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@@ -181,3 +181,21 @@ where $\rest{f}{\bar{A}}$ is total.
181</div> 181</div>
182</div> 182</div>
183 183
184<div class="exercise">
185Verify that for every monoid $R$ both $R\ca{Set}$ and $\ca{Set}R$ are categories of structured sets.
186
187<div class="proof">
188Since the two proofs are perfectly analogous we cover only the case $R\ca{Set}$.
189
190 1. For every $R$-Set $A \in R\ca{Set}$ there exists $\idarr{A}$
191
192 $\id$ on $A$ is indeed a structure preserving function ($\forall a \in A, r \in R \ldotp \id(ra) = ra = r \id(a)$).
193
194 2. There exists an associative partial binary operation $\circ$ on the arrows of $R\ca{Set}$
195
196 Given three R-Sets $A$, $B$, and $C$ and two continuous maps $g : A \to B$ and $f : B \to C$ the map $f \circ g : A \to C$ is an arrow in $R\ca{Set}$, that is to say $\forall a \in A, r \in R$:
197 $$(f \circ g)(ra) = f(r g(a)) = r (f \circ g)(a)$$
198
199The format of the proof was chosen to demonstrate that $R\ca{Set}$ is indeed a structured set.
200</div>
201</div>