@@ -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 $n \geq0$ 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 $n \geq0$, 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

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

\[

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

@@ -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 $n \geq0$ 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 $n \geq0$ 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)

\]

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 $n \geq0$. The canonical map

@@ -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 $n \geq0$, a canonical morphism of $n\Cat$