\chapter{Homollogy of ontractible $\oo$\nbd-categories its consequences}

\section{Contractible $\oo$-categories}

Recall that $\sD_0$ is the terminal object of $\oo\Cat$ that for any $oo$\nbd-category, we denote by $p_X : X \to\sD_0$ the canonical morphism.

\begin{definition}

An $\oo$\nbd-category $X$ is \emph{contractible} when there exists an object $x_0$ of $X$ and an oplax transformation

\[

\begin{tikzcd}

X \ar[r,"p_X"]\ar[rd,"1_X",""{name=A,above}]&\sD_0\ar[d,"\langle x_0\rangle"]\\

&X.

\ar[from=A,to=1-2,"\alpha"]

\end{tikzcd}

\]

\end{definition}

\begin{remark}

The previous definition admits many variations as we could use lax transformations instead oplax ones and we could change the direction of the (lax or oplax) transformation. All the results of this section could straightforwardly be adapted to all the variations of the definition of contractible $\oo$\nbd-category. Hopefully, the choice we made is consistent with the rest of the dissertation.

\end{remark}

\begin{paragr}

In other words, an $\oo$\nbd-category $X$ is contractible when there exists an object $x_0$ of $X$ such that $\langle x_0\rangle\sD_0\to X$ is a deformation retract (Paragraph \ref{paragr:defrtract}). It follows from Lemma \ref{lemma:oplacloc} that $p_X : X \to\sD_0$ is a Thomason weak equivalence. In particular, we have the following corollary.

\end{paragr}

\begin{corollary}

Let $X$ be a contractible $\oo$\nbd-category. The morphism of $\ho(\Ch)$

\[

\sH(X)\to\sH(\sD_0)

\]

induced by the canonical morphism $p_X : X \to\sD_0$ is an isomorphism.

\end{corollary}

\begin{paragr}

In addition to the previous result, it is immediate to check that $\sH(\sD_0)$ is nothing but $\mathbb{Z}$ seen as an object of $\ho(\Ch)$ concentrated in degree $0$.

We would like now to study the polygraphic homology of contractible $\oo$\nbd-categories. In order to do so, we need some technical results.