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%%--------------------------------------
\begin{document}
\title{Introduction to Category Theory}
\address{ $\dagger$ Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190 CH-8057 Zürich}
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\maketitle
\tableofcontents
%%--------------------------------------
\section{Instructions}
Welcome one and all to our shared exercise sheet! The purpose of this space is two-fold. Firstly, because the only way to learn is to do, and listening to your peer's talks is not enough to understand category theory. Secondly, because our goal is vast, and if we were to cover the whole book (plus our extra topics at the end) we would need twice as much time.
In class the speaker will always explain the most important topics, but you should always try, before or after the lecture, to read the relevant chapter on your own and attempt one of the questions below. These will allow you a chance to learn, and the speaker an excuse to skip the less interesting topics.
Questions, answers and comments will be posted the following way:
I don't know but it is not green.
\iam{An Other Student}
YELLOW!
Remember to sign you name using the \textbackslash iam\{\} command. Try to answer these questions without fear, ask your fellow students about their answers, or even post questions of your own!
If you would like to draw simple diagram you can do so here https://tikzcd.yichuanshen.de and insert them in the tex file as follows (see code)
\begin{center}
% https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAEEQBfU9TXfIRQAmclVqMWbAELdeIDNjwEio4ePrNWiEAGFu4mFADm8IqABmAJwgBbJGRA4ISAIzVNUnRbmWb9xHcnF0RRCS02Y18QazsHamckMM9tEGMAHXSAYywrLIACH2oGOgAjGAYABX5lIRArLGMACxwDLiA
\begin{tikzcd}
A \arrow[rr, "f"] \arrow[rrdd, "g\circ f"'] & & B \arrow[dd, "g"] \\
& & \\
& & C
\end{tikzcd}
\end{center}
\subsection{} The category \textbf{Rel} has as objects sets and morphisms $f:A\to B$ are \emph{relations}, that is subsets of the form $f\subseteq A\times B$. Composition in this category is given by
\begin{align*}
g\circ f = \{ (a,c)\in A\times C \,|\, \exists b\in B \,\,\text{with}\,\, (a,b)\in f, (b,c)\in g\}
\end{align*}
for $f\in A\times B$ and $g\in B\times C$. Show that \textbf{Rel} satisfies the axioms of a category. What is the identity morphism?\hfill \break
\subsection{} For a fixed set $X$, we call $\mathcal{P}(X)$ its \emph{power set}. This is the set of all subsets of $X$. Show how this has naturally the structure of a poset (so in particular a simple category) using subset inclusions.\hfill \break
\subsection{} Check that functors compose in an associative way, proving that $\mathbf{Cat}$ is a category. \hfill \break
\subsection{} Prove that inverses are unique. \hfill \break
\iam{Rızacan} Let $f$ be an arrow with Cod($f$)=$B$ and Dom($f$)=$A$. Suppose
$g,g'$ are both inverses of $f$. Then by associativity and properties of
the identity: $g'=\id_A\circ
\subsection{} Argue whether the following \emph{isomorphisms} of categories hold:
\begin{enumerate}
\item $\mathbf{Rel}\cong \mathbf{Rel}^{op}$;
\item $\mathbf{Set}\cong \mathbf{Set}^{op}$;
\item For a fixed set $X$, $\mathcal{P}(X)\cong \mathcal{P}(X)^{op}$.
\end{enumerate}\hfill \break
\subsection{} Show that in $\mathbf{Set}$ isomorphisms are bijections.\hfill \break
\subsection{} Show that in $\mathbf{Mon}$, the category of monoids, isomorphisms are bijective homomorphisms.\hfill \break
\subsection{} Show that in $\mathbf{Pos}$, the category of posets, isomorphisms are \emph{not} bijective homomorphisms.\hfill \break
\iam{Marius} Consider the posets $\mathcal{P} :=
\begin{tikzcd}[cramped, sep=scriptsize]
a \arrow[loop, distance=1.2em, in=115, out=65] & b \arrow[loop, distance=1.2em, in=115, out=65]
\end{tikzcd}$ and $\mathcal{Q} :=
\begin{tikzcd}[cramped, sep=scriptsize]
a' \arrow[loop, distance=1.2em, in=115, out=65] \arrow[r] & b' \arrow[loop, distance=1.2em, in=115, out=65]
\end{tikzcd}$, along with the map $f: \mathcal{P} \rightarrow \mathcal{Q}$ sending $a \mapsto a'$, and $b \mapsto b'$. Clearly, $f$ is bijective and monotone. However, it has no inverse: There are exactly two monotone maps from $\mathcal{Q}$ to $\mathcal{P}$, one sending both $a',b' \mapsto a$, and another sending both $a',b' \mapsto b$. Since neither is surjective, we can't recover the identity on $\mathcal{P}$ by pre-composing with $f$. Hence, $f$ is a bijective homomorphism which is not an isomorphism, as desired.
\subsection{} Construct the co-slice category $A\downarrow \mathbf{C}$, of objects of $\mathbf{C}$ ``under'' $A$, using the slice category $\mathbf{C}\downarrow A$ and the operation $(-)^{op}$. Prove your claim.\hfill \break
\subsection{} How many \emph{free categories} are there with 6 arrows? \emph{Hint:} Try to draw them on a piece of paper, and if you feel industrious you can report them here using this online tool https://tikzcd.yichuanshen.de/ \hfill \break
\subsection{} (\emph{Harder}) Prove the universal mapping property of free categories on graphs. \hfill \break