I need to go home

hmlgy.tex

@@ -888,7 +888,14 @@ The previous proposition admits the following corollary, which will be of great
 \begin{proof}
   The fact that $A$,$B$ and $C$ are free and one of the morphism $u$ or $f$ is a folk cofibration ensure that the square is folk homotopy cocartesian. The conclusion follows then from Proposition \ref{prop:criteriongoodcat}.
 \end{proof}
-\section{Polygraphic homology and truncation}
+\section{Equivalence of homologies in low dimension}
+\begin{paragr}
+  Recall that for every $n \geq 0$ we have taken the habit of identifying $n\Cat$ as a full subcategory of $\oo\Cat$ via the canonical fully faithful functor $\iota_n : n\Cat \to \oo\Cat$ (defined in \ref{paragr:defncat}) that sends an $n$\nbd-category $C$ to the $\oo$\nbd-category with the same $k$\nbd-cells as $C$ for $k\leq n$ and only unit cells for $k > n$. In particular, we abusively wrote
+  \[
+  C=\iota_n(C).
+  \]
+  Within this section, \emph{and only within this section}, we try not to make this abuse of notation and explicitly write $\iota_n$ whenever we should. 
+  \end{paragr}
 \begin{paragr}
  Let $\Ch^{\leq n}$ be the category of chain complexes in degree comprised between $0$ and $n$. This means that an object $K$ of $\Ch^{\leq n}$ is a diagram of abelian groups of the form
  \[
@@ -900,14 +907,14 @@ The previous proposition admits the following corollary, which will be of great
  \]
  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 $C$, seen as an $\oo$\nbd-category via the canonical inclusion $\iota_n : n\Cat \to \oo\Cat$, the chain complex $\lambda(C)$ is such that
+ Notice now that for an $n$\nbd-category $C$, the chain complex $\lambda(\iota_n(C))$ is such that
  \[
- \lambda(C)_k=0
+ \lambda(\iota_n(C))_k=0
  \]
- for every $k > n$ and hence $\lambda(C)$ 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
+ 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 \oo\Cat\\
-   C&\mapsto \lambda(C).
+   \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
  %% \[
@@ -918,11 +925,11 @@ The previous proposition admits the following corollary, which will be of great
  %% \]
  %% 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
+ 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
   \[
   x \sim y \text{ when there exists } z : x \to y \text{ in } C_{n+1}.
   \]
-  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 $\tau^{i}_{\leq n}(C) = C$.
+ % 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, for a chain complex $K$, write $\tau^{i}_{\leq n}(K)$ for the object of $\Ch^{\leq n}$ defined as
   \[
   K_0 \overset{\partial}{\longleftarrow} K_1 \overset{\partial}{\longleftarrow} K_2 \overset{\partial}{\longleftarrow} \cdots \overset{\partial}{\longleftarrow} K_{n-1} \overset{\partial}{\longleftarrow} K_{n}/{\partial(K_{n+1})}.
@@ -933,6 +940,20 @@ The previous proposition admits the following corollary, which will be of great
     \]
     which is left adjoint to the canonical inclusion $\iota_n : \Ch^{\leq n} \to \Ch$. 
 \end{paragr}
+\begin{lemma}
+  For every chain complex $K$, the unit map
+  \[
+  K \to \iota_n\tau^{i}_{\leq n}(K)
+  \]
+  induces isomorphisms
+  \[
+  H_k(K) \simeq H_k(\iota_n\tau^{i}_{\leq n}(K))
+  \]
+  for every $0 \leq k \leq n$.
+\end{lemma}
+\begin{proof}
+  
+\end{proof}
 \begin{lemma}
   The square
   \[
@@ -948,13 +969,13 @@ The previous proposition admits the following corollary, which will be of great
 \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}
-  Let $n \geq 0$ and $C$ be an $\oo$\nbd-category. If $C$ has a $k$\nbd-basis for every $ 0 \leq k < n$, then the canonical map of $\ho(\Ch)$
+  Let $n \geq 0$ 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 $\ho(\Ch)$
   \[
   \alpha^{\pol}_C : \sH^{\pol}(C) \to \lambda(C)
   \]
   induces isomorphisms
   \[
-  H_k^{\pol}(C) \to H_k(\lambda(C))
+  H_k^{\pol}(C) \simeq H_k(\lambda(C))
   \]
   for every $0 \leq k \leq n$.
 \end{proposition}
@@ -962,17 +983,17 @@ With this lemma at hand we can prove the important following proposition which b
   \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) the $0$-th and first homology groups of the chain complex $\lambda(C)$. This means that no cofibrant resolution of $C$ is needed to compute its first two polygraphic homology groups.
+  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 $n \geq 0$. The canonical map
   \[
-  \sH^{\pol}(C) \to \sH^{\pol}(\tau^{i}_{\leq n}(C))
+  \sH^{\pol}(C) \to \sH^{\pol}(\iota_n\tau^{i}_{\leq n}(C))
   \]
   induces isomorphisms
   \[
-  H^{\pol}_k(C) \simeq H^{\pol}_k(\tau^{i}_{\leq n}(C))
+  H^{\pol}_k(C) \simeq H^{\pol}_k(\iota_n\tau^{i}_{\leq n}(C))
   \]
   for every $0 \leq k \leq n$.
 \end{proposition}
@@ -997,15 +1018,59 @@ and for $n \in \mathbb{N}\cup \{\oo\}$ we write $c_n : \Psh{\Delta} \to n\Cat$ f
 \begin{proof}
 \todo{À écrire (facile)}  
 \end{proof}
-Slightly less trivial is the following lemma.
+\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 $n \geq 0$, a canonical morphism of $n\Cat$
+  \[
+  c_nN_{\oo}(C) \simeq \tau^{i}_{\leq n}c_{\oo}N_{\oo}(C) \to \tau^{i}_{\leq n}(C),
+  \]
+  which is natural in $C$.
+\end{paragr}
 \begin{lemma}
-  The following triangle of functors
+  For every $\oo$\nbd-category $C$, the canonical morphism of $\Cat$
   \[
-  \begin{tikzcd}
-    \oo\Cat \ar[r,"\tau^{i}_{\leq 1}"] \ar[d,"N_{\oo}"] & \Cat \\
-    \Psh{\Delta} \ar[ru,"c_1"'] &
-  \end{tikzcd}
+  c_1N_{\oo}(C) \to \tau^{i}_{\leq 1}(C)
   \]
-  is commutative (up to an isomorphism).
+  is an isomorphism.
 \end{lemma}
-\section{Homology and Homotopy of $\oo$-categories in low dimension}
+\begin{proof}
+  \todo{À écrire.}
+\end{proof}
+We can now prove the important following proposition.
+\begin{proposition}
+  For every $\oo$\nbd-category $C$, the canonical map of $\ho(\Ch)$
+  \[
+  \alpha^{\sing}: \sH^{\sing}(C) \to \lambda(C)
+  \]
+  induces isomorphisms
+  \[
+  H^{\sing}_k(C) \simeq H_k(\lambda(C))
+  \]
+  for $k \in \{0,1\}$.
+\end{proposition}
+\begin{proof}
+  \todo{À écrire}
+\end{proof}
+Finally, we obtain the result we were aiming for.
+\begin{proposition}
+  For every $\oo$\nbd-category $C$, the canonical comparison map
+  \[
+  \pi_C : \sH^{\sing}(C) \to \sH^{\pol}(C)
+  \]
+  induces isomorphisms
+  \[
+  H^{\sing}_k(C) \simeq H^{\pol}_k(C)
+  \]
+  for $k \in \{0,1\}$.
+\end{proposition}
+%% Slightly less trivial is the following lemma.
+%% \begin{lemma}
+%%   The following triangle of functors
+%%   \[
+%%   \begin{tikzcd}
+%%     \oo\Cat \ar[r,"\tau^{i}_{\leq 1}"] \ar[d,"N_{\oo}"] & \Cat \\
+%%     \Psh{\Delta} \ar[ru,"c_1"'] &
+%%   \end{tikzcd}
+%%   \]
+%%   is commutative (up to an isomorphism).
+%% \end{lemma}
+%%\section{Homology and Homotopy of $\oo$-categories in low dimension}