Commit fbd37cb9 authored by Leonard Guetta's avatar Leonard Guetta
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First complete version of the thesis !

parent d70ea50d
\chapter{Homotopy and homology type of free $2$-categories}
\section{Preliminaries: the case of free $1$-categories}
\section{Preliminaries: the case of free $1$-categories}\label{section:prelimfreecat}
In this section, we review some homotopical results on free
($1$-)categories that will be of great help in the sequel.
\begin{paragr}
......@@ -737,7 +737,7 @@ In practice, we will use the following corollary.
horizontal weak equivalences. The result follows then from Proposition
\ref{prop:bisimplicialcocontinuous}.
\end{proof}
\section{Bisimplicial nerve for 2-categories}
\section{Bisimplicial nerve for 2-categories}\label{section:bisimplicialnerve}
We shall now describe a ``nerve'' for $2$-categories with values in bisimplicial
sets and recall a few results that shows that this nerve is, in some sense,
equivalent to the nerve defined in \ref{paragr:nerve}.
......
......@@ -822,7 +822,7 @@ The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences
\end{align*}
\end{paragr}
\begin{proposition}(Folk Theorem $A$) Let
\begin{proposition}\label{prop:folkthmA}(Folk Theorem $A$) Let
\[
\begin{tikzcd}[column sep=small]
A \ar[rr,"u"] \ar[dr,"v"'] & &B \ar[dl,"w"] \\
......
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