The categories of finite algebras are countable accessible

Author: ykawase5048

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

@@ -29,6 +29,12 @@ VALUES
 	TRUE,
 	'This is a simple special case of Pontryagin duality: The functor $\hom(-,\mathbb{Q}/\mathbb{Z})$ provides the equivalence.'
 ),
+(
+	'FinAb',
+	'ℵ₁-accessible',
+	TRUE,
+	'The inclusion $\mathbf{FinAb} \hookrightarrow \mathbf{Ab}$ is closed under ℵ₁-filtered colimits, because ℵ₁-filtered colimits are computed in the same way as in $\mathbf{Set}$ and any ℵ₁-filtered colimit of finite sets is again finite. Since every finite abelian group is ℵ₁-presentable in $\mathbf{Ab}$, it is still ℵ₁-presentable in $\mathbf{FinAb}$, hence $\mathbf{FinAb}$ is ℵ₁-accessible, where every object is ℵ₁-presentable.'
+),
 (
 	'FinAb',
 	'small',

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

@@ -53,6 +53,12 @@ VALUES
 	TRUE,
 	'The category is Malcev and hence finitely complete, and it has all coequalizers. The regular epimorphisms coincide with the surjective group homomorphisms (see below), hence are clearly stable under pullbacks.'
 ),
+(
+	'FinGrp',
+	'ℵ₁-accessible',
+	TRUE,
+	'The inclusion $\mathbf{FinGrp} \hookrightarrow \mathbf{Grp}$ is closed under ℵ₁-filtered colimits, because ℵ₁-filtered colimits are computed in the same way as in $\mathbf{Set}$ and any ℵ₁-filtered colimit of finite sets is again finite. Since every finite group is ℵ₁-presentable in $\mathbf{Grp}$, it is still ℵ₁-presentable in $\mathbf{FinGrp}$, hence $\mathbf{FinGrp}$ is ℵ₁-accessible, where every object is ℵ₁-presentable.'
+),
 (
 	'FinGrp',
 	'subobject classifier',

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

@@ -41,6 +41,12 @@ VALUES
 	TRUE,
 	'The two-element set is a cogenerator.'
 ),
+(
+	'FinSet',
+	'ℵ₁-accessible',
+	TRUE,
+	'The inclusion $\mathbf{FinSet} \hookrightarrow \mathbf{Set}$ is closed under ℵ₁-filtered colimits, that is, any ℵ₁-filtered colimit of finite sets is again finite. Since every finite set is ℵ₁-presentable in $\mathbf{Set}$, it is still ℵ₁-presentable in $\mathbf{FinSet}$. Therefore, $\mathbf{FinSet}$ is ℵ₁-accessible, where every object is ℵ₁-presentable.'
+),
 (
 	'FinSet',
 	'small',