Commit 1f83b160 authored by Leonard Guetta's avatar Leonard Guetta
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parent 19ed555f
......@@ -317,11 +317,21 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp
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
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}.
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
......@@ -448,3 +458,30 @@ In particular, morphisms of op-prederivators \eqref{fromverttodiag} and \eqref{f
\todo{Écrire une preuve ?}
\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}.
\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
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
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 $C$.
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:
\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.
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