Commit 4b99831d authored by Leonard Guetta's avatar Leonard Guetta
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parent 3e739f45
......@@ -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
%% \[
%% \begin{tikzcd}
%% n\Cat \ar[d] \ar[r,"\lambda_{\leq n}"] & \Ch^{\leq n} \ar[d] \\
%% \oo\Cat \ar[r,"\lambda"] & \Ch.
%% \end{tikzcd}
%% \]
%% This functor being
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
\[
......@@ -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
\[
H_n(K)=\frac{\mathrm{Ker}(\partial : K_n \to K_{n-1})}{\mathrm{Im}(\partial : K_{n+1} \to K_n)}
\]
is isomorphic to
\[
H_n(\iota_n\tau^{i}_{\leq n}(K))=\mathrm{Ker}(\partial : K_n/{\partial(K_{n+1})} \to K_{n-1}).
\]
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
\begin{align*}
\lambda_{\leq n} : n\Cat &\to \Ch^{\leq n}\\
C&\mapsto \lambda(\iota_n(C)).
\end{align*}
We tautologically have that the square
\[
\begin{tikzcd}
n\Cat \ar[d,"\iota_n"] \ar[r,"\lambda_{\leq n}"] & \Ch^{\leq n} \ar[d,"\iota_n"] \\
\oo\Cat \ar[r,"\lambda"] & \Ch
\end{tikzcd}
\]
is commutative.
\end{paragr}
\begin{lemma}
The square
\[
......@@ -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 $H^{\pol}_k(C)$.
\begin{proposition}
......
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