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\documentclass{amsart}
\usepackage[foot]{amsaddr}
\usepackage{bbm}
\usepackage{mathbbol}
\usepackage{amsmath,amssymb,amsfonts, amsthm, graphicx}
\usepackage{mathtools}
%\usepackage{stmaryrd}
\usepackage{graphicx}%for \rotatebox
\usepackage[dvipsnames]{xcolor}
\usepackage{txfonts}
\usepackage{tikz}
\usetikzlibrary{arrows,shapes,snakes,automata,backgrounds,petri,positioning}
\usepackage{comment}
\usepackage[pdfborder=0,colorlinks=true,linktocpage,linkcolor=blue,urlcolor=cyan]{hyperref}
\usepackage{csquotes}
\usepackage{comment}
\usepackage[normalem]{ulem}
\usepackage{enumerate}
\usepackage{verbatim}
\usepackage{xcolor}
\usepackage{amsmath}
%\usepackage{fullpage}

\setlength{\textwidth}{\paperwidth}
\addtolength{\textwidth}{-2in}
\calclayout

%diagram stuff
%try:  https://tikzcd.yichuanshen.de/
%xymatrix
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt 
\usepackage[all,pdftex, cmtip]{xy}
\newdir{ >}{{}*!/-10pt/@{>}}
\newdir{> }{{}*!/10pt/@{>}}
\newcommand{\cd}[2][]{\vcenter{\hbox{\xymatrix#1{#2}}}}
\newcommand{\ltwocell}[3][0.5]{\ar@{}[#2] \ar@{=>}?(#1)+/r 0.2cm/;?(#1)+/l 0.2cm/_{#3}}

\newcommand{\hdash}{\rotatebox[origin=c]{90}{$\vdash$}}

%tikz-cd
\usepackage{amssymb}
\usepackage{tikz}
\usetikzlibrary{cd}
\usepackage{tikz-cd}
\tikzset{% for drawing adjunctions
    symbol/.style={%
        draw=none,
        every to/.append style={%
            edge node={node [sloped, allow upside down, auto=false]{$#1$}}}
    }
}

%roman enumeration
\usepackage{enumerate}
\newenvironment{enumroman}{\begin{enumerate}[\upshape (i)]}{\end{enumerate}}
\renewcommand{\theenumi}{\roman{enumi}}

%blackboard bold numbers and stuff
\newcommand{\bbefamily}{\fontencoding{U}\fontfamily{bbold}\selectfont}
\newcommand{\textbbe}[1]{{\bbefamily #1}}
\DeclareMathAlphabet{\mathbbe}{U}{bbold}{m}{n}

%theorem stuff
\theoremstyle{plain}
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{claim}[thm]{Claim}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}

\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{digression}[thm]{Digression}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{const}[thm]{Construction}

\theoremstyle{remark}
\newtheorem{ex}[thm]{Example}
\newtheorem{rmk}[thm]{Remark}
\newtheorem{notation}[thm]{Notation}

%numbering
\setcounter{tocdepth}{1}
\makeatletter
\let\c@equation\c@thm
\makeatother
\numberwithin{equation}{section}

%r arrows double headed with \simeq symbol
\newcommand{\refinement}{%
  \xrightarrow{\simeq}\mathrel{\mkern-14mu}\rightarrow
}
\usepackage{extarrows}

%bra-ket notation
\usepackage{mathtools}
\DeclarePairedDelimiter\bra{\langle}{\rvert}
\DeclarePairedDelimiter\ket{\lvert}{\rangle}
\DeclarePairedDelimiterX\braket[2]{\langle}{\rangle}{#1 \delimsize\vert #2}

%categories
\newcommand{\cat}[1]{\textup{\textsf{#1}}}% for categories
\newcommand{\Set}{\operatorname{\mathbf{Set}}}
\newcommand{\mat}{\operatorname{\mathbf{Mat}}}
\newcommand{\Cat}{\operatorname{\mathbf{Cat}}}
\newcommand{\Vect}[1]{\mathbf{Vect}_{#1}}%
\newcommand{\yonenr}[1]{\mathfrak Y_{#1}}%
\newcommand{\functorcat}[2]{[#1,#2]}%
\newcommand{\map}{\textbf{Map}}
\newcommand{\Map}{\operatorname{\mathbf{Map}}}
\newcommand{\Id}{\textrm{Id}}
\newcommand{\FVect}{\mathbf{FVect}}
\newcommand{\Field}{\operatorname{\mathbf{Field}}}

%blackboard bold
\newcommand{\ff}{\mathbb{F}}
\newcommand{\nn}{\mathbb{N}}
\newcommand{\rr}{\mathbb{R}}
\newcommand{\qq}{\mathbb{Q}}
\newcommand{\BB}{\mathbb{B}}

%big plus symbol
\usepackage{amsmath}
\DeclareMathOperator*{\foo}{\scalerel*{+}{\sum}}
\DeclareMathOperator*{\barr}{\scalerel*{+}{\textstyle\sum}}
\usepackage{scalerel}


%calligraphic
\newcommand{\Ca}{\mathcal{C}}
\newcommand{\Oa}{\mathrm{Sub}}
\newcommand{\Da}{\mathcal{C}}
\newcommand{\Ea}{\mathcal{E}}
\newcommand{\Ba}{\mathcal{B}}

%misc shortcuts
\newcommand{\lra}{\longrightarrow}
\newcommand{\xto}[1]{\xrightarrow{#1}}
\newcommand{\op}{^\text{op}}
\newcommand{\id}{\textbf{id}}
\newcommand{\inv}{^{-1}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Sub}{\operatorname{Sub}}
\newcommand{\Left}{\operatorname{Left}}
\newcommand{\Right}{\operatorname{Right}}
\newcommand{\Cov}{\operatorname{Cov}}
\DeclareMathOperator{\Hom}{Hom}

\allowdisplaybreaks

%use this to identify who is writing; LaTeX gurus please feel free to fiddle with the macro
\newcommand{\iam}[1]{\vspace{.1cm}\centerline{\fbox{\textbf{#1}}}\vspace{.1cm}}


\usepackage{float}
%%--------------------------------------


\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

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