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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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\tableofcontents
%%--------------------------------------
Welcome one and all to our shared exercise sheet. 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, and ask your fellow students about their answers!
\section{Week One}
\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
\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
