Commit 19ed555f authored by Leonard Guetta's avatar Leonard Guetta
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End of the working day

parent 9e598626
......@@ -317,15 +317,13 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp
\begin{paragr}
A \emph{bisimplicial set} is a presheaf over the category $\Delta \times \Delta$,
\[
(\Delta \times \Delta)^{op} \to \Set.
\]
The category of bisimplicial sets is denoted by $\Psh{\Delta\times\Delta}$.
Let $X$ be a bisimplicial set. In a similar fashion as for simplicial sets, we use the notation
X : (\Delta \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 $\Delta\times \Delta$.
for the image by $X$ of the object $([n],[m])$ of $\Delta\times \Delta$. The category of bisimplicial sets is denoted by $\Psh{\Delta\times\Delta}$.
Moreover for every $n \geq 0$, if we fix the first variable to $n$, we obtain a simplical set
\iffalse Moreover for every $n \geq 0$, if we fix the first variable to $n$, we obtain a simplical set
\begin{align*}
X_{n,\bullet} : \Delta^{op} &\to \Set \\
[m] &\mapsto X_{n,m}.
......@@ -344,7 +342,109 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp
\Psh{\Delta\times\Delta} \to \underline{\Hom}(\Delta^{op},\Psh{\Delta}),
\]
which are isomorphisms of categories. In other words, the category of bisimplicial sets can be identified with the category of functors $\underline{\Hom}(\Delta^{op},\Psh{\Delta})$ in two canonical ways.
\fi
\end{paragr}
\begin{paragr}
The functor
\begin{align*}
\delta : \Delta &\to \Delta\times\Delta \\
[n] &\mapsto ([n],[n])
\end{align*}
induces by pre-composition a functor
\[
\delta^* : \Psh{\Delta\times\Delta} \to \Psh{\Delta}.
\]
We say that a morphism a bisimplicial sets, $f : X \to Y$, is a \emph{diagonal weak equivalence} when $\delta^*(f)$ is a weak equivalence of simplicial sets. By definition, $\delta^*$ induces a morphism of op-prederivators
\[
\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}) \to \Ho(\Psh{\Delta}).
\]
Recall now 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 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}
Consider that $\Psh{\Delta\times\Delta}$ is equipped with the diagonal model structure. Then, the adjunction
\[
\begin{tikzcd}
\delta^* : \Psh{\Delta\times\Delta} \ar[r,shift left] & \Psh{\Delta} \ar[l,shift left]: \delta_*,
\end{tikzcd}
\]
is a Quillen equivalence.
\end{proposition}
\begin{proof}
\todo{Ça doit être contenu dans Moerdijk mais à voir.}
\end{proof}
In particular, the morphism of op-prederivators
\[
\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}) \to \Ho(\Psh{\Delta})
\]
is actually an equivalence of op-prederivators.
Diagonal weak equivalences are not the only interesting weak equivalences for bisimplicial sets as we shall now see.
\begin{paragr}
A morphism $f : X \to Y$ of bisimplicial sets is a \emph{vertical (resp.\ horizontal) weak equivalence} when for every $n \geq 0$, the induced morphism of simplicial sets
\[
f_{\bullet,n} : X_{\bullet,n} \to Y_{\bullet,n}
\]
(resp.
\[
f_{n,\bullet} : X_{n,\bullet} \to Y_{n,\bullet})
\]
is a weak equivalence (of simplicial sets). Recall now a very useful lemma.
\end{paragr}
\begin{lemma}\label{bisimpliciallemma}
Let $f : X \to Y$ be a morphism of bisimplicial sets. If $f$ is a vertical or horizontal weak equivalence then it is a diagonal weak equivalence.
\end{lemma}
\begin{proof}
\todo{Mettre référence}
\end{proof}
\begin{paragr}
In particular, the identity functor of the category of bisimplicial sets induces morphisms of op-prederivators:
\begin{equation}\label{fromverttodiag}
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to \Ho(\Psh{\Delta})
\end{equation}
and
\begin{equation}\label{fromhortodiag}
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to \Ho(\Psh{\Delta}).
\end{equation}
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}
Let
\[
\begin{tikzcd}
A \ar[r,"u"]\ar[d,"f"] & B \ar[d,"g"] \\
C \ar[r,"v"] & D
\end{tikzcd}
\]
be a square in the category of bisimplicial sets. If one of the following condition holds:
\begin{itemize}
\item For every $n\geq 0$, the square of simplicial sets
\[
\begin{tikzcd}
A_{n,\bullet} \ar[r,"{u_{n,\bullet}}"]\ar[d,"{f_{n,\bullet}}"] & B_{n,\bullet} \ar[d,"{g_{n,\bullet}}"] \\
C_{n,\bullet} \ar[r,"{v_{n,\bullet}}"] & D_{n,\bullet}
\end{tikzcd}
\]
is homotopy cocartesian.
\item For every $n\geq 0$, the square of simplicial sets
\[
\begin{tikzcd}
A_{\bullet,n} \ar[r,"{u_{\bullet,n}}"]\ar[d,"{f_{\bullet,n}}"] & B_{\bullet,n} \ar[d,"{g_{\bullet,n}}"] \\
C_{\bullet,n} \ar[r,"{v_{\bullet,n}}"] & D_{\bullet,n}
\end{tikzcd}
\]
is homotopy cocartesian.
\end{itemize}
Then the square
\[
\begin{tikzcd}
\delta^*(A) \ar[r,"\delta^*(u)"]\ar[d,"\delta^*(f)"] & \delta^*(B) \ar[d,"\delta^*(g)"] \\
\delta^*(C) \ar[r,"\delta^*(v)"] & \delta^*(D)
\end{tikzcd}
\]
is a homotopy cocartesian square of simplicial sets.
\end{corollary}
\begin{proof}
\todo{Écrire une preuve ?}
\end{proof}
......@@ -77,6 +77,15 @@ year={2020}
pages={195--244},
year={2003}
}
@article{cisinski2004localisateur,
title={Le localisateur fondamental minimal},
author={Cisinski, Denis-Charles},
journal={Cahiers de topologie et g{\'e}om{\'e}trie diff{\'e}rentielle cat{\'e}goriques},
volume={45},
number={2},
pages={109--140},
year={2004}
}
@article{cisinski2008proprietes,
title={Propri{\'e}t{\'e}s universelles et extensions de Kan d{\'e}riv{\'e}es},
author={Cisinski, Denis-Charles},
......@@ -218,6 +227,14 @@ year={2020}
year=2008,
publisher={International Press of Boston}
}
@incollection{moerdijk1989bisimplicial,
title={Bisimplicial sets and the group-completion theorem},
author={Moerdijk, Ieke},
booktitle={Algebraic K-theory: connections with geometry and topology},
pages={225--240},
year={1989},
publisher={Springer}
}
@unpublished{polybook,
title={Polygraphs : from {R}ewriting to {H}igher {C}ategories},
author={Ara and Burroni and Guiraud and Malbos and M{\'e}tayer and Mimram},
......
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