From 9deed79e33b9380ca90fbf3f2a64234f4e17945f Mon Sep 17 00:00:00 2001 From: Gregor Kleen Date: Wed, 3 Feb 2016 20:23:16 +0100 Subject: CatT 1.2.7 --- provider/posts/simmons-intro-to-cat-t/1.2.md | 30 ++++++++++++++++++++++++++++ 1 file changed, 30 insertions(+) (limited to 'provider') diff --git a/provider/posts/simmons-intro-to-cat-t/1.2.md b/provider/posts/simmons-intro-to-cat-t/1.2.md index 19d71e4..c714f81 100644 --- a/provider/posts/simmons-intro-to-cat-t/1.2.md +++ b/provider/posts/simmons-intro-to-cat-t/1.2.md @@ -148,3 +148,33 @@ $$ $$ + +
+Consider the category $\ca{Set_\bot}$ of pointed sets. + +Show that $\ca{Set_\bot}$ and $\ca{Pfn}$ are "essentially the same" category. + +
+The idea is to identify the undefined parts of a partial functions image with $\bot$: +$$ +\begin{aligned} +\Phi : \ca{Pfn} & \to \ca{Set_\bot} \\ +A & \mapsto A \cup \{ \bot_A \} \\ +\begin{tikzcd} +A \arrow[r, "f"] & B \\ +\bar{A} \arrow[u, hook] \arrow[ru, "\rest{f}{\bar{A}}" description] & +\end{tikzcd} & \mapsto \begin{aligned} +\Phi(f) : A \cup \{ \bot_A \} & \to B \cup \{ \bot_B \} \\ +\bot_A & \mapsto \bot_B \\ +a & \mapsto \begin{cases} +f(a) & \quad x \in \bar{A} \\ +\bot & \quad \text{else} +\end{cases} +\end{aligned} \\ +\Phi^\leftarrow : \ca{Set_\bot} & \to \ca{Pfn} \\ +A & \mapsto A - \{ \bot \} \\ +f & \mapsto \rest{f}{\left ( A - \{ \bot \} \right )} +\end{aligned} +$$ +
+
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