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-rw-r--r-- | provider/posts/simmons-intro-to-cat-t/1.2.md | 18 |
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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 faaf429..30d0492 100644 --- a/provider/posts/simmons-intro-to-cat-t/1.2.md +++ b/provider/posts/simmons-intro-to-cat-t/1.2.md | |||
@@ -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"> | ||
185 | Verify that for every monoid $R$ both $R\ca{Set}$ and $\ca{Set}R$ are categories of structured sets. | ||
186 | |||
187 | <div class="proof"> | ||
188 | Since 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 | |||
199 | The format of the proof was chosen to demonstrate that $R\ca{Set}$ is indeed a structured set. | ||
200 | </div> | ||
201 | </div> | ||