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author | Gregor Kleen <gkleen@yggdrasil.li> | 2016-02-03 13:49:04 +0100 |
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committer | Gregor Kleen <gkleen@yggdrasil.li> | 2016-02-03 13:49:04 +0100 |
commit | 35bbb98cbaa940a1266749a5f65a7a2451c44ee7 (patch) | |
tree | e76a15edd73c1cbdd145c18c22f43392f6285084 | |
parent | 7ff393170b2e542540ee5c0e044839e4df449947 (diff) | |
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CatT 1.2.2
-rw-r--r-- | provider/posts/simmons-intro-to-cat-t/1.2.md | 19 |
1 files changed, 19 insertions, 0 deletions
diff --git a/provider/posts/simmons-intro-to-cat-t/1.2.md b/provider/posts/simmons-intro-to-cat-t/1.2.md index eef7a0a..7759368 100644 --- a/provider/posts/simmons-intro-to-cat-t/1.2.md +++ b/provider/posts/simmons-intro-to-cat-t/1.2.md | |||
@@ -49,3 +49,22 @@ Let $\ca{Pno}$ be the category of objects $(A, \alpha, a)$ where $A$ is a set, $ | |||
49 | The morphism maps $\N$ to the transitive closure of $a$ under $\alpha$. | 49 | The morphism maps $\N$ to the transitive closure of $a$ under $\alpha$. |
50 | </div> | 50 | </div> |
51 | </div> | 51 | </div> |
52 | |||
53 | <div class="exercise"> | ||
54 | Consider objects of form $(A, a)$ where $A$ is a set and $a \subseteq A$. | ||
55 | For two such objects a morphism $\arr{(A, a)}{f}{(B, b)}$ is a function $f : A \to B$ that respects the selected subsets: | ||
56 | $$\forall \alpha \in a \ldotp f(\alpha) \in b$$ | ||
57 | |||
58 | Show that such objects and morphisms form a category $\ca{SetD}$ | ||
59 | |||
60 | 1. For every $(A, a) \in \ca{SetD}$ there exists $\idarr{(A, a)}$ | ||
61 | <div class="proof"> | ||
62 | $\id$ on $A$ is indeed a function which respects the distinguished subset ($\forall \alpha \in a \ldotp \id(\alpha) \in a$) and thus a morphism | ||
63 | </div> | ||
64 | |||
65 | 2. There exists a partial binary operation $\circ$ on the arrows of $\ca{SetD}$ | ||
66 | <div class="proof"> | ||
67 | Given three objects $(A, a), (B, b), (C, c)$ and two functions $g: A \to B$ and $f: B \to C$ the function $f \circ g: A \to C$ is an arrow in $\ca{SetD}$, that is to say, $\forall \alpha \in a$: | ||
68 | $$(f \circ g)(\alpha) = f(b) = c$$ | ||
69 | </div> | ||
70 | </div> | ||