### 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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