Update databases/catdat/data/003_category-property-assignments/CompHaus.sql

Co-authored-by: Script Raccoon <scriptraccoon@gmail.com>

Author: dschepler

databases/catdat/data/003_category-property-assignments/CompHaus.sql

@@ -120,6 +120,6 @@ VALUES
 	'CompHaus',
 	'accessible',
 	FALSE,
-	'For any small regular cardinal $\kappa$, consider the $\kappa$-directed system $\kappa \to \CompHaus$, where for ordinal $\alpha < \kappa$ we have $\alpha \mapsto [0, \alpha]$, and for $\alpha \le \beta < \kappa$ the morphism $\alpha \to \beta$ maps to the inclusion map $[0, \alpha] \hookrightarrow [0, \beta]$. Then the direct colimit of the underlying sets is $\kappa$, so the direct colimit in $\CompHaus$ is the Stone-Čech compatification $\tilde\kappa$ of $\kappa$.<br>
+	'For any small regular cardinal $\kappa$, consider the $\kappa$-directed system $\kappa \to \CompHaus$, where for ordinal $\alpha < \kappa$ we have $\alpha \mapsto [0, \alpha]$, and for $\alpha \le \beta < \kappa$ the morphism $\alpha \to \beta$ maps to the inclusion map $[0, \alpha] \hookrightarrow [0, \beta]$. Then the direct colimit of the underlying spaces is $\kappa$, so the direct colimit in $\CompHaus$ is the Stone-Čech compatification $\tilde\kappa$ of $\kappa$.<br>
 	We now claim that any nonempty object $K$ of $\CompHaus$ is not $\kappa$-presentable for any small regular cardinal $\kappa$. If so, then choose a point $x \in \tilde \kappa \setminus \kappa$, and consider the constant map $K \to \tilde \kappa$ with value $x$. By the assumption, this would have to factor through $[0, \alpha]$ for some ordinal $\alpha < \kappa$. But since the topological space $\kappa$ is Tychonoff, the map $\kappa \to \tilde \kappa$ is injective; thus, the map $[0, \alpha] \to \tilde\kappa$ is also injective. That means that the factor must be a constant map with value in $[0, \alpha]$, giving a contradiction.<br>
 	Now any colimit in $\CompHaus$ of empty spaces is empty, showing that $\CompHaus$ is not accessible.');