Commit d3221095 authored by Leonard Guetta's avatar Leonard Guetta
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on va boire une biere

parent 959fff37
......@@ -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 $n\geq 0$, the square of simplicial sets
\[
\begin{tikzcd}
......@@ -453,8 +453,8 @@ In particular, morphisms of op-prederivators \eqref{fromverttodiag} and \eqref{f
\end{tikzcd}
\]
is homotopy cocartesian.
\end{itemize}
Then the square
\end{enumerate}
Then, the square
\[
\begin{tikzcd}
\delta^*(A) \ar[r,"\delta^*(u)"]\ar[d,"\delta^*(f)"] & \delta^*(B) \ar[d,"\delta^*(g)"] \\
......@@ -486,35 +486,36 @@ We shall now describe a ``nerve'' for $2$-categories with value in bisimplicial
\end{notation}
\begin{paragr}
Each $2$-category $C$ defines a simplicial object in $\Cat$,
\[T(C): \Delta^{\op} \to \Cat,\]
\[H(C): \Delta^{\op} \to \Cat,\]
where, for each $n \geq 0$,
\[
T(C)_n:= \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}C(x_0,x_1) \times \cdots \times C(x_{n-1},x_n),
\]
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).
H(C)_n:= \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}C(x_0,x_1) \times \cdots \times C(x_{n-1},x_n).
\]
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 $n\geq 0$, $T(C)_n$ 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 $n\geq 0$, $H(C)_n$ 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
\[
\binerve(C)_{n,m}:=N(T(C)_n)_m,
\binerve(C)_{n,m}:=N(H(C)_n)_m,
\]
for all $n,m \geq 0$.
\end{definition}
\begin{paragr}\label{paragr:formulabisimplicialnerve}
In other words, the bisimplicial nerve of $C$ is obtained by ``uncurryfying'' the functor $NT(C) : \Delta^{op} \to \Psh{\Delta}$.
In other words, the bisimplicial nerve of $C$ is obtained by ``uncurryfying'' the functor $NH(C) : \Delta^{op} \to \Psh{\Delta}$.
Since the nerve $N$ commutes with products and sums, we obtain the formula
\[
\begin{equation}\label{fomulabinerve}
\binerve(C)_{n,m} = \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}N(C(x_0,x_1))_m \times \cdots \times N(C(x_{n-1},x_n))_m.
\]
\end{equation}
More intuitively, an element of $\binerve(C)_{n,m}$ consists of a ``pasting scheme'' in $C$ that look like
\[
m \underbrace{\left\{\begin{tikzcd}[column sep=huge,ampersand replacement=\&] \bullet \ar[r,"\vdots"]\ar[r,bend left=90,looseness=1.4,""{name=A,below}] \ar[r,bend left=35,""{name=B,above}] \ar[r,bend right=35,"\vdots",""{name=G,below}]\ar[r,bend right=90,looseness=1.4,""{name=H,above}] \& \bullet\ar[r,"\vdots"]\ar[r,bend left=90,looseness=1.4,""{name=C,below}] \ar[r,bend left=35,""{name=D,above}] \ar[r,bend right=35,"\vdots",""{name=I,below}]\ar[r,bend right=90,looseness=1.4,""{name=J,above}] \&\bullet\ar[r,phantom,description,"\cdots"]\&\bullet\ar[r,"\vdots"]\ar[r,bend left=90,looseness=1.4,""{name=E,below}] \ar[r,bend left=35,""{name=F,above}] \ar[r,bend right=35,"\vdots",""{name=K,below}]\ar[r,bend right=90,looseness=1.4,""{name=L,above}] \&\bullet
......@@ -527,45 +528,106 @@ More intuitively, an element of $\binerve(C)_{n,m}$ consists of a ``pasting sche
\end{tikzcd}\right.}_{ n }.
\]
\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 $k \geq 1$, 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.)
\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 $V(C)_0$ 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 $n \geq 0$, 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
\[
\overline{\binerve} : \Ho(2\Cat^{\Th}) \to \Ho(\Psh{\Delta\times\Delta}).
\binerve : 2\Cat \to \Psh{\Delta\times\Delta}
\]
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}.
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 $m \geq 0$, 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 $k \geq 0$,
\[V(F)_k : V(C)_k \to V(D)_k\] 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 $m \geq 0$,
\[
\binerve(C)_{\bullet,m}=N(V(C)_m).
\]
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
\[
\overline{\binerve} : \Ho(2\Cat^{\Th}) \to \Ho(\Psh{\Delta\times\Delta}).
\]
This morphism is in fact an \emph{equivalence} of op-prederivators as we shall soon see. In order to do that, consider the triangle of functors
\[
\begin{tikzcd}
2\Cat \ar[rr,"\binerve"] \ar[dr,"N"] & & \Psh{\Delta\times\Delta} \ar[ld,"\delta^*"] \\
......@@ -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
\[
\overline{\binerve} : \Ho(2\Cat^{\Th}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
\]
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.}).
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 $n \geq 0$, we have
\[
N(C^{(n)})=(\binerve(C))_{\bullet,n}.
\begin{proposition}
Let
\begin{equation}\tag{$\ast$}\label{coucou}\begin{tikzcd}
A \ar[r,"u"]\ar[d,"f"] & B \ar[d,"g"] \\
C \ar[r,"v"] & D
\end{tikzcd}\end{equation}
be a square in $2\Cat$ satisfying either of the following conditions:
\begin{enumerate}[label=(\alph*)]
\item for every $n\geq 0$, the square
\[
\begin{tikzcd}
V_{n}(A) \ar[r,"V_{n}(u)"]\ar[d,"V_{n}(f)"'] & V_n(B) \ar[d,"V_n(g)"] \\
V_n(C) \ar[r,"V_n(v)"] & V_n(D)
\end{tikzcd}
\]
\end{lemma}
is a Thomason homotopy cocartesian square of $\Cat$,
\item for every $n\geq 0$, the square
\[
\begin{tikzcd}
H_{n}(A) \ar[r,"H_{n}(u)"]\ar[d,"H_{n}(f)"'] & H_n(B) \ar[d,"H_n(g)"] \\
H_n(C) \ar[r,"H_n(v)"] & H_n(D)
\end{tikzcd}
\] 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}
\begin{corollary}\label{cor:criterionThomeqII}
Let $F : C \to D$ be a $2$-functor.
\end{corollary}
......@@ -68,6 +68,16 @@ year={2020}
year={1990},
publisher={Elsevier}
}
@article{bullejos2003geometry,
title={On the geometry of 2-categories and their classifying spaces},
author={Bullejos, Manuel and Cegarra, Antonio M},
journal={K-theory},
volume={29},
number={3},
pages={211--229},
year={2003},
publisher={Springer Netherlands}
}
@article{cisinski2003images,
title={Images directes cohomologiques dans les cat{\'e}gories de modeles},
author={Cisinski, Denis-Charles},
......
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