Commit 09626b31 by Leonard Guetta

### Il faut que je corrige 6.2.6 où je me suis emmêlé les pinceaux

parent 421eae7a
 ... ... @@ -374,9 +374,9 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp \] 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 $\overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}) \to \Ho(\Psh{\Delta}). \overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}}) \to \Ho(\Psh{\Delta}).$ Recall from \cite[Proposition 1.2]{moerdijk1989bisimplicial} that the category of bisimplicial sets can be equipped with a model structure whose weak equivalences are 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 $\delta^*$. 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 $\delta^*$. \end{paragr} \begin{proposition}\label{prop:diageqderivator} Consider that $\Psh{\Delta\times\Delta}$ is equipped with the diagonal model structure. Then, the adjunction ... ... @@ -388,7 +388,11 @@ We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition \ref{prop:hmtp is a Quillen equivalence. \end{proposition} \begin{proof} \todo{Ça doit être contenu dans Moerdijk mais à voir.} 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}. \end{proof} In particular, the morphism of op-prederivators ... ... @@ -467,7 +471,7 @@ In particular, morphisms of op-prederivators \eqref{fromverttodiag} and \eqref{f \todo{Écrire une preuve ?} \end{proof} \section{Bisimplicial nerve for 2-categories} We shall now describe a nerve'' for 2-categories with value in bisimplicial sets and recall a few results that shows that this nerve is, in some sense, equivalent to the Street nerve from section \todo{ref}. We shall now describe a nerve'' for 2-categories with values in bisimplicial sets and recall a few results that shows that this nerve is, in some sense, equivalent to the Street nerve from section \todo{ref}. \begin{notation} \begin{itemize} \item[-] Once again, we will use the notation N : \Cat \to \Psh{\Delta} instead of N_1 for the usual nerve of categories. Moreover, using the usual convention for the set of k-simplices of a simplicial set, if C is a (small) category, then ... ...  ... ... @@ -90,7 +90,7 @@ We end this section with an important result on slices \oo\nbd{}category (Para \[ \alpha_{\src_k(e_n)}=\begin{cases}\trgt_{k+1}(e_n), \text{ if } kn:] Since \sS_n is an n\nbd{}category, the image of \alpha_k is necessarily of the form \varphi(\alpha_k)=\1^k_{x} with x a cell of \sS_n of dimension non-greater than n. If the dimension of x is strictly lower than n, then everything works like in the case kn:] Since \sS_n is an n\nbd{}category, the image of \alpha_k is necessarily of the form \varphi(\alpha_k)=\1^k_{x} with x a cell of \sS_n of dimension non-greater than n. If the dimension of x is strictly lower than n, then everything works like in the case k  ... ... @@ -74,7 +74,7 @@ The functor \kappa : \Psh{\Delta} \to \Ch is left Quillen and sends weak equiv %% \end{paragr} \begin{remark} The adjective singular'' is there to avoid future confusion with another homological invariant for \oo\nbd{}categories that will be introduced later. As a matter of fact, the underlying point of view adopted in this thesis is that \emph{singular homology of \oo-categories} ought to be simply called \emph{homology of \oo\nbd{}categories} as it is the only correct'' definition of homology. This assertation will be justified later. \todo{Le faire !} The adjective singular'' is there to avoid future confusion with another homological invariant for \oo\nbd{}categories that will be introduced later. As a matter of fact, the underlying point of view adopted in this thesis is that \emph{singular homology of \oo-categories} ought to be simply called \emph{homology of \oo\nbd{}categories} as it is the only correct'' definition of homology. This assertion will be justified later. \todo{Le faire !} \end{remark} \begin{remark} We could also have defined the homology of \oo\nobreakdash-category with K : \Psh{\Delta}\to \Ch instead of \kappa : \Psh{\Delta} \to \Ch since these two functors are quasi-isomorphic (see \cite[Theorem 2.4]{goerss2009simplicial} for example). An advantage of the later one is that it is left Quillen. ... ... @@ -245,7 +245,7 @@ As we shall now see, when the \oo\nbd{}category C is \emph{free} the chain c \end{align*} \end{proof} \begin{paragr} Let C be a \emph{free} \oo-category and write \Sigma=(\Sigma_n)_{n \in \mathbb{N}} for its basis. For every n \geq 0 and every \alpha \in \Sigma_n, recall that we have proved in Proposition \ref{prop:countingfunction} the existence of a unique functionr w_{\alpha} : C_n \to \mathbb{N} such that: Let C be a \emph{free} \oo-category and write \Sigma=(\Sigma_n)_{n \in \mathbb{N}} for its basis. For every n \geq 0 and every \alpha \in \Sigma_n, recall that we have proved in Proposition \ref{prop:countingfunction} the existence of a unique function w_{\alpha} : C_n \to \mathbb{N} such that: \begin{enumerate}[label=(\alph*)] \item\label{cond:countingfunctionfirst} w_{\alpha}(\alpha)=1, \item\label{cond:countingfunctionsecond} w_{\alpha}(\beta)=0 for every \beta \in \Sigma_n such that \beta\neq \alpha, ... ...  ... ... @@ -319,7 +319,11 @@ From now on, we will consider that the category \Psh{\Delta} is equipped with \end{itemize} \end{paragr} \begin{example}\label{example:natisoplax} Let u,v : C \to D be two functors between (small) categories. Using the formulas of the previous paragraph, it is straightforward to check that the data of an oplax transformation from u to v consists exactly of the data of a natural transformation from u to v. When C and D are n\nbd{}categories with n finite and u,v :C \to D are two n\nbd{}functors, an oplax transformation \alpha : u \Rightarrow v amounts to the data of a (k+1)\nbd{}cell \alpha_x of D for each k\nbd{}cell x of C with 0 \leq k \leq n, with source and target as in the previous paragraph. These data being subject to the axioms of the previous paragraph. Note that when x is an n\nbd{}cell of C, \alpha_x is necessarily a unit, which can be expressed as the equality \[ \alpha_{t_{n-1}(x)}\comp_{n-1}\cdots\comp_1\alpha_{t_0(x)}\comp_0u(x) = v(x)\comp_0\alpha_{s_0(x)}\comp_1\cdots\comp_{n-1}\alpha_{s_{n-1}(x)} In particular, when $n=1$ and $C$ and $D$ are thus (small) categories, an oplax transformation $u \Rightarrow v$ is nothing but a natural transformation from $u$ to $v$. \end{example} \begin{paragr} Let $u : C \to D$ be an $\oo$-functor. There is an oplax transformation from $u$ to $u$, denoted by $1_u$, which is defined as ... ...
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