@@ -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

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 $n \geq0$, 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

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: