text.tex

% easychair.tex,v 3.5 2017/03/15

\documentclass{easychair}
%\documentclass[EPiC]{easychair}
%\documentclass[EPiCempty]{easychair}
%\documentclass[debug]{easychair}
%\documentclass[verbose]{easychair}
%\documentclass[notimes]{easychair}
%\documentclass[withtimes]{easychair}
%\documentclass[a4paper]{easychair}
%\documentclass[letterpaper]{easychair}

 
\input{common/preamble}
\input{common/math}
\input{common/tikzlibrary}
\input{common/lstlistinglibrary}
\usepackage{tikzit}
\input{common/drawing.tikzstyles}

% The categories we are dealing with
\newcommand{\catSET}{\mathbb{S}\mathbbm{et}}
\newcommand{\catINCL}{\mathbb{I}\mathbbm{ncl}}
\newcommand{\catCAT}{\mathbb{C}\mathbbm{at}}
\newcommand{\catPAR}{\mathbb{P}\mathbbm{ar}}
\newcommand{\catREL}{\mathbb{R}\mathbbm{el}}
\newcommand{\catSEN}{\mathbb{S}\mathbbm{en}}
\newcommand{\catMULT}{\mathbb{M}\mathbbm{ult}}
\newcommand{\catMON}{\mathbb{M}\mathbbm{on}}
\newcommand{\catGRAPH}{\mathbb{G}\mathbbm{raph}}
\newcommand{\catSIMPLEGRAPH}{\mathbb{SG}\mathbbm{raph}}
\newcommand{\catHYPERGRAPH}{\mathbb{H}\mathbbm{yp}}
\newcommand{\catALGOMEGA}{\mathbb{A}\mathbbm{lg}(\Omega)}
\newcommand{\catALGSIGMA}{\mathbb{A}\mathbbm{lg}(\Sigma)}
\newcommand{\catSYSSIGMA}{\mathbb{S}\mathbbm{ys}(\Sigma)}
\newcommand{\catExplanation}[4]{
\begin{description}[leftmargin=\parindent,labelindent=\parindent,itemsep=2pt,parsep=2pt]
\item[Objects] {#1}
\item[Morphisms] {#2}
\item[Composition] {#3}
\item[Identities] {#4}
\end{description}
}
%% Front Matter
%%
% Regular title as in the article class.
%
\title{Category Theory Compendium}


% Authors are joined by \and. Their affiliations are given by \inst, which indexes
% into the list defined using \institute
%
\author{
	Patrick Stünkel\inst{1}%\thanks{Designed and implemented the class style}
%	\and
%	Somebody else\inst{2}%\thanks{Did numerous tests and provided a lot of suggestions}
}

% Institutes for affiliations are also joined by \and,
\institute{
	Western Norway University of Applied Sciences,
	Bergen, Norway\\
	\email{past@hvl.no}
%	\and
%	Other University,
%	Other, Country\\
%	\email{other@university.edu}\\
}

%  \authorrunning{} has to be set for the shorter version of the authors' names;
% otherwise a warning will be rendered in the running heads. When processed by
% EasyChair, this command is mandatory: a document without \authorrunning
% will be rejected by EasyChair

\authorrunning{P. Stünkel}

% \titlerunning{} has to be set to either the main title or its shorter
% version for the running heads. When processed by
% EasyChair, this command is mandatory: a document without \titlerunning
% will be rejected by EasyChair
\titlerunning{Category Theory Compendium}

\begin{document}

\maketitle

\begin{abstract}
All the proofs I know
\end{abstract}


% The table of contents below is added for your convenience. Please do not use
% the table of contents if you are preparing your paper for publication in the
% EPiC Series or Kalpa Publications series


%\section{To mention}
%
%Processing in EasyChair - number of pages.
%
%Examples of how EasyChair processes papers. Caveats (replacement of EC
%class, errors).

%------------------------------------------------------------------------------
\section{Categories}
\label{sec:introduction}

\input{sections/foundations}

\section{Morphisms}
\label{sec:morphisms}

\input{sections/morphisms}


\section{Factorization Systems}
\label{sec:factorizationSystems}
\input{sections/factorization}

\section{Functors}
\label{sec:functors}

\input{sections/functors}






\section{Universal Constructions}
\label{sec:universalConstructions}

\input{sections/universal}


\section{Adjunctions}
\label{sec:adjunctions}

\input{sections/adjunctions}

\section{Algebras and Monads}
\label{sec:algebras}

\input{sections/algebras}

\section{Partial Arrows and Span Categories}
\label{sec:spans}

\input{sections/spans}


\section{Constructions}
\label{sec:constructions}

\input{sections/constructions}

\section{Toposes}
\label{sec:topoi}

\input{sections/topoi}

\section{Fibered Category Theory}
\label{sec:fibrations}

% TODO Def SLICE and CO-SLICE and ARROW category

\mkDefinition{Fibre Category}{def:fibre}{
Let $\cat{E}$ and $\cat{B}$ be two categories and $\arrow{\cat{E}}{P}{\cat{B}}$ be a functor.
For any object $I \in \catObjects{\cat{B}}$ we define the fibre $\cat{E}_I = \inverse{P}(I)$ over $I$ to be a category, which has 
\catExplanation{All objects $X \in \catObjects{\cat{E}}$ for which $P(X) = I$.}
{All morphisms $\arrow{X}{f}{Y}$ for which $P(f) = \identity{I}$.}
{Retained from composition in $\cat{E}$.}
{Retained from identities in $\cat{E}$.}
}

\mkDefinition{Cartesian and Opcartesian Morphisms}{def:cartesian}{
Let $\cat{E}$ and $\cat{B}$ be two categories and $\arrow{\cat{E}}{P}{\cat{B}}$ be a functor.
\begin{enumerate}
\item An arrow $\arrow{X}{f}{Y}$ with $P(X) = \arrow{I}{u}{J}$ is called \emph{cartesian} over $u$ iff. for any arrow $\arrow{Z}{g}{Y}$ where there exist $\arrow{P(Z)}{w}{I}$ with $\compose{w}{u} = P(g)$ there exists a unique $\arrow{Z}{h}{X}$ such that $\compose{h}{f} = g$ and $P(h) = w$.
\begin{equation}
\label{eq:cartesian}
\tag{Cartesian Arrow}
\xymatrix @C= 5em {
\cat{E} \ar[ddd]_{P}  & \textcolor{blue}{Z} \ar@[blue][dr]_{\textcolor{blue}{g}} \ar@[red]@{-->}[r]^{\textcolor{red}{h!}} & X \ar[d]^f \\
  & & Y \\
  &  \textcolor{blue}{P(Z)} \ar@[blue][r]^{\textcolor{blue}{w}\textcolor{red}{= P(h)}} \ar@[blue][dr]_{\textcolor{blue}{P(g)}} & I  \ar[d]^u \\
\cat{B} & & J
}
\end{equation}
\item  An arrow $\arrow{X}{f}{Y}$ with $P(X) = \arrow{I}{u}{J}$ is called \emph{cartesian} over $u$ iff. for any arrow $\arrow{Z}{g}{Y}$ iff. for any arrow $\arrow{X}{g}{Z}$ where there exists $\arrow{J}{w}{P(Z)}$ with $\compose{u}{w} = P(g)$ there exists a unique $\arrow{X}{h}{Z}$ such that $\compose{f}{g} = h$ and $P(h) = w$.
\begin{equation}
\label{eq:opcartesian}
\tag{Opcartesian Arrow}
\xymatrix @C= 5em {
	\cat{E} \ar[ddd]_{P}  & X \ar[d]^f \ar@[blue][r]^{\textcolor{blue}{g}} &  \textcolor{blue}{Z}  \\ 
	& Y \ar@[red]@{-->}[ur]_{\textcolor{red}{h!}} &  \\
	&  I  \ar[d]_u  \ar@[blue][r]^{\textcolor{blue}{P(g)}} & \textcolor{blue}{P(Z)}    \\
	\cat{B} & J \ar@[blue][ur]_{\textcolor{blue}{w}\textcolor{red}{= P(h)}} & 
}
\end{equation}
\end{enumerate}
A (op)cartesian arrow $f$ is called \emph{weakly (op-)cartesian} if the above conditions only hold for $g$'s being identities $\identity{X}$.
}

\mkProposition{Cartesian Liftings are unique up to isomorphisms}{prop:cartesianLiftingsAbstract}{
Let $\cat{E}$ and $\cat{B}$ be two categories and $\arrow{\cat{E}}{P}{\cat{B}}$ be a functor.
When $\arrow{X}{f}{Y}$ and $\arrow{X'}{f'}{Y'}$ are cartesian over the same map $\arrow{I}{u}{J} \in \catArrows{\cat{B}}$ then there exists a unique isomorphism $\arrow{X}{\isomorphic}{X'}$.
}

\mkDefinition{Fibrations and Opfibrations}{def:fibrations}{
A functor $\arrow{\cat{E}}{P}{\cat{B}}$ is called a \emph{fibration} iff. for every $Y \in \catObjects{\cat{E}}$ and $\arrow{I}{u}{P(Y)} \in \catArrows{\cat{B}}$ there exists a cartesian arrow $\arrow{X}{f}{Y}$ such that $P(f) = u$.\par
Dually, $P$ is called an \emph{opfibration} iff. for every $X \in \catObjects{E}$ and $\arrow{P(X)}{u}{J} \in \catArrows{\cat{B}}$ there exists a an opcartesian arrow $\arrow{X}{f}{Y}$ such that $P(f) = u$.
}

\mkProposition{Codomain Fibrations}{prop:codomainFibration}{
Let $\cat{C}$ be a category and $\arrow{\catArrows{\cat{C}}}{cod}{\cat{C}}$ be the codomain functor.
\begin{enumerate}
\item The fibre category $\cat{C}_I$ over $I \in \catObjects{\cat{C}}$ is equal to the slice category $\sliceCat{\cat{C}}{I}$.
\item Cartesian morphisms in $\catArrows{\cat{C}}$ are exactly the pullback squares in $\cat{C}$.
\item The functor $cod$ is fibration if and only if $\cat{C}$ has pullbacks. 
\end{enumerate}
}

\mkTheorem{Grothendieck Construction}{thm:grothendieck}{
Let $\cat{E}$ and $\cat{B}$ be two categories and $\arrow{\cat{E}}{P}{\cat{B}}$ be a functor.
For any object $B \in \catObjects{\cat{B}}$ there is an equivalence of 2-categories
\begin{equation}
	\mathbf{Fib}(B) \cong \left[\dual{B}, \catCAT \right]
\end{equation}
The functor going from right to left in this equivalence if called. the \emph{Grothendieck construction} (The reconstruction of a fibration from a pseudofunctor) and is denoted
\begin{equation}
 \int : \left[\dual{B}, \catCAT \right] \to \sliceCat{\catCAT}{B}
\end{equation}
}



\section{Adhesive Categories}
\label{sec:adhesive}

\input{sections/adhesive}

\section{Internal Category Theory}
\label{sec:internalCT}

% TODO working internal to a category with pullbacks

\section{Regular Categories and Allegories}
\label{sec:allegories}

\section{Higher Order Category Theory}
\label{sec:HO}

% TODO notion of enriched


\label{sect:bib}
\bibliographystyle{plain}
%\bibliographystyle{alpha}
%\bibliographystyle{unsrt}
%\bibliographystyle{abbrv}
\bibliography{../../git/literature/library}

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


%------------------------------------------------------------------------------
% Index
%\printindex

%------------------------------------------------------------------------------
\end{document}