qlkjfdlqkj

hmlgy.tex

@@ -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}