Sh(X,Ab) is not split abelian

Author: ScriptRaccoon

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 $\mathbf{Ab}$.'
 );
+