@@ -427,16 +427,16 @@ Diagonal weak equivalences are not the only interesting weak equivalences for bi

Recall now from \cite{cisinski2004localisateur} that the category of bisimplicial sets can be equipped with a model structure where the weak equivalences are the vertical (resp.\ horizontal) weak equivalences and the cofibrations are the monomorphisms. We refer respectively to this model structures as the \emph{vertical (resp.\ horizontal) model structure}. Since the identity functor on bisimplicial sets trivially preserves monomorphisms, it follows from Lemma \ref{bisimpliciallemma} that it induces a left Quillen functor from the vertical (resp.\ horizontal) model structure to the diagonal model structure.

In particular, morphisms of op-prederivators \eqref{fromverttodiag} and \eqref{fromhortodiag} preserve homotopy colimits. In practise, we will use the following corollary.

\end{paragr}

\begin{corollary}

\begin{corollary}\label{cor:bisimplicialsquare}

Let

\[

\begin{equation}\tag{$\ast$}\label{kiki}

\begin{tikzcd}

A \ar[r,"u"]\ar[d,"f"]& B \ar[d,"g"]\\

C \ar[r,"v"]& D

\end{tikzcd}

\]

be a square in the category of bisimplicial sets. If one of the following condition holds:

\begin{itemize}

\end{equation}

be a square in the category of bisimplicial sets satisfying either of the following conditions:

\begin{enumerate}[label=(\alph*)]

\item For every $n\geq0$, the square of simplicial sets

\[

\begin{tikzcd}

...

...

@@ -453,8 +453,8 @@ In particular, morphisms of op-prederivators \eqref{fromverttodiag} and \eqref{f

and where, similar to the nerve of categories, the face operators are induced by horizontal composition (of $2$-cells) and the degeneracy operators are induced by the units (for the horizontal composition).

Note that for $n=0$, the above formula reads $H(C)_0=\Ob(C)$.

The face operators$\partial_i : H(C)_{n}\to H(C)_{n-1}$ are induced by horizontal composition and the degeneracy operators $s_i : H(C)_{n}\to H(C)_{n+1}$ are induced by the units for the horizontal composition.

Post-composing $T(C)$ with the nerve functor $N : \Cat\to\Psh{\Delta}$, we obtain a functor

Post-composing $H(C)$ with the nerve functor $N : \Cat\to\Psh{\Delta}$, we obtain a functor

\[

NT(C) : \Delta^{\op}\to\Psh{\Delta}.

NH(C) : \Delta^{\op}\to\Psh{\Delta}.

\]

\end{paragr}

\begin{remark}

When $C$ is a $1$-category, the simplicial object $T(C)$ is nothing but the usual nerve of $C$ where, for each $n\geq0$, $T(C)_n$ is seen as a discrete category.

When $C$ is a $1$-category, the simplicial object $H(C)$ is nothing but the usual nerve of $C$ where, for each $n\geq0$, $H(C)_n$ is seen as a discrete category.

\end{remark}

\begin{definition}

The \emph{bisimplicial nerve} of a $2$-category $C$ is the bisimplicial set $\binerve(C)$ defined as

@@ -527,45 +528,106 @@ More intuitively, an element of $\binerve(C)_{n,m}$ consists of a ``pasting sche

\end{tikzcd}\right.}_{ n }.

\]

\todo{parenthèses moches dans le diagramme. Mettre dessins en petites dimensions des opérateurs de faces ?}

The bisimplicial nerve canonically defines a functor

\end{paragr}

In the definition of the bisimplicial nerve of a $2$-category we gave, we have priviledged one direction of the bisimplicial set over the other. We now give another definition of the bisimplicial nerve using the other direction.

\begin{paragr}

Let $C$ be a $2$-category. For every $k \geq1$, we define a $1$-category $V(C)_k$ in the following fashion:

\begin{itemize}

\item The objects of $V(C)_k$ are the objects of $C$.

\item A morphism $\alpha$ is a sequence

\[

\alpha=(\alpha_1,\alpha_2,\cdots,\alpha_k)

\]

of $2$-cells of $C$ that are vertically composable, i.e.\ such that for every $1\leq i \leq k-1$,

\[

\src(\alpha_i)=\trgt(\alpha_{i+1}).

\]

The source and target of alpha are given by

\[

\src(\alpha):=\src_0(\alpha_1)\text{ and }\trgt(\alpha):=\trgt_0(\alpha_1).

\]

(Note that we could have used any of the $\alpha_i$ instead of $\alpha_1$ since they all have the same $0$-source and $0$-target.)

For $k=0$, we define $V(C)_0$ to be the category obtained from $C$ by simply forgetting the $2$-cells (\todo{Faire le lien avec le tronqué bête.}). The correspondance $n \mapsto V(C)_n$ defines to a simplicial object in $\Cat$

\[

\binerve : 2\Cat\to\Psh{\Delta\times\Delta}.

V(C) : \Delta^{\op}\to\Cat,

\]

where the face operators are induced by the vertical composition and the degeneracy operators are induced by the units for the vertical composition.

\end{paragr}

\begin{lemma}\label{lemma:binverthom}

A $2$-functor $F : C \to D$ is a Thomason weak equivalence if and only if $\binerve(F)$ is a diagonal weak equivalence of bisimplicial sets.

\end{lemma}

\begin{lemma}\label{lemma:binervehorizontal}

Let $C$ be a $2$-category. For every $n \geq0$, we have

\[

N(V(C)_m)_n=(\binerve(C))_{n,m}.

\]

\end{lemma}

\begin{proof}

\todo{ref}

This is simply a reformulation of the formula given in Paragraph \ref{paragr:formulabisimplicialnerve}.

\end{proof}

\begin{paragr}

In particular, it follows from the previous lemma that the bisimplicial nerve induces a morphism of op-prederivators

The bisimplicial nerve canonically defines a functor

We shall see later that it is in fact an \emph{equivalence} of op-prederivators. Before that, we put here a useful sufficent criterion to detect Thomason weak equivalences, which follows from Lemma \ref{lemma:binverthom}.

which enables us to compare the homotopy theory of $2\Cat$ with the homotopy theory of bisimplicial sets.

\end{paragr}

\begin{lemma}\label{lemma:binervthom}

A $2$-functor $F : C \to D$ is a Thomason weak equivalence if and only if $\binerve(F)$ is a diagonal weak equivalence of bisimplicial sets.

\end{lemma}

\begin{proof}

It follows from what is shown in \cite[Section 2.1 and Theorem 2.7]{bullejos2003geometry} that there is weak equivalence of simplicial sets

\[

\delta^*(\binerve(C))\to N(C)

\]

which is natural in $C$. This implies what we wanted to show.

\end{proof}

From this lemma, we deduce two useful criteria to detect Thomason weak equivalences of $2$-categories.

\begin{corollary}\label{cor:criterionThomeqI}

Let $F : C \to D$ be a $2$-functor. If

\begin{itemize}

\item$F_0 : C_0\to D_0$ is an bijection,

Let $F : C \to D$ be a $2$-functor. If

\begin{enumerate}[label=\alph*)]

\item$F_0 : C_0\to D_0$ is an bijection,

\end{enumerate}

and

\begin{enumerate}[resume*]

\item for all objects $x,y$ of $C$, the functor

\[

C(x,y)\to D(F(x),F(y))

\]

induced by $F$ is a Thomason weak equivalence of $1$-categories,

\end{itemize}

\end{enumerate}

then $F$ is a Thomason weak equivalence of $2$-categories.

\end{corollary}

\begin{proof}

By definition, for every $2$-category $C$ and every $m \geq0$, we have

\[

(\binerve(C))_{\bullet,m}=T(C).

(\binerve(C))_{\bullet,m}=NH(C).

\]

The result follows then from Lemma \ref{bisimpliciallemma} and the fact that weak equivalences of simplicial sets are stable by coproduct and finite products.

The result follows then from Lemma \ref{bisimpliciallemma} and the fact that weak equivalences of simplicial sets are stable by coproducts and finite products.

\end{proof}

\begin{corollary}

Let $F : C \to D$ be a $2$-functor. If for every $k \geq0$,

\[V(F)_k : V(C)_k \to V(D)_k\] is a Thomason weak equivalence of $1$-categories, then $F$ is a Thomason weak equivalence of $2$-categories.

\end{corollary}

\begin{proof}

From Lemma \ref{lemma:binervehorizontal}, we now that for every $m \geq0$,

\[

\binerve(C)_{\bullet,m}=N(V(C)_m).

\]

The result follows them from Lemma \ref{bisimpliciallemma}.

\end{proof}

\begin{paragr}

So far, we have a triangle of functors

It also follows from Lemma \ref{lemma:binervthom} that the bisimplicial nerve induces a morphism of op-prederivators

@@ -585,66 +647,45 @@ In particular, it follows from the previous lemma that the bisimplicial nerve in

is commutative up to a canonical isomorphism.

\end{proposition}

\begin{proof}

\todo{Il faut aller faire de l'archéologie dans Cegarra pour avoir ce résultat.}

It is a consequence of the results contained in \cite[Section 2]{bullejos2003geometry}.

\end{proof}

\begin{corollary}

The morphism

\begin{paragr}

Since $\overline{\delta^*}$ and $\overline{N}$ are equivalences of op-prederivators (Proposition \ref{prop:diageqderivator} and Theorem \ref{thm:gagna} respectively), it follows from the previous proposition that the morphism

This follows from Proposition \ref{prop:streetvsbisimplicial}, the fact that $\overline{\delta^*}$ and $\overline{N}$ are equivalences of op-prederivators (Proposition \ref{prop:diageqderivator} and Theorem \ref{thm:gagna} respectively).

\end{proof}

Intuitively speaking, the two previous results tells us that the bisimplicial nerve and the Street nerve for $2$-categories are ``homotopically equivalent'' and define the same homotopy Theory on $2\Cat$.

We now review another point of view on the bisimplicial nerve, which will turn out to be very useful for the next section.

\begin{paragr}

Let $C$ be a $2$-category. For every $k \geq1$, we define a $1$-category $C^{(k)}$ in the following fashion:

\begin{itemize}

\item The objects of $C^{(k)}$ are the objects of $C$.

\item A morphism $\alpha$ is a sequence

\[

\alpha=(\alpha_1,\alpha_2,\cdots,\alpha_k)

\]

of $2$-cells of $C$ that are vertically composable, i.e.\ such that for every $1\leq i \leq k-1$,

\[

\src(\alpha_i)=\trgt(\alpha_{i+1}).

\]

The source and target of alpha are given by

\[

\src(\alpha):=\src_0(\alpha_1)\text{ and }\trgt(\alpha):=\trgt_0(\alpha_1).

\]

(Note that we could have used any of the $\alpha_i$ instead of $\alpha_1$ since they all have the same $0$-source and $0$-target.)

For $k=0$, we define $C^{(0)}$ to be the category obtained from $C$ by simply forgetting the $2$-cells (\todo{Faire le lien avec le tronqué bête.}).

From Proposition \ref{prop:streetvsbisimplicial}, we also deduce the proposition below which contains two useful critera to detect Thomason homotopy cocartesian square of $2\Cat$.

\end{paragr}

\begin{lemma}\label{lemma:binervehorizontal}

Let $C$ be a $2$-category. For every $n \geq0$, we have

\] is a Thomason homotopy cocartesian square of $\Cat$.

\end{enumerate}

Then, square \eqref{coucou} is a Thomason homotopy cocartesian in $2\Cat$.

\end{proposition}

\begin{proof}

This is simply a reformulation of the formula given in Paragraph \ref{paragr:formulabisimplicialnerve}.

This is an immediate consequence of Proposition \ref{prop:streetvsbisimplicial} and Corollary \ref{cor:bisimplicialsquare}.

\end{proof}

\begin{remark}

In fact, the previous lemma can be understood more conceptually by remarking that the correspondance $n \mapsto C^{(n)}$ can cannonically be extended to a functor $C^{(-)}: \Delta^{\op}\to\Cat$ and that the bisimplicial nerve of $C$ is simply obtained by uncurryfying the functor

\[

NC^{(-)} : \Delta^{\op}\to\Psh{\Delta}.

\]

\end{remark}

A simple consequence of Lemma \ref{lemma:binervehorizontal} is the following corollary which is to be compared with Corollary \ref{cor:criterionThomeqI}