Commit 977ae1e8 authored by Leonard Guetta's avatar Leonard Guetta
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gotta go

parent 09626b31
......@@ -371,35 +371,42 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp
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
\]
By the usual calculus of Kan extensions, $\delta^*$ admits a left adjoint
$\delta_!$ and a right adjoint $\delta_*$
\[
\delta_! \dashv \delta^* \dashv \delta_*.
\]
We say that a morphism a bisimplicial sets, $f : X \to Y$, is a \emph{diagonal
weak equivalence} (resp.\ \emph{diagonal fibration})when $\delta^*(f)$ is a
weak equivalence of simplicial sets (resp.\ fibration of simplicial sets). By definition, $\delta^*$ induces a morphism of op-prederivators
\[
\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}}) \to \Ho(\Psh{\Delta}).
\]
Recall from \cite[Proposition 1.2]{moerdijk1989bisimplicial} that the category of bisimplicial sets can be equipped with a model structure whose weak equivalences are the diagonal weak equivalences and whose fibrations are the diagonal fibrations in an obvious sense. We shall refer to this model structure as the \emph{diagonal model structure}. Let us write $\delta_*$ for the right adjoint of $\delta^*$.
Recall from \cite[Proposition 1.2]{moerdijk1989bisimplicial} that the category of bisimplicial sets can be equipped with a model structure whose weak equivalences are the diagonal weak equivalences and whose fibrations are the diagonal fibrations in an obvious sense. We shall refer to this model structure as the \emph{diagonal model structure}.
\end{paragr}
\begin{proposition}\label{prop:diageqderivator}
Consider that $\Psh{\Delta\times\Delta}$ is equipped with the diagonal model structure. Then, the adjunction
\[
\begin{tikzcd}
\delta^* : \Psh{\Delta\times\Delta} \ar[r,shift left] & \Psh{\Delta} \ar[l,shift left]: \delta_*,
\delta_! : \Psh{\Delta} \ar[r,shift left] & \Psh{\Delta\times\Delta} \ar[l,shift left]: \delta^*,
\end{tikzcd}
\]
is a Quillen equivalence.
\end{proposition}
\begin{proof}
We know from the second part of \cite[Proposition 1.2]{moerdijk1989bisimplicial} that $\delta^*$ induces an equivalence at the level of homotopy categories
\[
\overline{\delta^*} : \ho(\Psh{\Delta\times\Delta}) \overset{\sim}{\longrightarrow} \ho(\Psh{\Delta}).
\]
Hence, all we need to show is that the adjunction $\delta^* \dashv \delta_*$ is a Quillen adjunction. The fact that $\delta_*$ preserves weak equivalences follows easily from the above equivalence of homotopy categories and the fact that it preserves fibrations is \cite[Lemma 3.14]{goerss2009simplicial}.
By definition $\delta^*$ preserves weak equivalences and fibrations and thus,
the adjunction is a Quillen adjunction. The fact that $\delta^*$ induces an
equivalence at the level of homotopy categories is \cite[Proposition
1.2]{moerdijk1989bisimplicial}.
\end{proof}
\begin{paragr}
In particular, the morphism of op-prederivators
\[
\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}}) \to \Ho(\Psh{\Delta})
\]
is actually an equivalence of op-prederivators.
\end{paragr}
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
......@@ -411,26 +418,64 @@ Diagonal weak equivalences are not the only interesting weak equivalences for bi
f_{n,\bullet} : X_{n,\bullet} \to Y_{n,\bullet})
\]
is a weak equivalence (of simplicial sets). Recall now a very useful lemma.
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}
See for example \cite[Chapter XII,4.3]{bousfield1972homotopy} or \cite[Proposition 2.1.7]{cisinski2004localisateur}.
\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}
\[
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
\]
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.
\[
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}).
\]
\end{paragr}
\begin{proposition}
The morphisms of op-prederivators
\[
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
\]
and
\[
\Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
\]
are homotopy cocontinuous.
\end{proposition}
\begin{proof}
Recall 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 (see for example
\cite[Chapter IV]{goerss2009simplicial} or \cite{cisinski2004localisateur}).
We respectively refer
to these model structures as the \emph{vertical model structure} and
\emph{horizontal model structure}. Since the functor $\delta^* :
\Psh{\Delta\times\Delta} \to \Psh{\Delta}$ preserves monomorphisms, it follows from Lemma
\ref{bisimpliciallemma} that the adjunction
\[
\begin{tikzcd}
\delta^* : \Psh{\Delta\times\Delta} \ar[r,shift left] & \ar[l,shift left]
\Psh{\Delta} : \delta_*
\end{tikzcd}
\]
is a Quillen adjunction when $\Psh{\Delta\times\Delta}$ is equipped with
either the vertical model structure or the horizontal model structure. In
particular,
the induced morphisms of op-prederivators
\[
\overline{\delta^*} : \Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to \Ho(\Psh{\Delta})
\]
and
\[
\overline{\delta^*} : \Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to \Ho(\Psh{\Delta})
\]
are homotopy cocontinuous.
\end{proof}
\begin{corollary}\label{cor:bisimplicialsquare}
Let
\begin{equation}\tag{$\ast$}\label{kiki}
......@@ -777,3 +822,8 @@ For any $n \geq 0$, consider the following cocartesian square
\end{proposition}
\section{Zoology of $2$-categories: A stub of a criterion}
\section{``Bubble-free'' $2$-categories}
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......@@ -90,7 +90,13 @@ year={2020}
year={1990},
publisher={Elsevier}
}
@book{bousfield1972homotopy,
title={Homotopy limits, completions and localizations},
author={Bousfield, Aldridge Knight and Kan, Daniel Marinus},
volume={304},
year={1972},
publisher={Springer Science \& Business Media}
}
@article{buckley2016orientals,
title = "Orientals and cubes, inductively",
abstract = "We provide direct inductive constructions of the orientals and the cubes, exhibiting them as the iterated cones, respectively, the iterated cylinders, of the terminal strict globular ω-category.",
......
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