+);

database/data/007_special-morphisms/002_isomorphisms.sql

@@ -210,6 +210,11 @@ VALUES
 	'bijective $R$-linear maps',
 	'This characterization holds in every algebraic category.'
 ),
+(
+	'R-Mod_div',
+	'bijective $R$-linear maps',
+	'This characterization holds in every algebraic category.'
+),
 (
 	'real_interval',
 	'only the identity morphisms',

database/data/007_special-morphisms/003_monomorphisms.sql

@@ -205,6 +205,11 @@ VALUES
 	'injective $R$-linear maps',
 	'This holds in every finitary algebraic category: the forgetful functor to $\mathbf{Set}$ is faithful, hence reflects monomorphisms, and it is continuous (even representable), hence preserves monomorphisms.'
 ),
+(
+	'R-Mod_div',
+	'injective $R$-linear maps',
+	'This holds in every finitary algebraic category: the forgetful functor to $\mathbf{Set}$ is faithful, hence reflects monomorphisms, and it is continuous (even representable), hence preserves monomorphisms.'
+),
 (
 	'real_interval',
 	'every morphism',

database/data/007_special-morphisms/004_epimorphisms.sql

@@ -70,6 +70,11 @@ VALUES
 	'A functor $F : \mathcal{C} \to \mathcal{D}$ is an epimorphism iff $F$ is surjective on objects and for every morphism $s$ in $\mathcal{D}$ there is a zigzag over $U := F(\mathcal{C})$, meaning morphisms $u_1,\dotsc,u_{m+1} \in U$, $v_1,\dotsc,v_m \in U$, $x_1,\dotsc,x_m \in \mathcal{D}$ and $y_1,\dotsc,y_m \in \mathcal{D}$ such that $s = x_1 u_1$, $u_1 = v_1 y_1$, $x_{i-1} v_{i-1} = x_i u_i$, $u_i y_{i-1} = v_i y_i$, $x_m v_m = u_{m+1}$ and $u_{m+1} y_m = s$.',
 	'This is an extension of the <a href="https://en.wikipedia.org/wiki/Isbell''s_zigzag_theorem" target="_blank">corresponding theorem for monoids</a> and proven in <a href="https://www.jstor.org/stable/2373286" target="_blank">Epimorphisms and Dominions, III</a> by John R. Isbell.'
 ),
+(
+	'CAlg(R)',
+	'a homomorphism of algebras which is an epimorphism of commutative rings',
+	'The forgetful functor $\mathbf{CAlg}(R) \to \mathbf{Ring}$ is faithful and hence reflects epimorphisms, but it also preserves epimorphisms since it preserves pushouts (since $\mathbf{CAlg}(R) \cong R / \mathbf{Ring}$). For epimorphisms of commutative rings see their <a href="/category/CRing">detail page</a>.'
+),
 (
 	'CRing',
 	'A ring map $f : R \to S$ is an epimorphism iff $S$ equals the <i>dominion</i> of $f(R) \subseteq S$, meaning that for every $s \in S$ there is some matrix factorization $(s) = Y X Z$ with $X \in M_{n \times n}(R)$, $Y \in M_{1 \times n}(S)$, and $Z \in M_{n \times 1}(S)$.',
@@ -190,6 +195,11 @@ VALUES
 	'surjective $R$-linear maps',
 	'The forgetful functor to abelian groups is faithful and preserves colimits, hence reflects and preserves epimorphisms. Alternatively, use the same proof as for abelian groups.'
 ),
+(
+	'R-Mod_div',
+	'surjective $R$-linear maps',
+	'The forgetful functor to abelian groups is faithful and preserves colimits, hence reflects and preserves epimorphisms. Alternatively, use the same proof as for abelian groups.'
+),
 (
 	'real_interval',
 	'every morphism',