Commit d3221095 by Leonard Guetta

### 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 \tag{\ast}\label{coucou}\begin{tikzcd} A \ar[r,"u"]\ar[d,"f"] & B \ar[d,"g"] \\ C \ar[r,"v"] & D \end{tikzcd} 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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