@@ -902,7 +902,7 @@ The previous proposition admits the following corollary, which will be of great

\]

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}

\begin{proposition}\label{prop:fmsncat}

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 $\oo\Cat$,

...

...

@@ -1036,7 +1036,7 @@ As a consequence of this lemma, we have the analoguous of Proposition \ref{prop:

\]

is an isomorphism for every $k \leq n$. Since obviously $H_k(\iota_n\tau^{i}_{\leq n}(f))$ is also an isomorphism for $k > n$, this proves the result.

\end{proof}

We now investigate the relation between truncation and linearization.

We now investigate the relation between truncation and abelianization.

\begin{paragr}

Let $C$ be $n$\nbd-category. A straigtforward computation shows that the chain complex $\lambda(\iota_n(C))$ is such that

\[

...

...

@@ -1056,7 +1056,7 @@ We now investigate the relation between truncation and linearization.

With this lemma at hand we can prove the important following proposition which basically says that if an $\oo$\nbd-category $C$ is free up to dimension $n-1$, then for any $k$ such that $0\leq k \leq n$ there is no need to find a cofibrant replacement in order to compute $H^{\pol}_k(C)$.

\begin{proposition}

\begin{lemma}

The functor $\lambda_{\leq n} : n\Cat\to\Ch^{\leq n}$ is left Quillen when $n\Cat$ is equipped with the folk model structure and $\Ch^{\leq n}$ with the projective model structure.

\end{lemma}

\begin{proof}

Let $I$ and $J$ respectively be sets of generating cofibrations and generating trivial cofibrations of the folk model structure on $\oo\Cat$. We know from Proposition \ref{prop:fmsncat} that $\tau^{i}_{\leq n}(I)$ and $\tau^{i}_{\leq n}(J)$ respectively are sets of generating cofibrations and generating trivial cofibrations of the projective model structure on $\Ch^{\leq n}$. What we have to show is that for every $f$ in $I$ (resp. $J$), $\lambda_{\leq n}(\tau^{i}_{\leq n}(f))$ is a cofibration (resp.\ generating cofibration) for the folk model structure on $n\Cat$.

% Notice that we have $\tau^{i}_{\leq n } \circ \iota_n = \mathrm{id}_{n \Cat}$ and hence

From Lemma \ref{lemma:abelianizationtruncation}, we have

\[

\lambda_{\leq n }(\tau^{i}_{\leq n}(f))\simeq\tau^{i}_{\leq n}(\lambda(f)).

\]

Since $\lambda$ and $\tau^{i}_{\leq n}$ are both left Quillen functors, this proves the result.

\end{proof}

As an immediate consequence of the previous lemma, the functor $\lambda_{\leq n}$ is left derivable and we have the following key result.

Straightforward consequence of Lemma \ref{lemma:abelianizationtruncation} and the fact that the left derived functor of a composition of left Quillen functors is the composition of the left derived functors (see for example \cite[Theorem 1.3.7]{hovey2007model}.

is \emph{not} commutative. If it were, then for every $n$\nbd-category $C$ and every $k >n$, we would have $H_k^{pol}(\iota_n(C))=0$ for every $k >n$, which is not even true for the case $n=1$ in the following chapter.

\end{remark}

A useful consequence of Proposition \ref{prop:polhmlgytruncation} is the following corollary.

\begin{corollary}

Let $n \geq0$ and $C$ be an $\oo$\nbd-category. If $C$ has a $k$\nbd-basis for every $0\leq k \leq n-1$, then the canonical map of $\ho(\Ch)$

\[

\alpha^{\pol}_C : \sH^{\pol}(C)\to\lambda(C)

...

...

@@ -1134,28 +1170,36 @@ With this lemma at hand we can prove the important following proposition which b

H_k^{\pol}(C)\simeq H_k(\lambda(C))

\]

for every $0\leq k \leq n$.

\end{proposition}

\end{corollary}

\begin{proof}

\todo{À écrire}

\end{proof}

\begin{paragr}

Since every $\oo$\nbd-category $C$ admits $C_0$ as a $0$\nbd-base, it follows from the previous proposition that $H^{\pol}_0(C)$ and $H^{\pol}_1(C)$ respectively are (canonically isomorphic to) $H_0(\lambda(C))$ and $H_1(\lambda(C))$. This means that no cofibrant resolution of $C$ is needed to compute its first two polygraphic homology groups.

\end{paragr}

Somewhat related is the following proposition.

\begin{proposition}

Let $C$ be an $\oo$\nbd-category and $n \geq0$. The canonical map

Since every $\oo$\nbd-category trivially admits its set of $0$\nbd-cells as a $0$\nbd-base, it follows from the previous proposition that for every $\oo$\nbd-category $C$ we have

%% Let $f : P \to C$ be a cofibrant replacement for $C$. \todo{À finir}.

%% \end{proof}

We now turn to the relation between truncation and singular homology of $\oo$\nbd-categories. Recall that for any $n \geq0$, the nerve functor $N_n : n\Cat\to\Psh{\Delta}$ is defined as the following composition

@@ -1172,7 +1216,7 @@ and for $n \in \mathbb{N}\cup \{\oo\}$ we write $c_n : \Psh{\Delta} \to n\Cat$ f

is commutative (up to an isomorphism).

\end{lemma}

\begin{proof}

\todo{À écrire (facile)}

Straighforward consequence of the fact that $N_n = N_{\oo}\circ\iota_n$ and the fact that the composition of left adjoints if the left adjoint of the composition.

\end{proof}

\begin{paragr}

In particular, it follows from the previous lemma that the co-unit of the adjunction $c_{\oo}\dashv N_{\oo}$ induces, for every $\oo$\nbd-category $C$ and every $n \geq0$, a canonical morphism of $n\Cat$

...

...

@@ -1189,7 +1233,16 @@ and for $n \in \mathbb{N}\cup \{\oo\}$ we write $c_n : \Psh{\Delta} \to n\Cat$ f

is an isomorphism.

\end{lemma}

\begin{proof}

\todo{À écrire.}

Let $C$ be an $\oo$\nbd-category and $D$ a (small) category. By adjunction, we have

Now let $\Delta_{\leq2}$ be the full subcategory of $\Delta$ spanned by $[0]$, $[1]$ and $[2]$ and $i : \Delta_{\leq2}\to\Delta$ the canonical inclusion. This inclusion induces by pre-composition a functor $i^* : \Psh{\Delta}\to\Psh{\Delta_{\leq2}}$ which has a right-adjoint $i_* : \Psh{\Delta}\to\Psh{\Delta_{\leq2}}$. Recall that the nerve of a (small) category is $2$-coskelettal (see for example \cite[Theorem 5.2]{street1987algebra}), which means that for every category $D$, the unit morphism $ N_1(D)\to i_* i^*(N_1(D))$ is an isomorphism of simplicial sets. In particular, we have