### 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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