Commit 76a8b057 authored by Leonard Guetta's avatar Leonard Guetta
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......@@ -894,8 +894,51 @@ The previous proposition admits the following corollary, which will be of great
\[
C=\iota_n(C).
\]
Within this section, \emph{and only within this section}, we try not to make this abuse of notation and explicitly write $\iota_n$ whenever we should.
\end{paragr}
Within this section, \emph{and only within this section}, we try not to make this abuse of notation and explicitly write $\iota_n$ whenever we should.
We have already seen that $\iota_n$ has a left adjoint $\tau^{i}_{\leq n} : \oo\Cat \to n\Cat$, where for an $\oo$\nbd-category $C$, $\tau_{\leq n }^{i}(C)$ is the $n$\nbd-category whose set of $k$\nbd-cells is $C_k$ for $k<n$ and whose set of $n$\nbd-cells is the quotient of $C_n$ by the equivalence relation $\sim$ generated by
\[
x \sim y \text{ when there exists } z : x \to y \text{ in } C_{n+1}.
\]
In fact, we can use the adjunction $\tau_{\leq n}^{i} \dashv \iota_n$ to transport the folk model structure on $\oo\Cat$ to $n\Cat$.% Say that a morphism $f : C \to D$ of $n\Cat$ is a \emph{folk weak equivalence of $n$\nbd-categories} (resp. \emph{folk trivial fibration of $n$\nbd-categories}) if $\iota_n(f)$ is a weak equivalence (resp.\ fibration) for the folk model structure on $\oo\Cat$.
\end{paragr}
\begin{proposition}
There exists a model structure on $n\Cat$ such that:
\begin{itemize}[label=-]
\item weak equivalences are exactely those morphisms $f : C \to D$ such that $\iota_n(f)$ is a weak equivalence for the folk model structure on $\oo\Cat$,
\item fibrations are exactly those morphisms $f : C \to D$ such that $\iota_n(f)$ is a fibrations for the folk model structure on $\oo\Cat$.
\end{itemize}
Moreover, there exists a set $I$ of generating cofibrations (resp.\ a set $J$ of generating trivial cofibrations) for the folk model structure on $\oo\Cat$ such that the image by $\tau^{i}_{\leq n}$ of $I$ (resp.\ $J$) is a set of generating cofibrations (resp.\ generating trivial cofibrations) of the above model structure on $n\Cat$.
\end{proposition}
\begin{proof}
This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generating cofibrations is not made explicit in the statement of the theorem, it is contained in proof.)
\end{proof}
\begin{paragr}
We refer to the model structure of the above proposition as the \emph{folk model structure on $n\Cat$}. By definition, the functor $\iota_n : n\Cat \to \oo\Cat$ preserves weak equivalences and fibrations when $\oo\Cat$ and $n\Cat$ are equipped with the folk model structure. In particular, the adjunction $\tau^{i}_{\leq n } \dashv \iota_n$ is a Quillen adjunction. As it happens, the functor $\tau^{i}_{\leq n}$ also preserves weak equivalences.
\end{paragr}
\begin{lemma}
The functor $\tau^{i}_{\leq n} : \oo\Cat \to n\Cat$ sends weak equivalences of the folk model structure on $\oo\Cat$ to weak equivalences of the folk model structure on $n\Cat$.
\end{lemma}
\begin{proof}
% Let $C$ be an $\oo$-category and let $\eta_C : C \to \iota_n\tau^{i}_{\leq n}(C)$ be the unit morphism of the adjunction $\tau^{i}_{\leq n} \dashv \iota_n$. Let us prove that for every pair $(x,y)$ of $k$\nbd-cells of $C$, if $k \leq n$ then $x \simeq_{\oo} y$ (see \ref{paragr:ooequivalence}) if and only if $\eta_C(x)\simeq_{\oo}\eta_C(y)$.
Let $f : C \to D$ be a weak equivalence for the folk model structure on $\oo\Cat$. By definition of weak equivalences for the the folk model structure on $n\Cat$, we have to show that
\[
\iota_n\tau^{i}_{\leq n}(f) : \iota_n\tau^{i}_{\leq n}(C) \to \iota_n\tau^{i}_{\leq n}(D)
\]
is again a weak equivalence for the folk model structure on $\oo\Cat$. Consider the following commutative square
\[
\begin{tikzcd}
C \ar[d,"\eta_C"] \ar[r,"f"] & D \ar[d,"\eta_D"] \\
\iota_n\tau^{i}_{\leq n}(C) \ar,"\iota_n\tau^{i}_{\leq n}(f)"] & \iota_n\tau^{i}_{\leq n}(D)
\end{tikzcd},
\]
where $\eta$ is the unit of the adjunction $\tau^{i}_{\leq n} \dashv \iota_n$.
\begin{enumerate}[label=(\roman*)]
\item Let $y$ be a $0$\nbd-cell of $\iota_n\tau^{i}_{\leq n}(D)$. The map $\eta_D$ being surjective on $0$\nbd-cells (even if $n=0$), there exists $y'$ such that $\eta_D(y')=y$. Since $f$ is a folk weak equivalence, there exists $x' \in C_0$ such that $x'\simeq_{\oo} y'$ and then $x:=\eta_C(x') \simeq_{\oo} y'$.
\item Let $x$ and $y$ be parallel
\end{enumerate}
\toto{À finir}
\end{proof}
\begin{paragr}
Let $\Ch^{\leq n}$ be the category of chain complexes in degree comprised between $0$ and $n$. This means that an object $K$ of $\Ch^{\leq n}$ is a diagram of abelian groups of the form
\[
......@@ -907,22 +950,23 @@ The previous proposition admits the following corollary, which will be of great
\]
This functor is fully faithful and $\Ch^{\leq n}$ may be identified with the full subcategory of $\Ch$ spanned by chain complexes $K$ such that $K_k = 0$ for every $k >n$.
Recall from \ref{paragr:defncat} that for every $n \geq 0$, the canonical inclusion $\iota_n : n\Cat \to \oo\Cat $ has a left adjoint $\tau^{i}_{\leq n} : \oo\Cat \to n\Cat$, where for an $\oo$\nbd-category $C$, $\tau_{\leq n }^{i}(C)$ is the $n$\nbd-category whose set of $k$\nbd-cells is $C_k$ for $k<n$ and whose set of $n$\nbd-cells is the quotient of $C_n$ by the equivalence relation $\sim$ generated by
\[
x \sim y \text{ when there exists } z : x \to y \text{ in } C_{n+1}.
\]
% Note that when $C$ is an $n$\nbd-category, seen as an $\oo$\nbd-category with only unit cells in dimension higher that $n$ via the canonical inclusion $\iota_n : n\Cat \to \oo\Cat$, then $\tau^{i}_{\leq n}(C) = C$.
Similarly, for a chain complex $K$, write $\tau^{i}_{\leq n}(K)$ for the object of $\Ch^{\leq n}$ defined as
Similarly to the case of $n$\nbd-categories, the functor $\iota_n : \Ch^{\leq n} \to \Ch$ has a left adjoint $\tau^{i}_{\leq n} : \Ch \to \Ch^{\leq}$, where for a chain complex $K$, $\tau^{i}_{\leq n}(K)$ for the object of $\Ch^{\leq n}$ defined as
\[
K_0 \overset{\partial}{\longleftarrow} K_1 \overset{\partial}{\longleftarrow} K_2 \overset{\partial}{\longleftarrow} \cdots \overset{\partial}{\longleftarrow} K_{n-1} \overset{\partial}{\longleftarrow} K_{n}/{\partial(K_{n+1})}.
\]
This defines a functor
\[
\tau^{i}_{\leq n} : \Ch^{\leq n} \mapsto \Ch
\]
which is left adjoint to the canonical inclusion $\iota_n : \Ch^{\leq n} \to \Ch$.
Likewise $n$\nbd-categories again, we can use the adjunction $\iota_n : \Ch^{\leq n} \overset{\longrightarrow}{\longleftarrow} \Ch : \tau^{i}_{\leq n}$ to create a model structure on $\Ch^{\leq n}$.
\end{paragr}
\begin{proposition}
There exists a model structure on $\Ch^{\geq n}$ such that:
\begin{itemize}
\item weak equivalences are exactly those morphisms $f : K \to K'$ such that $\iota_n(f)$ is a weak equivalence for the projective model structure on $\Ch$ (i.e.\ a quasi-isomorphism),
\item fibrations are exactly those morphisms $f : K \to K'$ such that $\iota_n(f)$ is a fibration for the projective model structure on $\Ch$.
\end{itemize}
\end{proposition}
\begin{proof}
This a typical example of a transfer a cofibrantely generated model structure along a right adjoint as in
\end{proof}
\begin{lemma}
For every chain complex $K$, the unit map
\[
......@@ -945,6 +989,7 @@ The previous proposition admits the following corollary, which will be of great
\]
The isomorphism being obviously induced by the unit map $K \to \iota_n\tau^{i}_{\leq n}(K)$.
\end{proof}
As a consequence of this lemma, we have the analoguous of
\begin{paragr}
Let $C$ be $n$\nbd-category $C$. An straigtforward computation shows that the chain complex $\lambda(\iota_n(C))$ is such that
\[
......
......@@ -498,7 +498,7 @@ From now on, we will consider that the category $\Psh{\Delta}$ is equipped with
All the results we have seen in this section are still true if we replace ``oplax'' by ``lax'' everywhere.
\end{remark}
\section{Equivalence of $\omega$-categories and the folk model structure}
\begin{paragr}
\begin{paragr}\label{paragr:ooequivalence}
Let $C$ be an $\omega$-category. We define the equivalence relation $\sim_{\omega}$ on the set $C_n$ by co-induction on $n \in \mathbb{N}$. For $x, y \in C_n$, we have $x \sim_{\omega} y $ when:
\begin{itemize}
\item[-] $x$ and $y$ are parallel,
......
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