Commit 18fede95 authored by Leonard Guetta's avatar Leonard Guetta
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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*}
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...@@ -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}
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...@@ -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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