@@ -771,3 +771,5 @@ For any $n \geq 0$, consider the following cocartesian square

\begin{proposition}

Let $m,n \geq0$ and consider the $2$-category $A_{(m,n)}$. If $m\neq0$ or $n\neq0$, then $A_{(m,n)}$ is \good{} and has the homotopy type of a point. If $n=m=0$, then $A_{(0,0)}$ is not \good{} and has the homotopy type of a $K(\mathbb{Z},2)$.

\end{proposition}

\section{Zoology of $2$-categories: A stub of a criterion}

@@ -56,7 +56,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{

For an $\oo$-category $X$, $\sH(X)$ is the \emph{homology of $X$}.

\end{definition}

\begin{paragr}

In simpler words, the homology of an $\oo$-category $X$ is, by definition, the homology of its nerve. We will sometimes call this the \emph{``true'' homology of $X$} in order to distinguish it from other homological invariants that we shall introduce later.

In simpler words, the homology of an $\oo$-category $X$ is, by definition, the homology of its nerve. We will sometimes call this the \emph{Street homology of $X$} in order to distinguish it from other homological invariants that we shall introduce later.

Recall that we consider localization functors as identity on objects (see \ref{paragr:loc}). Hence, the homology of $X$ is simply $\kappa(N_{\oo}(X))$, only considered as a chain complex up to quasi-isomorphism, i.e. an object of $\ho(\Ch)$. The homology groups of $X$ are the homology groups of the chain complex $\kappa(N_{\oo}(X))$. However, with our definition, the \emph{homology of $X$} means something more precise than the mere sequence of homology groups. An alternative terminology would be to call $\sH(X)$ the \emph{homology type of $X$}, in reference to the homotopy type of a topological space.

\end{paragr}

\begin{remark}

...

...

@@ -214,8 +214,8 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{

From Theorem \ref{thm:cisinskiII}, this morphism of op-prederivators is cocontinuous. This property will be extremely important in the sequel.

\section{``True'' homology as derived abelianization}

We have seen in the previous section that the polygraphic homology is the left derived of the abelianization functor with respect to the canonical weak equivalences on $\oo\Cat$.\todo{Uniformiser appellations} As it turns out, the abelianization functor is also left derivable with respect to the Thomason weak equivalences and the left derived functor is the ``true'' homology functor for $\oo$-categories. In order to prove this result, we need a few lemmas.

\section{Street homology as derived abelianization}

We have seen in the previous section that the polygraphic homology is the left derived of the abelianization functor with respect to the canonical weak equivalences on $\oo\Cat$.\todo{Uniformiser appellations} As it turns out, the abelianization functor is also left derivable with respect to the Thomason weak equivalences and the left derived functor is the Street homology functor for $\oo$-categories. In order to prove this result, we need a few lemmas.

\begin{lemma}\label{lemma:nuhomotopical}

Let $\nu : \Ch\to\oo\Cat$ be the right adjoint to the abelianization functor (see Lemma \ref{lemma:adjlambda}). This functor sends weak equivalences of chain complexes to Thomason weak equivalences.

\end{lemma}

...

...

@@ -250,7 +250,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{

\[

\LL\lambda : \Ho(\oo\Cat^{\Th})\to\Ho(\Ch)

\]

is isomorphic to the ``true'' homology

is isomorphic to the Street homology

\[

\sH : \Ho(\oo\Cat^{\Th})\to\Ho(\Ch).

\]

...

...

@@ -293,7 +293,7 @@ In particular, this means that we have a morphism of localizers $\kappa : (\Psh{

\overline{\kappa}\overline{N_{\oo}} : \Ho(\oo\Cat^{\Th})\ar[r,shift left]&\ar[l,shift left]\Ho(\Ch) : M \overline{N_{\oo}}\overline{\nu}\simeq\overline{\nu}.

\end{tikzcd}

\]

From \todo{Critère de Gonzalez version dérivateur à écrire !}, we conclude that $\lambda : \oo\Cat\to\Ch$ is strongly left derivable and that $\LL^{\Th}\lambda\simeq\overline{\kappa}\overline{N_{\oo}}$, which is, by definition, the ``true'' homology.

From \todo{Critère de Gonzalez version dérivateur à écrire !}, we conclude that $\lambda : \oo\Cat\to\Ch$ is strongly left derivable and that $\LL^{\Th}\lambda\simeq\overline{\kappa}\overline{N_{\oo}}$, which is, by definition, the Street homology.

\end{proof}

\begin{remark}

Beware that neither $c_{\oo} : \Psh{\Delta}\to\oo\Cat$ nor $\lambda : \oo\Cat\to\Ch$ preserve weak equivalences \todo{Uniformiser appellations}, but this does not contradict the fact that $\lambda c_{\oo} : \Psh{\Delta}\to\Ch$ does.

...

...

@@ -437,7 +437,7 @@ We shall now proceed to give an abstract criterion to find \good{} $\oo$-categor

\[

\sH^{\pol} : \Ho(\oo\Cat^{\folk})\to\Ho(\Ch)

\]

and the ``true'' homology

and the Street homology

\[

\sH : \Ho(\oo\Cat^{\Th})\to\Ho(\Ch)

\]

...

...

@@ -496,3 +496,5 @@ 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{Homology and Homotopy of $\oo$-categories in low dimension}

\abstract{In this dissertation, we study the homology of strict $\oo$-categories. More precisely, we intend to compare the ``classical'' homology of an $\oo$-category (defined as the homology of its Street nerve) with its polygraphic homology. Along the way, we prove several important result concerning free strict $\oo$\nbd-categories over polygraphs-or-computad and concerning the homotopy theory of strict $\oo$\nbd-categories. }