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@@ -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">
54Consider objects of form $(A, a)$ where $A$ is a set and $a \subseteq A$.
55For 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
58Show 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>