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-rw-r--r--provider/posts/simmons-intro-to-cat-t/1.1.md10
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diff --git a/provider/posts/simmons-intro-to-cat-t/1.1.md b/provider/posts/simmons-intro-to-cat-t/1.1.md
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--- 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
4tags: Category Theory 4tags: Category Theory
5--- 5---
6 6
7<div class="corollary"> 7<div class="exercise">
8Sets and functions do form a category $\ca{Set}$. 8Sets and functions do form a category $\ca{Set}$.
9 9
10<div class="proof"> 10<div class="proof">
@@ -16,21 +16,21 @@ $\circ$ is indeed associative.
16</div> 16</div>
17</div> 17</div>
18 18
19<div class="corollary"> 19<div class="exercise">
20Each [poset](https://en.wikipedia.org/wiki/Partially_ordered_set) is a category. 20Each [poset](https://en.wikipedia.org/wiki/Partially_ordered_set) is a category.
21 21
22<div class="proof"> 22<div class="proof">
23The 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 $a \leq b$. 23The objects of each poset are its elements and we construct an arrow $\arr{a}{\leq}{b}$ for every pair of objects $(a, b)$ iff $a \leq b$.
24 24
25Reflexivity ($a \leq a$) and transitivity ($a \leq b \land b \leq c \implies a \leq c$) provide a construction for $\idarr{a}$ and $\circ$, respectively. 25Reflexivity ($a \leq a$) and transitivity ($a \leq b \land b \leq c \implies a \leq c$) provide a construction for $\idarr{a}$ and $\circ$, respectively.
26</div> 26</div>
27</div> 27</div>
28 28
29<div class="corollary"> 29<div class="exercise">
30Each [monoid](https://en.wikipedia.org/wiki/Monoid) is a category. 30Each [monoid](https://en.wikipedia.org/wiki/Monoid) is a category.
31 31
32<div class="proof"> 32<div class="proof">
33The objects of each monoid are its elements and we construct for every pair of elements $(a, b)$ an arrow $\begin{tikzcd} a \arrow[r, "\cdot b"] & a \cdot b \end{tikzcd}$. 33The objects of each monoid are its elements and we construct for every pair of elements $(a, b)$ an arrow $\arr{a}{\cdot b}{a \cdot b}$.
34 34
35The existence of an identity element and associativity, as required by the monoid structure, immediately provide us with a construction for $\id$ and associativity of $\circ$. 35The existence of an identity element and associativity, as required by the monoid structure, immediately provide us with a construction for $\id$ and associativity of $\circ$.
36</div> 36</div>