From 7b4df716eb30be99d31258156d6f388fff82b237 Mon Sep 17 00:00:00 2001 From: Gregor Kleen Date: Tue, 2 Feb 2016 19:14:55 +0100 Subject: more concise arrows & exercise envs --- provider/posts/simmons-intro-to-cat-t/1.1.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/provider/posts/simmons-intro-to-cat-t/1.1.md b/provider/posts/simmons-intro-to-cat-t/1.1.md index 6c69c62..1b6c16d 100644 --- a/provider/posts/simmons-intro-to-cat-t/1.1.md +++ b/provider/posts/simmons-intro-to-cat-t/1.1.md @@ -4,7 +4,7 @@ published: 2016-01-29 tags: Category Theory --- -
+
Sets and functions do form a category $\ca{Set}$.
@@ -16,21 +16,21 @@ $\circ$ is indeed associative.
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+
Each [poset](https://en.wikipedia.org/wiki/Partially_ordered_set) is a category.
-The objects of each poset are its elements and we construct an arrow $\begin{tikzcd} a \arrow[r, "\leq"] & b \end{tikzcd}$ for every pair of objects $(a, b)$ iff $a \leq b$. +The objects of each poset are its elements and we construct an arrow $\arr{a}{\leq}{b}$ for every pair of objects $(a, b)$ iff $a \leq b$. Reflexivity ($a \leq a$) and transitivity ($a \leq b \land b \leq c \implies a \leq c$) provide a construction for $\idarr{a}$ and $\circ$, respectively.
-
+
Each [monoid](https://en.wikipedia.org/wiki/Monoid) is a category.
-The objects of each monoid are its elements and we construct for every pair of elements $(a, b)$ an arrow $\begin{tikzcd} a \arrow[r, "\cdot b"] & a \cdot b \end{tikzcd}$. +The objects of each monoid are its elements and we construct for every pair of elements $(a, b)$ an arrow $\arr{a}{\cdot b}{a \cdot b}$. The existence of an identity element and associativity, as required by the monoid structure, immediately provide us with a construction for $\id$ and associativity of $\circ$.
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