CatT 1.2.3 & 1.2.4

2 files changed, 48 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 7759368..3e83b5f 100644
--- a/provider/posts/simmons-intro-to-cat-t/1.2.md
+++ b/provider/posts/simmons-intro-to-cat-t/1.2.md
@@ -68,3 +68,48 @@ Show that such objects and morphisms form a category $\ca{SetD}$
     $$(f \circ g)(\alpha) = f(b) = c$$
     </div>
 </div>
+
+<div class="exercise">
+Consider pairs $(A, \odot)$ where $A$ is a set and $\odot \subseteq A \times A$ is a binary relation on $A$.
+
+Show that these pairs are objects of a category finding a sensible notion of morphism.
+
+<div class="proof">
+For two such objects $(A, \odot), (B, \oplus)$ a morphism shall be a function $f : A \to B$ that respects the binary relation:
+$$\forall a \odot a^\prime \ldotp f(a) \oplus f(a^\prime)$$
+We call the category comprised of such objects and arrows $\ca{RelH}$.
+
+ 1. For every $(A, \odot) \in \ca{RelH}$ there exists $\idarr{(A, \odot)}$
+
+    $\id$ 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
+    
+ 2. There exists a partial binary operation $\circ$ on the arrows of $\ca{RelH}$
+
+    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, $\forall a \odot a^\prime$:
+    $$a \odot a^\prime \implies g(a) \oplus g(a^\prime) \implies (g \circ f)(a) \otimes (g \circ f)(a^\prime)$$
+</div>
+</div>
+
+<div class="exercise">
+Topological 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 $\ca{Top}$.
+
+ 1. For every $(S, \mathcal{O}_S) \in \ca{Top}$ there exists $\idarr{(S, \mathcal{O}_S)}$
+    <div class="proof">
+    $\id$ on $S$ is indeed a continuous map and thus a morphism
+    </div>
+    
+ 2. There exists a partial binary operation $\circ$ on the arrows of $\ca{Top}$
+    <div class="proof">
+    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:
+    $$V \in \mathcal{O}_U \implies g^\leftarrow(V) \in \mathcal{O}_T \implies (f \circ g)^\leftarrow(V) \in \mathcal{O}_S$$
+    </div>
+</div>
+
+<div class="exercise">
+Let $A$ be an object of a category $\ca{C}$.
+Show that $\hom{\ca{C}}{A}{A}$ is a monoid under composition.
+
+<div class="proof">
+
+</div>
+</div>
diff --git a/provider/templates/preamble.tex b/provider/templates/preamble.tex
index 6ca168f..0d76503 100644
--- a/provider/templates/preamble.tex
+++ b/provider/templates/preamble.tex
@@ -16,3 +16,6 @@
 
 \newcommand{\N}{\ensuremath{\mathbb{N}}}
 \newcommand{\arr}[3]{\begin{tikzcd}[ampersand replacement=\&]{#1} \rar{#2} \& {#3}\end{tikzcd}}
+\renewcommand{\hom}[3]{\ensuremath{\mathrm{hom}_{#1} \left [ #2, #3 \right ]}}
+
+\newcommand{\powerset}{\raisebox{.15\baselineskip}{\Large\ensuremath{\wp}}}