From 5553a3223d07f73955465952d7765d47d3558d7b Mon Sep 17 00:00:00 2001 From: Gregor Kleen Date: Wed, 3 Feb 2016 21:33:51 +0100 Subject: CatT 1.2.8 --- provider/posts/simmons-intro-to-cat-t/1.2.md | 18 ++++++++++++++++++ 1 file changed, 18 insertions(+) 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 faaf429..30d0492 100644 --- a/provider/posts/simmons-intro-to-cat-t/1.2.md +++ b/provider/posts/simmons-intro-to-cat-t/1.2.md @@ -181,3 +181,21 @@ where $\rest{f}{\bar{A}}$ is total. +
+Verify that for every monoid $R$ both $R\ca{Set}$ and $\ca{Set}R$ are categories of structured sets. + +
+Since the two proofs are perfectly analogous we cover only the case $R\ca{Set}$. + + 1. For every $R$-Set $A \in R\ca{Set}$ there exists $\idarr{A}$ + + $\id$ on $A$ is indeed a structure preserving function ($\forall a \in A, r \in R \ldotp \id(ra) = ra = r \id(a)$). + + 2. There exists an associative partial binary operation $\circ$ on the arrows of $R\ca{Set}$ + + Given three R-Sets $A$, $B$, and $C$ and two continuous maps $g : A \to B$ and $f : B \to C$ the map $f \circ g : A \to C$ is an arrow in $R\ca{Set}$, that is to say $\forall a \in A, r \in R$: + $$(f \circ g)(ra) = f(r g(a)) = r (f \circ g)(a)$$ + +The format of the proof was chosen to demonstrate that $R\ca{Set}$ is indeed a structured set. +
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