### 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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