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-rw-r--r--provider/posts/simmons-intro-to-cat-t/1.2.md27
-rw-r--r--provider/templates/preamble.tex9
2 files changed, 31 insertions, 5 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 b91cac4..19d71e4 100644
--- a/provider/posts/simmons-intro-to-cat-t/1.2.md
+++ b/provider/posts/simmons-intro-to-cat-t/1.2.md
@@ -91,7 +91,7 @@ We call the category comprised of such objects and arrows $\ca{RelH}$.
91</div> 91</div>
92 92
93<div class="exercise"> 93<div class="exercise">
94Topological 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 $\ca{Top}$. 94Topological 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 $\ca{Top}$.
95 95
96 1. For every $(S, \mathcal{O}_S) \in \ca{Top}$ there exists $\idarr{(S, \mathcal{O}_S)}$ 96 1. For every $(S, \mathcal{O}_S) \in \ca{Top}$ there exists $\idarr{(S, \mathcal{O}_S)}$
97 <div class="proof"> 97 <div class="proof">
@@ -123,3 +123,28 @@ We further define $m \circ m = m \ast m$.
123</div> 123</div>
124</div> 124</div>
125</div> 125</div>
126
127<div class="exercise">
128Show that $\circ$ in $\ca{Pfn}$ is associative.
129
130<div class="proof">
131Consider the composition $g \circ f = \rest{g}{\bar{B}} \circ \rest{f}{\bar{\bar{A}}}$ of two partial functions $f : A \to B$ and $g : B \to C$:
132$$
133\begin{tikzcd}
134A \arrow[r, "f"] & B \arrow[r, "g"] & C \\
135\bar{A} \arrow[u, hook] \arrow[ru, "\rest{f}{\bar{A}}" description] & \bar{B} \arrow[u, hook] \arrow[ru, "\rest{g}{\bar{B}}" description] & \\
136\bar{\bar{A}} \arrow[u, hook] \arrow[ru, "\rest{f}{\bar{\bar{A}}}" description] & & \\
137\end{tikzcd}
138$$
139
140Extending the above to three partial functions $f : A \to B$, $g : B \to C$, and $h : C \to D$:
141$$
142\begin{aligned}
143(h \circ g) \circ f &= \rest{\left (\rest{h}{\bar{C}} \circ \rest{g}{\bar{\bar{B}}} \right )}{\bar{B}} \circ \rest{f}{\bar{\bar{A}}} \\
144&= \rest{h}{\bar{C}} \circ \rest{g}{\bar{\bar{B}}} \circ \rest{f}{\bar{\bar{A}}} \\
145&= \rest{h}{\bar{C}} \circ \rest{\left (\rest{g}{\bar{\bar{B}}} \circ \rest{f}{\bar{\bar{A}}} \right)}{\bar{\bar{A}}} \\
146&= h \circ (g \circ f)
147\end{aligned}
148$$
149</div>
150</div>
diff --git a/provider/templates/preamble.tex b/provider/templates/preamble.tex
index 1f2c1e4..541b415 100644
--- a/provider/templates/preamble.tex
+++ b/provider/templates/preamble.tex
@@ -8,15 +8,16 @@
8 8
9\usetikzlibrary{cd} 9\usetikzlibrary{cd}
10 10
11\newcommand*{\ca}[1]{\ensuremath{\mathbf{#1}}}
12\newcommand*{\idarr}[1]{\ensuremath{\mathrm{id}_{#1}}}
13
14\newcommand*{\id}{\ensuremath{\mathrm{id}}} 11\newcommand*{\id}{\ensuremath{\mathrm{id}}}
15\renewcommand{\implies}{\ensuremath{\rightarrow}} 12\renewcommand{\implies}{\ensuremath{\rightarrow}}
16 13
17\newcommand*{\N}{\ensuremath{\mathbb{N}}} 14\newcommand*{\N}{\ensuremath{\mathbb{N}}}
15\newcommand*{\powerset}[1]{\raisebox{.15\baselineskip}{\Large\ensuremath{\wp}} \left ( #1 \right )}
16
17\newcommand*{\ca}[1]{\ensuremath{\mathbf{#1}}}
18\newcommand*{\idarr}[1]{\ensuremath{\mathrm{id}_{#1}}}
18\newcommand*{\arr}[3]{\begin{tikzcd}[ampersand replacement=\&]{#1} \rar{#2} \& {#3}\end{tikzcd}} 19\newcommand*{\arr}[3]{\begin{tikzcd}[ampersand replacement=\&]{#1} \rar{#2} \& {#3}\end{tikzcd}}
19\newcommand*{\Hom}[3]{\ensuremath{\mathrm{Hom}_{#1} \left [ #2, #3 \right ]}} 20\newcommand*{\Hom}[3]{\ensuremath{\mathrm{Hom}_{#1} \left [ #2, #3 \right ]}}
20\newcommand*{\End}[2]{\ensuremath{\mathrm{End}_{#1} \left [ #2 \right ]}} 21\newcommand*{\End}[2]{\ensuremath{\mathrm{End}_{#1} \left [ #2 \right ]}}
21 22
22\newcommand*{\powerset}{\raisebox{.15\baselineskip}{\Large\ensuremath{\wp}}} 23\newcommand*{\rest}[2]{\ensuremath{#1 \upharpoonright #2}}