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@@ -68,3 +68,48 @@ Show that such objects and morphisms form a category $\ca{SetD}$
68 $$(f \circ g)(\alpha) = f(b) = c$$ 68 $$(f \circ g)(\alpha) = f(b) = c$$
69 </div> 69 </div>
70</div> 70</div>
71
72<div class="exercise">
73Consider pairs $(A, \odot)$ where $A$ is a set and $\odot \subseteq A \times A$ is a binary relation on $A$.
74
75Show that these pairs are objects of a category finding a sensible notion of morphism.
76
77<div class="proof">
78For two such objects $(A, \odot), (B, \oplus)$ a morphism shall be a function $f : A \to B$ that respects the binary relation:
79$$\forall a \odot a^\prime \ldotp f(a) \oplus f(a^\prime)$$
80We call the category comprised of such objects and arrows $\ca{RelH}$.
81
82 1. For every $(A, \odot) \in \ca{RelH}$ there exists $\idarr{(A, \odot)}$
83
84 $\id$ on $A$ is indeed a function which respects the binary relation ($\forall a \odot a^\prime \ldotp \id(a) \odot \id(a^\prime)$) and thus a morphism
85
86 2. There exists a partial binary operation $\circ$ on the arrows of $\ca{RelH}$
87
88 Given three objects $(A, \odot), (B, \oplus), (C, \otimes)$ and two functions $g: A \to B$ and $f: B \to C$ the function $f \circ g: A \to C$ is an arrow in $\ca{RelH}$, that is to say, $\forall a \odot a^\prime$:
89 $$a \odot a^\prime \implies g(a) \oplus g(a^\prime) \implies (g \circ f)(a) \otimes (g \circ f)(a^\prime)$$
90</div>
91</div>
92
93<div class="exercise">
94Topological spaces $(S, \mathcal{O}_S)$, where $S$ is a Set and $\mathcal{O}_S \subseteq \powerset(S)$, and continuous maps $f : S \to P$, that is $\forall V \in \mathcal{O}_T \ldotp f^\leftarrow(V) \in \mathcal{O}_S$, form a Category $\ca{Top}$.
95
96 1. For every $(S, \mathcal{O}_S) \in \ca{Top}$ there exists $\idarr{(S, \mathcal{O}_S)}$
97 <div class="proof">
98 $\id$ on $S$ is indeed a continuous map and thus a morphism
99 </div>
100
101 2. There exists a partial binary operation $\circ$ on the arrows of $\ca{Top}$
102 <div class="proof">
103 Given three spaces $(S, \mathcal{O}_S), (T, \mathcal{O}_T), (U, \mathcal{O}_U)$ and two continuous maps $g : S \to T$ and $f : T \to U$ the map $f \circ g : S \to U$ is an arrow in $\ca{Top}$, that is to say, it is contiuous:
104 $$V \in \mathcal{O}_U \implies g^\leftarrow(V) \in \mathcal{O}_T \implies (f \circ g)^\leftarrow(V) \in \mathcal{O}_S$$
105 </div>
106</div>
107
108<div class="exercise">
109Let $A$ be an object of a category $\ca{C}$.
110Show that $\hom{\ca{C}}{A}{A}$ is a monoid under composition.
111
112<div class="proof">
113
114</div>
115</div>