Note that for every $n\geq0$, we have simplicial sets

\[

X_{\bullet,n} :

\]

The category of bisimplicial sets is denoted by $\Psh{\Delta\times\Delta}$.

\iffalse Moreover for every $n \geq0$, 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: