assign multi-completeness

Author: ykawase5048

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

@@ -53,6 +53,12 @@ VALUES
 	TRUE,
 	'If $f : X \to Y$ is a surjective map of finite sets, it is the coequalizer of the two projections $p_1, p_2 : X \times_Y X \rightrightarrows X$ in $\mathbf{FinSet}$, but also in $\mathbf{FS}$. Notice that $p_1,p_2$ are surjective. Even though $X \times_Y X$ is not a pullback in $\mathbf{FS}$, we can use this finite set here.'
 ),
+(
+	'FS',
+	'multi-terminal object',
+	TRUE,
+	'The empty set and a singleton give a multi-terminal object.'
+),
 (
 	'FS',
 	'small',
@@ -117,6 +123,12 @@ VALUES
 	$(E_1 \vee E_2) \wedge (E_1 \vee E_3) = \top \wedge \top = \top$.
 	<p>*For thin categories, the properties codistributive and distributive <a href="/category-implication/distributive_duality">are equivalent</a>.'
 ),
+(
+	'FS',
+	'multi-initial object',
+	FALSE,
+	'If a multi-initial object exists, then the connected component consisting of non-empty finite sets has an initial object $X$. Then, any non-empty finite set cannot have a cardinality strictly greater than $X$,  which is a contradiction.'
+),
 (
 	'FS',
 	'countable',

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

@@ -89,6 +89,19 @@ VALUES
 	TRUE,
 	'Since the inclusion $\mathbf{Set}_{\neq \varnothing} \hookrightarrow \mathbf{Set}$ is closed under non-empty colimits, it is also closed under filtered colimits. Therefore, non-empty finite sets are still finitely presentable in $\mathbf{Set}_{\neq \varnothing}$, and every non-empty set is written as a filtered colimit of them.'
 ),
+(
+	'Setne',
+	'generalized variety',
+	TRUE,
+	'Since the inclusion $\mathbf{Set}_{\neq \varnothing} \hookrightarrow \mathbf{Set}$ is closed under non-empty colimits, it is also closed under sifted colimits. Therefore, non-empty finite sets are still strongly finitely presentable in $\mathbf{Set}_{\neq \varnothing}$, and every non-empty set is written as a sifted colimit of them.'
+),
+
+(
+	'Setne',
+	'multi-complete',
+	TRUE,
+	'Let $D$ be a diagram in $\mathbf{Set}_{\neq \varnothing}$, and let $L$ be a limit of $D$ in $\mathbf{Set}$. If $L$ is non-empty, it gives a limit in $\mathbf{Set}_{\neq \varnothing}$ as well. If $L$ is the empty set, there is no cone over $D$ in $\mathbf{Set}_{\neq \varnothing}$; hence the empty set of cones gives a multi-limit of $D$ in $\mathbf{Set}_{\neq \varnothing}$.'
+),
 (
 	'Setne',
 	'sequential limits',