From 60b2206e179317f9576fc9e89031ba9fd3892ec6 Mon Sep 17 00:00:00 2001 From: Gregor Kleen Date: Fri, 29 Jan 2016 18:49:11 +0100 Subject: Simmons 1.1 --- provider/posts/simmons-intro-to-cat-t/1.1.md | 38 ++++++++++++++++++++++++++++ 1 file changed, 38 insertions(+) create mode 100644 provider/posts/simmons-intro-to-cat-t/1.1.md diff --git a/provider/posts/simmons-intro-to-cat-t/1.1.md b/provider/posts/simmons-intro-to-cat-t/1.1.md new file mode 100644 index 0000000..41aef99 --- /dev/null +++ b/provider/posts/simmons-intro-to-cat-t/1.1.md @@ -0,0 +1,38 @@ +--- +title: Exercises in Category Theory — 1.1 +published: 2016-01-29 +tags: Category Theory +--- + +
+Sets and functions do form a category $\ca{Set}$. + +
+The objects of $\ca{Set}$ are sets and its arrows are functions. + +For any set $A$ we construct $id_A$ by restricting $id$ to $A$: +$$id_A = id \upharpoonright A$$ + +$\circ$ 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 $\leq: a \to b$ 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 $id_a$ and $\circ$, respectively. +
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+ +
+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 $\cdot b : a \to a \cdot b$. + +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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