Commit 1f83b160 by Leonard Guetta

### security commit

parent 19ed555f
 ... ... @@ -317,11 +317,21 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp \begin{paragr} A \emph{bisimplicial set} is a presheaf over the category $\Delta \times \Delta$, $X : (\Delta \times \Delta)^{op} \to \Set. X : \Delta^{op} \times \Delta^{op} \to \Set.$ In a similar fashion as for simplicial sets, we use the notation $X_{n,m} := X([n],[m])$ for the image by $X$ of the object $([n],[m])$ of $\Delta\times \Delta$. The category of bisimplicial sets is denoted by $\Psh{\Delta\times\Delta}$. In a similar fashion as for simplicial sets (\ref{paragr:simpset}), for $n,m \geq 0$, we use the notation \begin{align*} X_{n,m} &:= X([n],[m]) \\ \partial_i^h &:=X(\delta^i,\mathrm{id}_{[m]}) : X_{n+1,m} \to X_{n,m}\\ \partial_j^v &:=X(\mathrm{id}_{[n]},\delta_j) : X_{n,m+1} \to X_{n,m}\\ s_i^h &:=X(\sigma^i,\mathrm{id}_{[m]})): X_{n+1,m} \to X_{n,m}\\ s_j^v&:=X(\mathrm{id}_n,\sigma^j) : X_{n,m+1} \to X_{n,m}. \end{align*} Note that for every $n\geq 0$, we have simplicial sets $X_{\bullet,n} :$ The category of bisimplicial sets is denoted by $\Psh{\Delta\times\Delta}$. \iffalse Moreover for every $n \geq 0$, if we fix the first variable to $n$, we obtain a simplical set \begin{align*} ... ... @@ -448,3 +458,30 @@ In particular, morphisms of op-prederivators \eqref{fromverttodiag} and \eqref{f \begin{proof} \todo{Écrire une preuve ?} \end{proof} \section{Bisimplicial nerve for 2-categories} We shall now describe a nerve'' for $2$-categories with value in bisimplicial sets and recall a few results that shows that this nerve is, in some sense, equivalent to the Street nerve from section \todo{ref}. \begin{notation} \begin{itemize} \item[-] Once again, we will use the notation $N : \Cat \to \Psh{\Delta}$ instead of $N_1$ for the usual nerve of categories. Moreover, using the usual convention for the set of $k$-simplices of a simplicial set, if $C$ is a (small) category, then $N(C)_k$ is the set of $k$-simplices of the nerve of $C$. \item[-] Similarly, we will use the notation $N : 2\Cat \to \Psh{\Delta}$ instead of $N_2$ for the nerve of $2$-categories \todo{ref interne}. This makes sense since the nerve for categories is the restriction of the nerve for $2$-categories. \item[-] For $2$-categories, we refer to the $\comp_0$-composition of $2$-cells as the \emph{horizontal composition} and the $\comp_1$-composition of $2$-cells as the \emph{vertical composition}. \todo{Uniformiser} \item[-] For a $2$-category $C$ and $x$ and $y$ objects of $C$, we denote by $C(x,y)$ the category whose objects are the $1$-cells of $C$ with $x$ as source and $y$ as target, and whose arrows are the $2$-cells of $C$ with $x$ as $0$-source and $y$ as $0$-target. Composition is induced by vertical composition in $C$. \end{itemize} \end{notation} \begin{paragr} Let $C$ be a $2$-category. The \emph{bisimplicial nerve} of $C$ is the bisimplicial set $\nu(C)$ \todo{notation ?} defined in the following fashion: \begin{itemize} \item[-] For $n,m \geq 0$, we have $\nu(C)_{n,m} = \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times n+1}}N(C(x_0,x_1))_m\times\cdots\times N(C(x_{n-1},x_n))_m.$ \end{itemize} \end{paragr}
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