Skip to content
Snippets Groups Projects
Exercises.tex 9.65 KiB
Newer Older
Nicola Capacci's avatar
Nicola Capacci committed
\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
Rizacan Çiloglu's avatar
Rizacan Çiloglu committed
\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt
Nicola Capacci's avatar
Nicola Capacci committed
\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
Nicola Capacci's avatar
Nicola Capacci committed
\newcommand{\iam}[1]{\vspace{.1cm}{\fbox{\textbf{#1}}}\vspace{.1cm}}
Nicola Capacci's avatar
Nicola Capacci committed


\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

%%--------------------------------------
Nicola Capacci's avatar
Nicola Capacci committed
\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: 
Nicola Capacci's avatar
Nicola Capacci committed

Nicola Capacci's avatar
Nicola Capacci committed
\subsection{} What is the most creative color?\hfill \break
Nicola Capacci's avatar
Nicola Capacci committed

Rizacan Çiloglu's avatar
Rizacan Çiloglu committed
\iam{Student}
Nicola Capacci's avatar
Nicola Capacci committed
I don't know but it is not green.

\iam{An Other Student}
YELLOW!

Nicola Capacci's avatar
Nicola Capacci committed
\subsection{} The next Question...\hfill \break
Nicola Capacci's avatar
Nicola Capacci committed

Nicola Capacci's avatar
Nicola Capacci committed
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!
Nicola Capacci's avatar
Nicola Capacci committed

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}

Nicola Capacci's avatar
Nicola Capacci committed
\section{Week One}

Rizacan Çiloglu's avatar
Rizacan Çiloglu committed
\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
Nicola Capacci's avatar
Nicola Capacci committed
\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

Nicola Capacci's avatar
Nicola Capacci committed
\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
Nicola Capacci's avatar
Nicola Capacci committed

Nicola Capacci's avatar
Nicola Capacci committed
\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
Nicola Capacci's avatar
Nicola Capacci committed

Rizacan Çiloglu's avatar
Rizacan Çiloglu committed
\iam{Rızacan} Let $f$ be an arrow with Cod($f$)=$B$ and Dom($f$)=$A$. Suppose
Rizacan Çiloglu's avatar
Rizacan Çiloglu committed
$g,g'$ are both inverses of $f$. Then by associativity and properties of
the identity: $g'=\id_A\circ
Rizacan Çiloglu's avatar
Rizacan Çiloglu committed
g'=(g\circ f)\circ g'= g\circ(f\circ g')=g\circ \id_B=g$.
Nicola Capacci's avatar
Nicola Capacci committed

Rizacan Çiloglu's avatar
Rizacan Çiloglu committed

Nicola Capacci's avatar
Nicola Capacci committed
\subsection{} Argue whether the following \emph{isomorphisms} of categories hold:
Nicola Capacci's avatar
Nicola Capacci committed
\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

Nicola Capacci's avatar
Nicola Capacci committed
\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
Nicola Capacci's avatar
Nicola Capacci committed

Nicola Capacci's avatar
Nicola Capacci committed
\subsection{} Show that in $\mathbf{Pos}$, the category of posets, isomorphisms are \emph{not} bijective homomorphisms.\hfill \break
Nicola Capacci's avatar
Nicola Capacci committed

Marius Furter's avatar
Marius Furter committed
\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.

Nicola Capacci's avatar
Nicola Capacci committed
\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
Nicola Capacci's avatar
Nicola Capacci committed

Nicola Capacci's avatar
Nicola Capacci committed
\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
Nicola Capacci's avatar
Nicola Capacci committed

Nicola Capacci's avatar
Nicola Capacci committed
\subsection{} (\emph{Harder}) Prove the universal mapping property of free categories on graphs. \hfill \break
Nicola Capacci's avatar
Nicola Capacci committed

Nicola Capacci's avatar
Nicola Capacci committed


\end{document}