@@ -907,23 +907,6 @@ 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 $k >n$.

Notice now that for an $n$\nbd-category $C$, the chain complex $\lambda(\iota_n(C))$ is such that

\[

\lambda(\iota_n(C))_k=0

\]

for every $k > n$ and hence $\lambda(\iota_n(C))$ can be seen as an object of $\Ch^{\leq n}$. Thus, we can define a functor $\lambda_{\leq n } : n\Cat\to\Ch^{\leq n}$ as

\begin{align*}

\lambda_{\leq n} : n\Cat&\to\Ch^{\leq n}\\

C&\mapsto\lambda(\iota_n(C)).

\end{align*}

%% this means that there exists a unique functor $\lambda_{\leq n} : n\Cat \to \Ch^{\leq n}$ such that the following square is commutative

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

\[

...

...

@@ -952,8 +935,35 @@ The previous proposition admits the following corollary, which will be of great

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

\end{lemma}

\begin{proof}

For $0\leq k < n-1$, this is trivial. For $k = n-1$, this follows easily from the fact that the image of $\partial : K_k/{\partial(K_{k+1})}\to K_{k-1}$ is equal to the image of $\partial : K_k \to K_{k+1}$. Finally for $k = n$, it is straightforward to check that

The isomorphism being obviously induced by the unit map $K \to\iota_n\tau^{i}_{\leq n}(K)$.

\end{proof}

\begin{paragr}

Notice now that for an $n$\nbd-category $C$, the chain complex $\lambda(\iota_n(C))$ is such that

\[

\lambda(\iota_n(C))_k=0

\]

for every $k > n$ and hence $\lambda(\iota_n(C))$ can be seen as an object of $\Ch^{\leq n}$. Thus, we can define a functor $\lambda_{\leq n } : n\Cat\to\Ch^{\leq n}$ as

@@ -965,7 +975,7 @@ The previous proposition admits the following corollary, which will be of great

is commutative (up to a canonical isomorphism).

\end{lemma}

\begin{proof}

\todo{À écrire}

Let $C$ be an $\oo$\nbd-category. For $k<n$, we obviously have that

\end{proof}

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 $H^{\pol}_k(C)$.