Commit 959fff37 authored by Leonard Guetta's avatar Leonard Guetta
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I need to wrap up Corollary 6.3.15. Moreover, I thinking of moving up...

I need to wrap up Corollary 6.3.15. Moreover, I thinking of moving up 6.3.12-13-14-15 before Lemma 6.3.6
parent 241c7c49
......@@ -378,7 +378,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp
\]
Recall from \cite[Proposition 1.2]{moerdijk1989bisimplicial} that the category of bisimplicial sets can be equipped with a model structure whose weak equivalences are diagonal weak equivalences and whose cofibrations are monomorphisms. We shall refer to this model structure as the \emph{diagonal model structure}. Since $\delta^*$ preserves monomorphisms (and diagonal weak equivalences trivially), it is left Quillen. In fact, the following proposition tells us more. We denote by $\delta_*$ the right adjoint of $\delta^*$.
\end{paragr}
\begin{proposition}
\begin{proposition}\label{prop:diageqderivator}
Consider that $\Psh{\Delta\times\Delta}$ is equipped with the diagonal model structure. Then, the adjunction
\[
\begin{tikzcd}
......@@ -392,7 +392,7 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp
\end{proof}
In particular, the morphism of op-prederivators
\[
\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}) \to \Ho(\Psh{\Delta})
\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}}) \to \Ho(\Psh{\Delta})
\]
is actually an equivalence of op-prederivators.
......@@ -508,22 +508,143 @@ The \emph{bisimplicial nerve} of a $2$-category $C$ is the bisimplicial set $\bi
\]
for all $n,m \geq 0$.
\end{definition}
%(Or, more conceptually, by ``uncurryfying'' $NT(C)$).
\begin{paragr}
\begin{paragr}\label{paragr:formulabisimplicialnerve}
In other words, the bisimplicial nerve of $C$ is obtained by ``uncurryfying'' the functor $NT(C) : \Delta^{op} \to \Psh{\Delta}$.
Since the nerve $N$ commutes with products and sums, we obtain the formula
\[
\binerve(C)_{n,m} = \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}N(C(x_0,x_1))_m \times \cdots \times N(C(x_{n-1},x_n))_m.
\]
More intuitively, an element of $\binerve(C)_{n,m}$ consists of a ``pasting scheme'' in $C$ that look like
\[
m \text{ times }\underbrace{\left\{\begin{tikzcd}[column sep=huge,ampersand replacement=\&] \bullet \ar[r,"\vdots"]\ar[r,bend left=90,looseness=1.4,""{name=A,below}] \ar[r,bend left=35,""{name=B,above}] \ar[r,bend right=35,"\vdots",""{name=G,below}]\ar[r,bend right=90,looseness=1.4,""{name=H,above}] \& \bullet\ar[r,"\vdots"]\ar[r,bend left=90,looseness=1.4,""{name=C,below}] \ar[r,bend left=35,""{name=D,above}] \ar[r,bend right=35,"\vdots",""{name=I,below}]\ar[r,bend right=90,looseness=1.4,""{name=J,above}] \&\bullet\ar[r,phantom,description,"\cdots"]\&\bullet\ar[r,"\vdots"]\ar[r,bend left=90,looseness=1.4,""{name=E,below}] \ar[r,bend left=35,""{name=F,above}] \ar[r,bend right=35,"\vdots",""{name=K,below}]\ar[r,bend right=90,looseness=1.4,""{name=L,above}] \&\bullet
m \underbrace{\left\{\begin{tikzcd}[column sep=huge,ampersand replacement=\&] \bullet \ar[r,"\vdots"]\ar[r,bend left=90,looseness=1.4,""{name=A,below}] \ar[r,bend left=35,""{name=B,above}] \ar[r,bend right=35,"\vdots",""{name=G,below}]\ar[r,bend right=90,looseness=1.4,""{name=H,above}] \& \bullet\ar[r,"\vdots"]\ar[r,bend left=90,looseness=1.4,""{name=C,below}] \ar[r,bend left=35,""{name=D,above}] \ar[r,bend right=35,"\vdots",""{name=I,below}]\ar[r,bend right=90,looseness=1.4,""{name=J,above}] \&\bullet\ar[r,phantom,description,"\cdots"]\&\bullet\ar[r,"\vdots"]\ar[r,bend left=90,looseness=1.4,""{name=E,below}] \ar[r,bend left=35,""{name=F,above}] \ar[r,bend right=35,"\vdots",""{name=K,below}]\ar[r,bend right=90,looseness=1.4,""{name=L,above}] \&\bullet
\ar[from=A,to=B,Rightarrow]
\ar[from=C,to=D,Rightarrow]
\ar[from=E,to=F,Rightarrow]
\ar[from=G,to=H,Rightarrow]
\ar[from=I,to=J,Rightarrow]
\ar[from=K,to=L,Rightarrow]
\end{tikzcd}\right.}_{ n \text{ times }}
\end{tikzcd}\right.}_{ n }.
\]
\todo{parenthèses moches dans le diagramme}
\todo{parenthèses moches dans le diagramme. Mettre dessins en petites dimensions des opérateurs de faces ?}
The bisimplicial nerve canonically defines a functor
\[
\binerve : 2\Cat \to \Psh{\Delta\times\Delta}.
\]
\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{proof}
\todo{ref}
\end{proof}
\begin{paragr}
In particular, it follows from the previous lemma that the bisimplicial nerve induces a morphism of op-prederivators
\[
\overline{\binerve} : \Ho(2\Cat^{\Th}) \to \Ho(\Psh{\Delta\times\Delta}).
\]
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}.
\end{paragr}
\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,
\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}
then $F$ is a Thomason weak equivalence of $2$-categories.
\end{corollary}
\begin{proof}
By definition, for every $2$-category $C$ and every $m \geq 0$, we have
\[
(\binerve(C))_{\bullet,m} = T(C).
\]
The result follows then from Lemma \ref{bisimpliciallemma} and the fact that weak equivalences of simplicial sets are stable by coproduct and finite products.
\end{proof}
\begin{paragr}
So far, we have a triangle of functors
\[
\begin{tikzcd}
2\Cat \ar[rr,"\binerve"] \ar[dr,"N"] & & \Psh{\Delta\times\Delta} \ar[ld,"\delta^*"] \\
&\Psh{\Delta}
\end{tikzcd}.
\]
This triangle is \emph{not} commutative. However, the next proposition tells us that it becomes commutative (up to an isomorphism) after localization.
\end{paragr}
\begin{proposition}\label{prop:streetvsbisimplicial}
The triangle of morphisms of op-prederivators
\[
\begin{tikzcd}
\Ho(2\Cat^{\Th}) \ar[rr,"\overline{\binerve}"] \ar[dr,"\overline{N}"] & & \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}) \ar[ld,"\overline{\delta^*}"] \\
&\Ho(\Psh{\Delta})
\end{tikzcd}
\]
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.}
\end{proof}
\begin{corollary}
The morphism
\[
\overline{\binerve} : \Ho(2\Cat^{\Th}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
\]
is an \emph{equivalence} of op-prederivators.
\end{corollary}
\begin{proof}
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 $k \geq 1$, 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.)
\item Composition is given by
\[
(\alpha_1,\alpha_2,\cdots,\alpha_k)\circ(\beta_1,\beta_2,\cdots,\beta_k):=(\alpha_1\comp_0\beta_1,\alpha_2\comp_0\beta_2,\cdots,\alpha_k\comp_0\beta_k)
\]
and the unit on an object $x$ is the sequence
\[
(1^2_x,\cdots, 1^2_x).
\]
\end{itemize}
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.}).
\end{paragr}
\begin{lemma}\label{lemma:binervehorizontal}
Let $C$ be a $2$-category. For every $n \geq 0$, we have
\[
N(C^{(n)})=(\binerve(C))_{\bullet,n}.
\]
\end{lemma}
\begin{proof}
This is simply a reformulation of the formula given in Paragraph \ref{paragr:formulabisimplicialnerve}.
\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}
\begin{corollary}\label{cor:criterionThomeqII}
Let $F : C \to D$ be a $2$-functor.
\end{corollary}
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