CatT 1.2.8

1 files changed, 18 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 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.
 </div>
 </div>
 
+<div class="exercise">
+Verify that for every monoid $R$ both $R\ca{Set}$ and $\ca{Set}R$ are categories of structured sets.
+
+<div class="proof">
+Since the two proofs are perfectly analogous we cover only the case $R\ca{Set}$.
+
+ 1. For every $R$-Set $A \in R\ca{Set}$ there exists $\idarr{A}$
+
+    $\id$ on $A$ is indeed a structure preserving function ($\forall a \in A, r \in R \ldotp \id(ra) = ra = r \id(a)$).
+    
+ 2. There exists an associative partial binary operation $\circ$ on the arrows of $R\ca{Set}$
+
+    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 $\forall a \in A, r \in R$:
+    $$(f \circ g)(ra) = f(r g(a)) = r (f \circ g)(a)$$
+
+The format of the proof was chosen to demonstrate that $R\ca{Set}$ is indeed a structured set.
+</div>
+</div>