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-rw-r--r-- | provider/posts/simmons-intro-to-cat-t/1.1.md | 10 |
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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 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 | |||
4 | tags: Category Theory | 4 | tags: Category Theory |
5 | --- | 5 | --- |
6 | 6 | ||
7 | <div class="corollary"> | 7 | <div class="exercise"> |
8 | Sets and functions do form a category $\ca{Set}$. | 8 | Sets 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"> |
20 | Each [poset](https://en.wikipedia.org/wiki/Partially_ordered_set) is a category. | 20 | Each [poset](https://en.wikipedia.org/wiki/Partially_ordered_set) is a category. |
21 | 21 | ||
22 | <div class="proof"> | 22 | <div class="proof"> |
23 | 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 $a \leq b$. | 23 | 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 $a \leq b$. |
24 | 24 | ||
25 | 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. | 25 | 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. |
26 | </div> | 26 | </div> |
27 | </div> | 27 | </div> |
28 | 28 | ||
29 | <div class="corollary"> | 29 | <div class="exercise"> |
30 | Each [monoid](https://en.wikipedia.org/wiki/Monoid) is a category. | 30 | Each [monoid](https://en.wikipedia.org/wiki/Monoid) is a category. |
31 | 31 | ||
32 | <div class="proof"> | 32 | <div class="proof"> |
33 | The objects of each monoid are its elements and we construct for every pair of elements $(a, b)$ an arrow $\begin{tikzcd} a \arrow[r, "\cdot b"] & a \cdot b \end{tikzcd}$. | 33 | The objects of each monoid are its elements and we construct for every pair of elements $(a, b)$ an arrow $\arr{a}{\cdot b}{a \cdot b}$. |
34 | 34 | ||
35 | 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 $\circ$. | 35 | 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 $\circ$. |
36 | </div> | 36 | </div> |