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author | Gregor Kleen <gkleen@yggdrasil.li> | 2016-01-29 21:40:39 +0100 |
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committer | Gregor Kleen <gkleen@yggdrasil.li> | 2016-01-29 21:40:39 +0100 |
commit | 2b133152c5af28d0f419e5ce78f529b99159500f (patch) | |
tree | 4c476ea9ddc18a1fbda7d814ee3c9c011d020274 /provider/posts/simmons-intro-to-cat-t | |
parent | 60b2206e179317f9576fc9e89031ba9fd3892ec6 (diff) | |
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Tikz
Diffstat (limited to 'provider/posts/simmons-intro-to-cat-t')
-rw-r--r-- | provider/posts/simmons-intro-to-cat-t/1.1.md | 12 |
1 files changed, 6 insertions, 6 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 41aef99..f95c068 100644 --- a/provider/posts/simmons-intro-to-cat-t/1.1.md +++ b/provider/posts/simmons-intro-to-cat-t/1.1.md | |||
@@ -10,8 +10,8 @@ Sets and functions do form a category $\ca{Set}$. | |||
10 | <div class="proof"> | 10 | <div class="proof"> |
11 | The objects of $\ca{Set}$ are sets and its arrows are functions. | 11 | The objects of $\ca{Set}$ are sets and its arrows are functions. |
12 | 12 | ||
13 | For any set $A$ we construct $id_A$ by restricting $id$ to $A$: | 13 | For any set $A$ we construct \idarr{A} by restricting \id to $A$: |
14 | $$id_A = id \upharpoonright A$$ | 14 | $$\idarr{A} = \id \upharpoonright A$$ |
15 | 15 | ||
16 | $\circ$ is indeed associative. | 16 | $\circ$ is indeed associative. |
17 | </div> | 17 | </div> |
@@ -21,9 +21,9 @@ $\circ$ is indeed associative. | |||
21 | Each [poset](https://en.wikipedia.org/wiki/Partially_ordered_set) is a category. | 21 | Each [poset](https://en.wikipedia.org/wiki/Partially_ordered_set) is a category. |
22 | 22 | ||
23 | <div class="proof"> | 23 | <div class="proof"> |
24 | 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 $a \leq b$. | 24 | 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 $a \leq b$. |
25 | 25 | ||
26 | 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. | 26 | 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. |
27 | </div> | 27 | </div> |
28 | </div> | 28 | </div> |
29 | 29 | ||
@@ -31,8 +31,8 @@ Reflexivity ($a \leq a$) and transitivity ($a \leq b \land b \leq c \implies a \ | |||
31 | Each [monoid](https://en.wikipedia.org/wiki/Monoid) is a category. | 31 | Each [monoid](https://en.wikipedia.org/wiki/Monoid) is a category. |
32 | 32 | ||
33 | <div class="proof"> | 33 | <div class="proof"> |
34 | The objects of each monoid are its elements and we construct for every pair of elements $(a, b)$ an arrow $\cdot b : a \to a \cdot b$. | 34 | The objects of each monoid are its elements and we construct for every pair of elements $(a, b)$ an arrow $\begin{tikzcd} a \arrow[r, "\cdot b"] & a \cdot b \end{tikzcd}$. |
35 | 35 | ||
36 | 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 | 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 $\circ$. |
37 | </div> | 37 | </div> |
38 | </div> | 38 | </div> |