Exercises.tex

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\begin{document}
\title{Introduction to Category Theory}




\address{ $\dagger$ Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190 CH-8057 Zürich}



%\begin{abstract}
%\end{abstract}


\maketitle

\tableofcontents

%%--------------------------------------


Welcome one and all to our shared exercise sheet. Questions, answers and comments will be posted the following way: 

\subsection{Question 0.1} What is the most creative color?\hfill \break

\iam{Student} 
I don't know but it is not green.

\iam{An Other Student}
YELLOW!

\subsection{Question 0.2} The next Question...\hfill \break

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{Question 1.1} 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{Question 1.2} 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{Question 1.3} 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{Question 1.4} Show that in $\mathbf{Set}$ isomorphisms are bijections.\hfill \break

\subsection{Question 1.5} Show that in $\mathbf{Mon}$, the category of monoids isomorphisms are bijective homomorphisms.\hfill \break

\subsection{Question 1.6} Show that in $\mathbf{Pos}$, the category of posets, isomorphisms are \emph{not} bijective homomorphisms.\hfill \break

\subsection{Question 1.7} 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{Question 1.8} How many 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{Question 1.9} (\emph{Harder}) Prove the universal mapping property of free categories on graphs. \hfill \break

\subsection{Question 1.10}  Check that functors compose in an associative way, proving that $\mathbf{Cat}$ is a category. \hfill \break
\end{document}