Commit b5de6425 authored by Leonard Guetta's avatar Leonard Guetta
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First commit. The writing of my dissertation has officially begun.

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\documentclass[12pt,a4paper]{report}
\usepackage{mystyle}
\title{Homology of $\omega$-categories}
\author{Léonard Guetta}
\begin{document}
\maketitle
\include{omegacat}
\end{document}
\ProvidesPackage{mystyle}
% Layout
\usepackage[utf8]{inputenc}
% Maths packages
\usepackage{amsmath,amssymb,amsthm}
\usepackage{tikz-cd}
% Theorems
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{scholia}[theorem]{Scholia}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{fact}[theorem]{Fact}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem*{convention}{Convention}
\newtheorem*{notation}{Notation}
\newtheorem{paragr}[theorem]{}
\newtheorem{example}[theorem]{Example}
\theoremstyle{plain}
\newtheorem*{theorem*}{Theorem} % Unnumbered theorem
\newtheorem{theoremintro}{Theorem} % Theorem environment for the introduction. Numbering not following the sections
\theoremstyle{definition}
\newtheorem*{definition*}{Definition} %Unnumbered definition
%%%%%% MACROS %%%%%%%
% oo-categories
\newcommand{\oo}{\omega}
\newcommand{\Cat}{\mathbf{Cat}}
\newcommand{\ooCat}{\mathbf{\oo Cat}}
\newcommand{\nCat}{\mathbf{n Cat}}
% Source and target
\newcommand{\s}[1]{s^{(#1)}}
\newcommand{\sk}{\s{k}}
\newcommand{\t}[1]{t^{(#1)}}
\newcommand{\tk}{\t{k}}
% ad hoc
\newcommand{\nbar}{\mathbb{N}\cup\{ \oo \}}
\chapter{$\oo$-Category theory}
\section{$\oo$-graphs, $\oo$-magmas and $\oo$-categories}
\begin{paragr}
An \emph{$\oo$-graph} $C$ consists of a sequence $(C_k)_{k \in \mathbb{N}}$ of sets together with maps
\[ \begin{tikzcd}
C_{k-1} &\ar[l,"s",shift left] \ar[l,"t"',shift right] C_{k}
\end{tikzcd}
\]
for every $k > 0$, satisfying the \emph{globular identities}:
\begin{equation*}
\left\{
\begin{aligned}
s \circ s &= s \circ t, \\
t \circ t &= t \circ s.
\end{aligned}
\right.
\end{equation*}
Elements of $C_k$ are called \emph{$k$-cells}.
For $x$ a $k$-cell with $k>0$, $s(x)$ is the \emph{source} of $x$ and $t(x)$ is the \emph{target} of $x$.
%Similarly, for $n \in \mathbb{N}$, an \emph{$n$-graph} $C$ is a finite sequence $(C_k)_{0 \leq k \leq n}$ of sets together with maps
% \[ \begin{tikzcd}
% C_{k-1} &\ar[l,"s",shift left] \ar[l,"t"',shift right] C_{k}
% \end{tikzcd}
% \]
% for every $0 <k < n$, satisfying the same globular identities.
\end{paragr}
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