@@ -916,9 +916,9 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati

\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.

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}

\end{proposition}

\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

...

...

@@ -951,23 +951,52 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati

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 $k >n$.

% 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 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

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 $K$, $\tau^{i}_{\leq n}(K)$ is the object of $\Ch^{\leq n}$ defined as

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 $\Ch^{\leq n}$.

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),

There exists a model structure on $\Ch^{\leq n}$ such that:

\begin{itemize}[label=-]

\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 $\Ch$,

\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

This a typical example of a transfer a cofibrantely generated model structure along a right adjoint as in \cite[Proposition 2.3]{beke2001sheafifiableII}. The only \emph{a priori} non-trivial hypothesis to check is that there exists a set $J$ of generating trivial cofibrations of the projective model structure on $\Ch$ such that for every $j : A \to B$ in $J$ and every cocartesian square

\[

\begin{tikzcd}

\tau^{i}_{\leq n}(A)\ar[r]\ar[d,"\tau^{i}_{\leq n}(j)"']& X \ar[d,"g"]\\

\tau^{i}_{\leq n}(B)\ar[r]& Y

\ar[from=1-1, to=2-2, phantom, "\ulcorner",very near end]

\end{tikzcd},

\]

the morphism $\iota_n(g)$ is a weak equivalence of $\Ch$. As explained in \cite[Proposition 7.19]{dwyer1995homotopy}, there exists a set of generating trivial cofibrations of the projective model structure on $\Ch$ consisting of the maps

\[

0\to D_k

\]

for each $k >0$, where $D_k$ is the following chain complex concentrated in degree $k$ and $k-1$

What is left to show then is that for every $k > 0$ and every object $X$ of $\Ch^{\leq n}$, the canonical inclusion map

\[

X \to X \oplus\tau^{i}_{\leq n}(D_k)

\]

is sent by $\iota_n$ to a weak equivalence of $\Ch$. This follows immediatly from the fact that homology groups commute with direct sums.

\end{proof}

\begin{lemma}

\begin{paragr}

We refer to the model structure of the previous proposition as the \emph{projective model structure on $\Ch^{\leq n}$}.

\end{paragr}

\begin{lemma}\label{lemma:unitajdcomp}

For every chain complex $K$, the unit map

\[

K \to\iota_n\tau^{i}_{\leq n}(K)

...

...

@@ -989,9 +1018,27 @@ This is \cite[Theorem 5]{lafont2010folk}. (Although the part concerning generati

\]

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

As a consequence of this lemma, we have the analoguous of Proposition \ref{prop:truncationhomotopical}.

\begin{proposition}

The functor $\tau^{i}_{\leq n} : \Ch\to\Ch^{\leq n}$ sends weak equivalences of the projective model structure on $\Ch$ to weak equivalences of the projective model structure on $\Ch^{\leq n}$.

\end{proposition}

\begin{proof}

Let $f : K \to K'$ be a weak equivalence for the projective model structure on $\Ch$ and consider the naturality square

\[

\begin{tikzcd}[column sep=huge]

K \ar[d,"\eta_K"]\ar[r,"f"]& K' \ar[d,"\eta_K'"]\\