CatT 1.2.2

1 files changed, 19 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 eef7a0a..7759368 100644
--- a/provider/posts/simmons-intro-to-cat-t/1.2.md
+++ b/provider/posts/simmons-intro-to-cat-t/1.2.md
@@ -49,3 +49,22 @@ Let $\ca{Pno}$ be the category of objects $(A, \alpha, a)$ where $A$ is a set, $
     The morphism maps $\N$ to the transitive closure of $a$ under $\alpha$.
     </div>
 </div>
+
+<div class="exercise">
+Consider objects of form $(A, a)$ where $A$ is a set and $a \subseteq A$.
+For two such objects a morphism $\arr{(A, a)}{f}{(B, b)}$ is a function $f : A \to B$ that respects the selected subsets:
+$$\forall \alpha \in a \ldotp f(\alpha) \in b$$
+
+Show that such objects and morphisms form a category $\ca{SetD}$
+
+ 1. For every $(A, a) \in \ca{SetD}$ there exists $\idarr{(A, a)}$
+    <div class="proof">
+    $\id$ on $A$ is indeed a function which respects the distinguished subset ($\forall \alpha \in a \ldotp \id(\alpha) \in a$) and thus a morphism
+    </div>
+    
+ 2. There exists a partial binary operation $\circ$ on the arrows of $\ca{SetD}$
+    <div class="proof">
+    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, $\forall \alpha \in a$:
+    $$(f \circ g)(\alpha) = f(b) = c$$
+    </div>
+</div>