diff --git a/database/data/001_categories/200_comments.sql b/database/data/001_categories/200_comments.sql index f0f75d87..f29387a4 100644 --- a/database/data/001_categories/200_comments.sql +++ b/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 generic 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 generic 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', diff --git a/database/data/004_property-assignments/Sh(X,Ab).sql b/database/data/004_property-assignments/Sh(X,Ab).sql index 303b4b12..8f37bbe4 100644 --- a/database/data/004_property-assignments/Sh(X,Ab).sql +++ b/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 here), 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 $\mathbf{Ab}$.' ); + diff --git a/database/data/004_property-assignments/Z.sql b/database/data/004_property-assignments/Z.sql index aab24b31..cd9a98b4 100644 --- a/database/data/004_property-assignments/Z.sql +++ b/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 MO/390611 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 MO/390611 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 $\kappa$.' +); \ No newline at end of file diff --git a/database/data/007_special-morphisms/002_isomorphisms.sql b/database/data/007_special-morphisms/002_isomorphisms.sql index 7f08ac83..ad35047e 100644 --- a/database/data/007_special-morphisms/002_isomorphisms.sql +++ b/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', diff --git a/database/data/007_special-morphisms/003_monomorphisms.sql b/database/data/007_special-morphisms/003_monomorphisms.sql index 505ce188..d4ff8f88 100644 --- a/database/data/007_special-morphisms/003_monomorphisms.sql +++ b/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', diff --git a/database/data/007_special-morphisms/004_epimorphisms.sql b/database/data/007_special-morphisms/004_epimorphisms.sql index 656e856f..f594003e 100644 --- a/database/data/007_special-morphisms/004_epimorphisms.sql +++ b/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 corresponding theorem for monoids and proven in Epimorphisms and Dominions, III 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 detail page.' +), ( 'CRing', 'A ring map $f : R \to S$ is an epimorphism iff $S$ equals the dominion 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',