show that Set_* and Top_* have cocartesian cofiltered limits

Author: ScriptRaccoon

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

@@ -53,6 +53,14 @@ VALUES
 	TRUE,
 	'Malcev categories are closed under slice categories by Prop. 2.2.14 in <a href="https://ncatlab.org/nlab/show/Malcev,+protomodular,+homological+and+semi-abelian+categories" target="_blank">Malcev, protomodular, homological and semi-abelian categories</a>. It follows that co-Malcev categories are closed under coslice categories, and $\mathbf{Set}_*$ is a coslice category of $\mathbf{Set}$, which is co-Malcev since every elementary topos is co-Malcev.'
 ),
+(
+	'Set*',
+	'cocartesian cofiltered limits',
+	TRUE,
+	'Let $X$ be a pointed set and $(Y_i)$ be a filtered diagram of pointed sets. Base points will be denoted by $0$. The canonical map $X \vee \lim_i Y_i \to \lim_i (X \vee Y_i)$ is injective since the wedge sum naturally embeds into the product and the natural map $X \vee \prod_i Y_i \to \prod_i (X \times Y_i)$ is injective. Now let $z = (z_i) \in \lim_i (X \vee Y_i)$.
+	<br>Case 1: There is some index $i$ with $z_i \in X \setminus \{0\}$. We claim $z_j \in X$ for any index $j$ and $z_j = z_i$ in $X$, so that $z$ has a preimage in $X$. To see this, choose an index $k \geq i,j$. Since $X \vee Y_i \to X \vee Y_k$ maps $z_i \mapsto z_k$ and is the identity on $X$, we see that $z_k \in X$ and $z_k = z_i$ in $X$. Since $X \vee Y_j \to X \vee Y_k$ maps $z_j \mapsto z_k$, we see that $z_j \notin Y_j$, since otherwise $z_k \in Y_k \cap X = \{0\}$. Hence, $z_j \in X \setminus \{0\}$, and then $z_j = z_k = z_i$.
+	<br>Case 2: We have $z_i \in Y_i$ for all $i$. Then clearly $(z_i) \in \lim_i Y_i$ is a preimage.'
+),
 (
 	'Set*',
 	'skeletal',

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

@@ -77,6 +77,13 @@ VALUES
 	TRUE,
 	'Since embeddings are regular monomorphisms in this category (see below) and hence strong monomorphisms, it suffices to prove that the canonical morphism $X \vee Y \hookrightarrow X \times Y$ is an embedding. For a proof, see <a href="https://math.stackexchange.com/questions/4055988" target="_blank">MSE/4055988</a>.'
 ),
+(
+	'Top*',
+	'cocartesian cofiltered limits',
+	TRUE,
+	'We continue the proof for <a href="/category/Set*">$\mathbf{Set}_*$</a> by showing that the natural bijective map <br>$\alpha : X \vee \lim_i Y_i \to \lim_i (X \vee Y_i)$<br>
+	is open. It suffices to consider open sets of two types: (1) If $U \subseteq X$ is open, the $\alpha$-image of $U \vee \lim_i Y_i$ is $p_{i_0}^{-1}(U \vee Y_{i_0})$ for any chosen index $i_0$, hence open. (2) If $i$ is an index and $V_i \subseteq Y_i$ is open, then the $\alpha$-image of $X \vee (p_i^{-1}(V_i) \cap \lim_i Y_i)$ is $p_i^{-1}(X \vee V_i)$, hence open.'
+),
 (
 	'Top*',
 	'coregular',