Commit 18fede95 by Leonard Guetta

Il faut que je finisse de nettoyer la section 5.3 (à partir de corollaire 5.3.9)

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 ... @@ -138,7 +138,7 @@ We write $\Ab$ for category of abelian groups and for an abelian group $G$, we w ... @@ -138,7 +138,7 @@ We write $\Ab$ for category of abelian groups and for an abelian group $G$, we w \begin{proof} \begin{proof} The category of chain complexes is equivalent to the category $\omega\Cat(\Ab)$ of $\omega$-categories internal to abelian groups (see \cite[Theorem 3.3]{bourn1990another}) and with this identification, the functor $\lambda : \omega\Cat \to \omega\Cat(\Ab)$ is nothing but the left adjoint of the canonical forgetful functor $\omega\Cat(\Ab) \to \omega\Cat$. The category of chain complexes is equivalent to the category $\omega\Cat(\Ab)$ of $\omega$-categories internal to abelian groups (see \cite[Theorem 3.3]{bourn1990another}) and with this identification, the functor $\lambda : \omega\Cat \to \omega\Cat(\Ab)$ is nothing but the left adjoint of the canonical forgetful functor $\omega\Cat(\Ab) \to \omega\Cat$. \end{proof} \end{proof} As we shall now see, when the $\oo$\nbd-categoru $C$ is \emph{free} the chain complex $\lambda(C)$ admits a nice expression. As we shall now see, when the $\oo$\nbd-category $C$ is \emph{free} the chain complex $\lambda(C)$ admits a nice expression. \begin{paragr} \begin{paragr} Let $n\geq 0$. Recall that for every monoid $M$ (supposed commutative if $n \geq 1$) we have defined in Section \ref{sec:suspmonoids} an $n$\nbd-category $B^nM$ whose set of $n$\nbd-cells is isomorphic to the underlying set of $M$. And the correspondence $M \mapsto B^nM$ defines a functor in the expected way. By considering abelian groups as particular cases of (commutative) monoids, we obtain a functor for each $n\geq 0$ Let $n\geq 0$. Recall that for every monoid $M$ (supposed commutative if $n \geq 1$) we have defined in Section \ref{sec:suspmonoids} an $n$\nbd-category $B^nM$ whose set of $n$\nbd-cells is isomorphic to the underlying set of $M$. And the correspondence $M \mapsto B^nM$ defines a functor in the expected way. By considering abelian groups as particular cases of (commutative) monoids, we obtain a functor for each $n\geq 0$ \begin{align*} \begin{align*} ... ...
 ... @@ -701,7 +701,7 @@ We now turn to the most important way of obtaining op-prederivators. ... @@ -701,7 +701,7 @@ We now turn to the most important way of obtaining op-prederivators. \begin{proof} \begin{proof} See for example \cite[Proposition 3.4]{cisinski2003images}. See for example \cite[Proposition 3.4]{cisinski2003images}. \end{proof} \end{proof} \begin{paragr} \begin{paragr}\label{paragr:projmod} The model structure of the previous proposition is referred to as the \emph{projective model structure on $\M(A)$}. The model structure of the previous proposition is referred to as the \emph{projective model structure on $\M(A)$}. \end{paragr} \end{paragr} \begin{proposition} \begin{proposition} ... @@ -725,7 +725,7 @@ By definition of the projective model structure, $u^*$ preserve weak equivalence ... @@ -725,7 +725,7 @@ By definition of the projective model structure, $u^*$ preserve weak equivalence \] \] is a Quillen adjunction. Since $\hocolim_A$ is the left derived of $\colim_A$, we have the following immediate corollary of the previous proposition. is a Quillen adjunction. Since $\hocolim_A$ is the left derived of $\colim_A$, we have the following immediate corollary of the previous proposition. \end{paragr} \end{paragr} \begin{corollary} \begin{corollary}\label{cor:cofprojms} Let $(\M,\W,\Cof,\Fib)$ be a cofibrantely generated model category, $A$ a small category and $X : A \to \M$ a diagram. If $X$ is cofibrant for the projective model structure on $\C(A)$, then the canonical morphism of $\ho(\C)$ Let $(\M,\W,\Cof,\Fib)$ be a cofibrantely generated model category, $A$ a small category and $X : A \to \M$ a diagram. If $X$ is cofibrant for the projective model structure on $\C(A)$, then the canonical morphism of $\ho(\C)$ $\[ \hocolim_A(X) \to \colim_A(X) \hocolim_A(X) \to \colim_A(X) ... @@ -741,7 +741,7 @@ By definition of the projective model structure, u^* preserve weak equivalence ... @@ -741,7 +741,7 @@ By definition of the projective model structure, u^* preserve weak equivalence \end{proof} \end{proof} Below is a particular case for which the previous corollary applies. Below is a particular case for which the previous corollary applies. \begin{proposition} \begin{proposition}\label{prop:sequentialhmtpycolimit} Let (\M,\W,\Cof,\Fib) be a cofibrantely generated model category and let X be a sequential diagram in \M Let (\M,\W,\Cof,\Fib) be a cofibrantely generated model category and let X be a sequential diagram in \M \[ \[ X_0 \to X_1 \to X_2 \to \cdots X_0 \to X_1 \to X_2 \to \cdots ... @@ -753,7 +753,7 @@ By definition of the projective model structure, u^* preserve weak equivalence ... @@ -753,7 +753,7 @@ By definition of the projective model structure, u^* preserve weak equivalence \end{proof} \end{proof} Another setting for which a category of diagrams \M(A) can be equipped with a model structure whose weak equivalences are pointwise equivalences and for which the A-colimit functor is left Quillen is when the category A is a \emph{Reedy category}. Rather that recalling this theory, we simply put here the only practical result that we shall need in the sequel. Another setting for which a category of diagrams \M(A) can be equipped with a model structure whose weak equivalences are pointwise equivalences and for which the A-colimit functor is left Quillen is when the category A is a \emph{Reedy category}. Rather that recalling this theory, we simply put here the only practical result that we shall need in the sequel. \begin{lemma}\label{lemma:hmptycocartesianreedy} \begin{lemma}\label{lemma:hmtpycocartesianreedy} Let (\M,\W,\Cof,\Fib) be a model category and let Let (\M,\W,\Cof,\Fib) be a model category and let \[ \[ \begin{tikzcd} \begin{tikzcd} ... ... No preview for this file type  ... @@ -1071,7 +1071,7 @@ We can now prove the expected result. ... @@ -1071,7 +1071,7 @@ We can now prove the expected result. \todo{À finir ? Il faudrait peut-être dire qu'en rajoutant des hypothèses on doit pouvoir quand même avoir une forme normale. Dans le cas n=2, quand on fait l'hypothèse que le 2-polygraphe est positif, je sais que c'est vrai.} \todo{À finir ? Il faudrait peut-être dire qu'en rajoutant des hypothèses on doit pouvoir quand même avoir une forme normale. Dans le cas n=2, quand on fait l'hypothèse que le 2-polygraphe est positif, je sais que c'est vrai.} \end{paragr} \end{paragr} \section{Discrete Conduché \oo-functors} \section{Discrete Conduché \oo-functors}\label{section:conduche} \begin{paragr} \begin{paragr} Recall that given a category \C and a class M of arrows of \C, we say that an arrow f : X \to Y of \C is \emph{left orthogonal} to M if for every m : A \to B in M and every solid arrow square Recall that given a category \C and a class M of arrows of \C, we say that an arrow f : X \to Y of \C is \emph{left orthogonal} to M if for every m : A \to B in M and every solid arrow square ... @@ -1135,7 +1135,7 @@ We can now prove the expected result. ... @@ -1135,7 +1135,7 @@ We can now prove the expected result. means exactly that x=1^{n}_y. means exactly that x=1^{n}_y. \end{paragr} \end{paragr} \begin{definition}\label{def:conduche} \begin{definition}\label{def:conduche} An \oo-functor F : C \to D is an \emph{discrete Conduché \oo-functor} if it is right orthogonal to the \oo-functors An \oo-functor F : C \to D is an \emph{discrete Conduché \oo\nbd-functor} if it is right orthogonal to the \oo-functors \[ \[ \kappa^n_k : \sD_n \to \sD_k \kappa^n_k : \sD_n \to \sD_k$ \] ... ...
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