\chapter{Homology and abelianization of \texorpdfstring{$\oo$}{ω}-categories}
\chaptermark{Homology of $\omega$-categories}
\section{Homology via the nerve}
\begin{paragr}
We denote by $\Ch$ the category of non-negatively graded chain complexes of abelian groups. Recall that $\Ch$ can be equipped with a cofibrantly generated model structure, known as the \emph{projective model structure on $\Ch$}, where:
\begin{itemize}
\item[-] the weak equivalences are the quasi-isomorphisms, i.e.\ morphisms of chain complexes that induce an isomorphism on homology groups,
\item[-] the cofibrations are the morphisms of chain complexes $f: X\to Y$ such that for every $n\geq 0$, $f_n : X_n \to Y_n$ is a monomorphism with projective cokernel,
\item[-] the fibrations are the morphisms of chain complexes $f : X \to Y$ such that for every $n>0$, $f_n : X_n \to Y_n$ is an epimorphism.
\end{itemize}
(See for example \cite[Section 7]{dwyer1995homotopy}.)
From now on, we will implicitly consider that the category $\Ch$ is equipped with this model structure.
\end{paragr}
\begin{paragr}
Let $X$ be a simplicial set. We denote by $K_n(X)$ the abelian group of $n$\nbd{}chains of $X$, i.e.\ the free abelian group on the set $X_n$. For $n>0$, let $\partial : K_n(X) \to K_{n-1}(X)$ be the linear map defined for $x \in X_n$ by
\[
\partial(x):=\sum_{i=0}^n(-1)^i\partial_i(x).
\]
It follows from the simplicial identities (see \cite[section 2.1]{gabriel1967calculus}) that $\partial \circ \partial = 0$. Hence, the previous data defines a chain complex $K(X)$ and this defines a functor
\begin{align*}
K : \Psh{\Delta} &\to \Ch\\
X &\mapsto K(X)
\end{align*}
in the expected way.
\end{paragr}
\begin{paragr}
Recall that an $n$-simplex $x$ of a simplicial set $X$ is \emph{degenerate} if there exists an epimorphism $\varphi : [n] \to [m]$ with $m0$. Hence, there is an induced differential which we still denote by $\partial$:
\[
\partial : \kappa_n(X) \to \kappa_{n-1}(X).
\]
This defines a chain complex $\kappa(X)$, which we call the \emph{normalized chain complex of $X$}. This yields a functor
\begin{align*}
\kappa : \Psh{\Delta} &\to \Ch \\
X &\mapsto \kappa(X).
\end{align*}
\end{paragr}
\begin{lemma}\label{lemma:normcompquil}
The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends the weak equivalences of simplicial sets to quasi-isomorphisms.
\end{lemma}
\begin{proof}
Recall that the Quillen model structure on simplicial sets admits the set of inclusions
\[
\{\partial\Delta_n \hookrightarrow \Delta_n \vert n \in \mathbb{N} \}
\]
as generating cofibrations and the set of inclusions
\[
\{\Lambda^i_n \hookrightarrow \Delta_n \vert n \in \mathbb{N}, 0 \leq i \leq n\}
\]
as generating trivial cofibrations (see for example \cite[Section I.1]{goerss2009simplicial} for the notations). A quick computation, which we leave to the reader, shows that the image by $\kappa$ of $\partial\Delta_n \hookrightarrow \Delta_n$ is a monomorphism with projective cokernel and the image by $\kappa$ of $\Lambda^i_n \hookrightarrow \Delta_n$ is a quasi-isomorphism. This proves that $\kappa$ is left Quillen. Since all simplicial sets are cofibrant, it follows from Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model} that $\kappa$ also preserves weak equivalences.
\end{proof}
\begin{remark}
The previous lemma admits also a more conceptual proof as follows. From the Dold--Kan equivalence, we know that $\Ch$ is equivalent to the category $\Ab(\Delta)$ of simplicial abelian groups and with this identification the functor $\kappa : \Psh{\Delta} \to \Ch$ is left adjoint of the canonical forgetful functor
\[
U : \Ch \simeq \Ab(\Delta) \to \Psh{\Delta}
\]
induced by the forgetful functor from abelian groups to sets. The fact that $U$ is right Quillen follows then from \cite[Lemma 2.9 and Corollary 2.10]{goerss2009simplicial}.
\end{remark}
\begin{paragr}
In particular, $\kappa$ induces a morphism of localizers \[\kappa : (\Psh{\Delta},\W_{\Delta}) \to (\Ch,\W_{\Ch}),\]
where we wrote $\W_{\Ch}$ for the class of quasi-isomorphisms.
\end{paragr}
\begin{definition}\label{def:hmlgycat}
The \emph{singular homology functor for $\oo$\nbd{}categories} $\sH^{\sing}$ is defined as the following composition
\[
\sH^{\sing} : \ho(\oo\Cat^{\Th}) \overset{\overline{N_{\omega}}}{\longrightarrow} \ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \ho(\Ch).
\]
For an $\oo$\nbd{}category $C$, $\sH^{\sing}(C)$ is the \emph{singular homology of $C$}.
\end{definition}
\begin{paragr}\label{paragr:singularhmlgygroup}
In other words, the singular homology of $C$ is the chain complex $\kappa(N_{\oo}(C))$ seen as an object of $\ho(\Ch)$ (see Remark \ref{remark:localizedfunctorobjects}). For $k \geq 0$, the $k$\nbd{}th singular homology group of an $\oo$\nbd{}category $C$ is defined as
\[
H_k^{\sing}(C):=H_k(\sH^{\sing}(C))=H_k(\kappa(N_{\oo}(C))),
\]
where $H_k : \ho(\Ch) \to \Ab$ is the usual functor that associates to an object of $\ho(\Ch)$ its $k$\nbd{}th homology group.
\end{paragr}
%% \begin{paragr}
%% In simpler words, the homology of an $\oo$\nbd{}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}\label{remark:singularhmlgyishmlgy}
The adjective ``singular'' is there to avoid future confusion with another
homological invariant for $\oo$\nbd{}categories that will be introduced later.
As a matter of fact, the underlying point of view adopted in this thesis is
that \emph{singular homology of $\oo$\nbd{}categories} ought to be simply called
\emph{homology of $\oo$\nbd{}categories} as it is the only ``correct''
definition of homology. This assertion will be justified in Remark \ref{remark:polhmlgyisnotinvariant}.
\end{remark}
\begin{remark}
We could also have defined the homology of $\oo$\nbd{}categories with $K : \Psh{\Delta}\to \Ch$ instead of $\kappa : \Psh{\Delta} \to \Ch$ since these two functors are quasi-isomorphic (see \cite[Theorem 2.4]{goerss2009simplicial} for example). An advantage of the latter one is that it is left Quillen.
\end{remark}
\begin{paragr}
We will also denote by $\sH^{\sing}$ the morphism of op-prederivators defined as the following composition
\[
\sH^{\sing} : \Ho(\oo\Cat^{\Th}) \overset{\overline{N_{\omega}}}{\longrightarrow} \Ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \Ho(\Ch).
\]
\end{paragr}
\begin{proposition}\label{prop:singhmlgycocontinuous}
The singular homology \[\sH^{\sing} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)\] is homotopy cocontinuous.
\end{proposition}
\begin{proof}
This follows from the fact that $\overline{N_{\oo}}$ and $\overline{\kappa}$ are both homotopy cocontinuous. For $\overline{N_{\oo}}$, this is because it is an equivalence of op\nbd{}prederivators and thus we can apply Lemma \ref{lemma:eqisadj} and Lemma \ref{lemma:ladjcocontinuous}. For $\overline{\kappa}$, this is because $\kappa$ is left Quillen and thus we can apply Theorem \ref{thm:cisinskiII}.
\end{proof}
\section{Abelianization}
We write $\Ab$ for the category of abelian groups and for an abelian group $G$, we write $\vert G \vert$ for the underlying set of $G$.
\begin{paragr}
Let $C$ be an $\oo$\nbd{}category. For every $n\geq 0$, we define $\lambda_n(C)$ as the abelian group obtained by quotienting $\mathbb{Z}C_n$ (the free abelian group on $C_n$) by the congruence generated by the relations
\[
x \comp_k y \sim x+y
\]
for all $x,y \in C_n$ that are $k$\nbd{}composable for some $k0$, consider the linear map
\begin{align*}
\mathbb{Z}C_n &\to \mathbb{Z}C_{n-1}\\
x \in C_n &\mapsto t(x)-s(x).
\end{align*}
The axioms of $\oo$\nbd{}categories imply that it induces a map
\[
\partial : \lambda_{n}(C) \to \lambda_{n-1}(C)
\]
which is natural in $C$. Furthermore, it satisfies the equation $\partial \circ \partial = 0$. Thus, for every $\oo$\nbd{}category $C$, we have defined a chain complex $\lambda(C)$:
\[
\lambda_0(C) \overset{\partial}{\longleftarrow} \lambda_1(C) \overset{\partial}{\longleftarrow} \lambda_2(C) \overset{\partial}{\longleftarrow} \cdots
\]
and for every $f : C \to D$ a morphism of chain complexes
\[
\lambda(f) : \lambda(C) \to \lambda(D).
\]
Altogether, this defines a functor
\[
\lambda : \omega\Cat \to \Ch,
\]
which we call the \emph{abelianization functor}.
\end{paragr}
\begin{lemma}\label{lemma:adjlambda}
The functor $\lambda$ is a left adjoint.
\end{lemma}
\begin{proof}
The category $\Ch$ is equivalent to the category $\omega\Cat(\Ab)$ of $\oo$\nbd{}categories internal to abelian groups (see \cite[Theorem 3.3]{bourn1990another}) and with this identification, the functor $\lambda : \omega\Cat \to \omega\Cat(\Ab)$ is nothing but the left adjoint of the canonical forgetful functor $\omega\Cat(\Ab) \to \omega\Cat$.
\end{proof}
As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain complex $\lambda(C)$ admits a nice expression.
\begin{paragr}
Let $n\geq 0$. Recall that for every monoid $M$ (supposed commutative if $n \geq 1$) we have defined in Section \ref{sec:suspmonoids} an $n$\nbd{}category $B^nM$ whose set of $n$\nbd{}cells is isomorphic to the underlying set of $M$. And the correspondence $M \mapsto B^nM$ defines a functor in the expected way. By considering abelian groups as particular cases of (commutative) monoids, we obtain a functor for each $n\geq 0$
\begin{align*}
B^n : \Ab &\to n\Cat \\
G &\mapsto B^nG,
\end{align*}
where $\Ab$ is the category of abelian groups.
Besides, let us write $\lambda_n$ again for the functor
\begin{align*}
\lambda_n : n\Cat &\to \Ab\\
C&\mapsto \lambda_n(C).
\end{align*}
(That is the restriction of $\lambda_n : \oo\Cat \to \Ab$ to $n\Cat$.)
\end{paragr}
\begin{lemma}\label{lemma:adjlambdasusp}
Let $n \geq 0$. The functor $\lambda_n : n\Cat \to \Ab$ is left adjoint to the functor $B^n : \Ab \to n\Cat$.
\end{lemma}
\begin{proof}
The case $n=0$ is immediate since the functor $\lambda_0 : 0\Cat = \Set \to \Ab$ is the ``free abelian group'' functor and the functor $B^0 : \Ab \to 0\Cat=\Set$ is the ``underlying set'' functor.
Suppose now that $n >0$. From Lemma \ref{lemma:nfunctortomonoid} we know that for every abelian group $G$ and every $n$\nbd{}category $C$, the map
\begin{align*}
\Hom_{n\Cat}(C,B^nG) &\to \Hom_{\Set}(C_n,\vert G \vert)\\
F &\mapsto F_n,
\end{align*}
is injective and its image consists of those functions $f : C_n \to \vert G \vert$ such that:
\begin{enumerate}[label=(\roman*)]
\item\label{cond:comp} for every $0 \leq k 0$, notice that the map $\partial : \mathbb{Z}\Sigma_n \to \mathbb{Z}\Sigma_{n-1}$ given in the statement of the proposition is nothing but the composition
\[
\mathbb{Z}\Sigma_n \overset{\phi_n}{\longrightarrow} \lambda_n(C) \overset{\partial}{\longrightarrow} \lambda_{n-1}(C) \overset{\psi_{n-1}}{\longrightarrow} \mathbb{Z}\Sigma_{n-1}.
\]
The first part of the proposition follows then from Lemma \ref{lemma:abelpol}.
As for the second part, it suffices to notice that if we identify $\lambda_n(C)$ with $\mathbb{Z}\Sigma_n$ via $\phi_n$ for every free $\oo$\nbd{}category $C$, then map $\mathbb{Z}\Sigma_n \to \mathbb{Z}\Sigma'_n$ (where $\Sigma'_n$ is the $n$-basis of $C'$) induced by $F$ is given by the composition
\[
\mathbb{Z}\Sigma_n \overset{\phi_n}{\longrightarrow} \lambda_n(C) \overset{\lambda_n(F)}{\longrightarrow} \lambda_n(C') \overset{\psi_n}{\longrightarrow} \mathbb{Z}\Sigma'_n.\qedhere
\]
\end{proof}
\section{Polygraphic homology}\label{section:polygraphichmlgy}
\begin{paragr}\label{paragr:chainhmtpy}
Let $f,g : K \to K'$ be two morphisms of non-negatively graded chain
complexes. Recall that a \emph{chain homotopy} from $f$ to $g$ consists of a
sequence of linear maps $(h_n \colon K_n \to K'_{n+1})_{n \in \mathbb{N}}$
such that
\[
\partial \circ h_0 = g_0-f_0
\]
and such that for every $n > 0$, we have
\[
\partial \circ h_n + h_{n-1} \circ \partial = g_n - f_n.
\]
Recall also that if there is a chain homotopy from $f$ to $g$, then the
localization functor $\gamma^{\Ch} : \Ch \to \ho(\Ch)$ identifies $f$ and
$g$, which means that \[\gamma^{\Ch}(f)=\gamma^{\Ch}(g).\]
\end{paragr}
%For the definition of \emph{homotopy of chain complexes} see for example \cite[Definition 1.4.4]{weibel1995introduction} (where it is called \emph{chain homotopy}).
\begin{lemma}\label{lemma:abeloplax}
Let $u, v : C \to D$ be two $\oo$\nbd{}functors. If there is an oplax
transformation $\alpha : u \Rightarrow v$, then there is a chain homotopy from $\lambda(u)$ to $\lambda(v)$.
\end{lemma}
\begin{proof}
For an $n$-cell $x$ of $C$ (resp.\ $D$), let us use the notation $[x]$ for the image of $x$ in $\lambda_n(C)$ (resp.\ $\lambda_n(D)$).
Let $h_n$ be the map
\[
\begin{aligned}
h_n : \lambda_n(C) &\to \lambda_{n+1}(D)\\
[x] & \mapsto [\alpha_x].
\end{aligned}
\]
The formulas for oplax transformations from Paragraph
\ref{paragr:formulasoplax} imply that $h_n$ is linear and that for every
$n$-cell $x$ of $C$, if $n=0$, we have
\[
\partial(h_0(x))=[v(x)]-[u(x)],
\]
and if $n>0$, we have
\[
\partial (h_n(x)) + h_{n-1}(\partial(x)) = [v(x)] - [u(x)].
\]
Details are left to the reader.
\end{proof}
\begin{proposition}
The abelianization functor $\lambda : \oo\Cat \to \Ch$ is left Quillen with respect to the folk model structure on $\oo\Cat$.
\end{proposition}
\begin{proof}
The fact that $\lambda$ is a left adjoint is Lemma \ref{lemma:adjlambda}.
A simple computation using Lemma \ref{prop:abelianizationfreeoocat} shows that for every $n\in \mathbb{N}$,
\[
\lambda(i_n) : \lambda(\sS_{n-1}) \to \lambda(\sD_{n})
\]
is a monomorphism with projective cokernel. Hence $\lambda$ sends folk cofibrations to cofibrations of chain complexes.
Then, we know from \cite[Sections 4.6 and 4.7]{lafont2010folk} and \cite[Remarque B.1.16]{ara2016joint} (see also \cite[Paragraph 3.11]{ara2019folk}) that there exists a set of generating trivial cofibrations $J$ of the folk model structure on $\omega\Cat$ such that every $j : X \to Y$ in $J$ is a deformation retract (see Paragraph \ref{paragr:defrtract}).
From Lemma \ref{lemma:abeloplax}, we conclude that $\lambda$ sends folk trivial cofibrations to trivial cofibrations of chain complexes.
\end{proof}
In particular, $\lambda$ is totally left derivable (when $\oo\Cat$ is equipped with folk weak equivalences). This motivates the following definition.
\begin{definition}\label{de:polhom}
The \emph{polygraphic homology functor}
\[
\sH^{\pol} : \ho(\oo\Cat^{\folk}) \to \ho(\Ch)
\]
is the total left derived functor of $\lambda : \oo\Cat \to \Ch$ (where $\oo\Cat$ is equipped with folk weak equivalences). For an $\oo$\nbd{}category $C$, $\sH^{\pol}(C)$ is the \emph{polygraphic homology of $C$}.
\end{definition}
\begin{paragr}
Similarly to singular homology groups, for $k\geq0$ the $k$\nbd{}th polygraphic homology group of an $\oo$\nbd{}category $C$ is defined as
\[
H^{\pol}_k(C):=H_k(\sH^{\pol}(C))
\]
where $H_k : \ho(\Ch) \to \Ab$ is the usual functor that associate to an object of $\ho(\Ch)$ its $k$-th homology group. In practice, this means that one has to find a cofibrant replacement of $C$, that is to say a free $\oo$\nbd{}category $P$ and a folk trivial fibration
\[
P \to C,
\]
and then the polygraphic homology groups of $C$ are those of $\lambda(P)$ which are computed using Proposition \ref{prop:abelianizationfreeoocat}.
\end{paragr}
\begin{paragr}
For later reference, let us recall here that since $\sH^{\pol}$ is the left derived functor of $\lambda$, it comes equipped with a universal natural transformation (see \ref{paragr:defleftderived})
\[
\begin{tikzcd}
\oo\Cat \ar[d,"\gamma^{\folk}"'] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
\ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\pol}"'] & \ho(\Ch).
\ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >= 1em, Rightarrow]
\end{tikzcd}
\]
\end{paragr}
As we shall now see, oplax homotopy equivalences (Definition \ref{def:oplaxhmtpyequiv}) induce isomorphisms in polygraphic homology. In order to prove that, we first need a couple of technical lemmas.
\begin{lemma}\label{lemma:liftingoplax}
Let
\[
\begin{tikzcd}
C' \ar[r,"f_{\epsilon}'"] \ar[d,"u"] & D' \ar[d,"v"]\\
C \ar[r,"f_{\epsilon}"] & D
\end{tikzcd}
\]
be commutative squares in $\omega\Cat$ for $\epsilon\in\{0,1\}$.
If $C'$ is a free $\omega$-category and $v$ a folk trivial fibration, then for every oplax transformation \[\alpha : f_0 \Rightarrow f_1,\] there exists an oplax transformation \[\alpha' : f_0' \Rightarrow f_1'\] such that
\[
v \star \alpha' = \alpha \star u.
\]
\end{lemma}
\begin{proof}
Notice first that because of the natural isomorphism \[(\sD_0\amalg \sD_0) \otimes C \simeq C \amalg C\] we have that $\alpha : f_0 \Rightarrow f_1$ can be encoded in a functor $\alpha : \sD_1 \otimes C \to D$ such that the diagram
\[
\begin{tikzcd}
(\sD_0\amalg \sD_0) \otimes C \simeq C \amalg C \ar[d,"i_1 \otimes C"'] \ar[dr,"{\langle u, v \rangle}"] &\\
\sD_1 \otimes C \ar[r,"\alpha"'] & D
\end{tikzcd}
\]
(where $i_1 : \sD_0 \amalg \sD_0 \simeq \sS_0 \to \sD_1$ is the morphism introduced in \ref{paragr:inclusionsphereglobe}) is commutative.
Now, the hypotheses of the lemma yield the following commutative square
\[
\begin{tikzcd}
(\sD_0 \amalg \sD_0)\otimes C' \ar[d,"{i_1\otimes C'}"'] \ar[rr,"{\langle f'_0, f_1' \rangle}"] && D' \ar[d,"v"] \\
\sD_1\otimes C'\ar[r,"\sD_1 \otimes u"'] & \sD_1\otimes C \ar[r,"\alpha"] & D
\end{tikzcd}
\]
and since $i_1$ is a folk cofibration and $C'$ is cofibrant, it follows
that the left vertical morphism of the previous square is a folk
cofibration (see \cite[Proposition 5.1.2.7]{lucas2017cubical} or
\cite{ara2019folk}). By hypothesis, $v$ is a folk trivial fibration, and
so the above square admits a lift
\[
\alpha' : \sD_1\otimes C' \to D'.
\]
The commutativity of the two induced triangles shows what we needed to prove.
\end{proof}
From now on, for an $\oo$\nbd{}functor $u$, we write $\sH^{\pol}(u)$ instead of $\sH^{\pol}(\gamma^{\folk}(u))$ (where $\gamma^{\folk}$ is the localization functor $\oo\Cat \to \ho(\oo\Cat^{\folk})$) for the morphism induced by $u$ at the level of polygraphic homology.
\begin{lemma}\label{lemma:oplaxpolhmlgy}
Let $u,v : C \to D$ be two $\oo$\nbd{}functors. If there exists an oplax transformation $u\Rightarrow v$, then
\[
\sH^{\pol}(u)=\sH^{\pol}(v).
\]
\end{lemma}
\begin{proof}
In the case that $C$ and $D$ are both folk cofibrant, this follows immediately
from Lemma \ref{lemma:abeloplax} and the fact that the localization functor
$\Ch \to \ho(\Ch)$ identifies chain homotopic maps (\ref{paragr:chainhmtpy}).
In the general case, let
\[
p : C' \to C
\]
and
\[
q : D' \to D
\]
be trivial fibrations for the canonical model structure with $C'$ and $D'$ cofibrant. Using that $q$ is a trivial fibration and $C'$ is cofibrant, we know that there exists $u' : C' \to D'$ and $v' : C' \to D'$ such that the squares
\[
\begin{tikzcd}
C' \ar[d,"p"] \ar[r,"u'"] & D' \ar[d,"q"] \\
C \ar[r,"u"] & D
\end{tikzcd}
\text{ and }
\begin{tikzcd}
C' \ar[d,"p"] \ar[r,"v'"] & D' \ar[d,"q"] \\
C \ar[r,"v"] & D
\end{tikzcd}
\]
are commutative. From Lemma \ref{lemma:liftingoplax}, we deduce the existence of an oplax transformation $u' \Rightarrow v'$. Since $C'$ and $D'$ are cofibrant, we have already proved that
\[\sH^{\pol}(u')=\sH^{\pol}(v').\]
The commutativity of the two previous squares and the fact that $p$ and $q$ are folk weak equivalences imply the desired result.
\end{proof}
The following proposition is an immediate consequence of the previous lemma.
\begin{proposition}\label{prop:oplaxhmtpypolhmlgy}
Let $u : C \to D$ be an $\oo$\nbd{}functor. If $u$ is an oplax homotopy equivalence, then the induced morphism
\[
\sH^{\pol}(u) : \sH^{\pol}(C) \to \sH^{\pol}(D)
\]
is an isomorphism.
\end{proposition}
\begin{paragr}\label{paragr:polhmlgythomeq}
Oplax homotopy equivalences being particular cases of Thomason equivalences, one may wonder whether it is true that \emph{every} Thomason equivalence induce an isomorphism in polygraphic homology. As we shall see later (Proposition \ref{prop:polhmlgynotinvariant}), it is not the case.
\end{paragr}
\begin{remark}
Lemma \ref{lemma:liftingoplax}, Lemma \ref{lemma:oplaxpolhmlgy} and Proposition \ref{prop:oplaxhmtpypolhmlgy} are also true if we replace ``oplax'' by ``lax'' everywhere.
\end{remark}
\begin{paragr}
The functor $\lambda$ being left Quillen, it is strongly derivable (Definition \ref{def:strnglyder}) and hence also induces a morphism of op-prederivators, which we again denote by $\sH^{\pol}$:
\[
\sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch).
\]
Moreover, we also have a universal $2$-morphism which we again denote by $\alpha^{\pol}$:
\[
\begin{tikzcd}
\oo\Cat \ar[d,"\gamma^{\folk}"'] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
\Ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\pol}"'] & \Ho(\Ch).
\ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >= 1em, Rightarrow]
\end{tikzcd}
\]
\end{paragr}
The following proposition is an immediate consequence of Theorem \ref{thm:cisinskiII}.
\begin{proposition}\label{prop:polhmlgycocontinuous}
The polygraphic homology
\[
\sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch)
\]
is homotopy cocontinuous.
\end{proposition}
\section{Singular homology as derived abelianization}\label{section:singhmlgyderived}
We have seen in the previous section that the polygraphic homology functor is the total left derived functor of $\lambda : \oo\Cat \to \Ch$ when $\oo\Cat$ is equipped with the folk weak equivalences. As it turns out, the abelianization functor is also totally left derivable when $\oo\Cat$ is equipped with the Thomason equivalences and the total left derived functor is the singular homology functor. In order to prove this result, we first need a few technical lemmas.
\begin{lemma}\label{lemma:nuhomotopical}
Let $\nu : \Ch \to \oo\Cat$ be the right adjoint of the abelianization functor (see Lemma \ref{lemma:adjlambda}). This functor sends weak equivalences of chain complexes to Thomason equivalences.
\end{lemma}
\begin{proof}
We have already seen that $\lambda : \oo\Cat \to \Ch$ is left Quillen with respect to the folk model structure on $\oo\Cat$. By adjunction, this means that $\nu$ is right Quillen for this model structure. In particular, it sends trivial fibrations of chain complexes to folk trivial fibrations. From Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model} and the fact that all chain complexes are fibrant, it follows that $\nu$ sends weak equivalences of chain complexes to weak equivalences of the folk model structure, which are in particular Thomason equivalences (Lemma \ref{lemma:nervehomotopical}).
\end{proof}
\begin{remark}
The proof of the previous lemma shows the stronger result that $\nu$ sends weak equivalences of chain complexes to weak equivalences for the folk model structure on $\oo\Cat$. This will be of no use in the sequel.
\end{remark}
Recall that we write $c_{\oo} : \Psh{\Delta} \to \oo\Cat$ for the left adjoint of the nerve functor ${N_{\oo} : \oo\Cat \to \Psh{\Delta}}$ (see Paragraph \ref{paragr:nerve}).
\begin{lemma}\label{lemma:abelor}
The triangle of functors
\[
\begin{tikzcd}
\Psh{\Delta} \ar[r,"c_{\oo}"] \ar[dr,"\kappa"']& \oo\Cat\ar[d,"\lambda"]\\
&\Ch
\end{tikzcd}
\]
is commutative (up to a canonical isomorphism).
\end{lemma}
\begin{proof}
All the functors involved are cocontinuous, hence it suffices to prove that the triangle is commutative when pre-composed by the Yoneda embedding $\Delta \to \Psh{\Delta}$. This follows immediately from the description of the orientals in \cite{steiner2004omega}.
\end{proof}
Recall now that the notions of adjunction and equivalence are valid in every $2$-category and in particular in the $2$\nbd{}category of pre-derivators (see \ref{paragr:prederequivadjun}). We omit the proof of the following lemma, which is the same as when the ambient $2$-category is the $2$-category of categories.
\begin{lemma}\label{lemma:adjeq}
Let $\begin{tikzcd} f : y \ar[r,shift left]&z :g\ar[l,shift left] \end{tikzcd}$ be an adjunction and $h : x \to y$ an equivalence with quasi-inverse $k : y \to x$. Then $fh$ is left adjoint to $kg$.
\end{lemma}
We can now state and prove the promised result.
\begin{theorem}\label{thm:hmlgyderived}
Consider that $\oo\Cat$ is equipped with the Thomason equivalences. The abelianization functor $\lambda : \oo\Cat \to \Ch$ is strongly left derivable and the left derived morphism of op\nbd{}prederivators
\[
\LL \lambda^{\Th} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)
\]
is isomorphic to the singular homology
\[
\sH^{\sing} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch).
\]
\end{theorem}
\begin{proof}
Let $\nu$ be the right adjoint of the abelianization functor (see Lemma \ref{lemma:adjlambda}) and consider the following adjunctions
\[
\begin{tikzcd}
\Psh{\Delta} \ar[r,shift left,"c_{\omega}"] &\oo\Cat \ar[l,shift left,"N_{\omega}"] \ar[r,shift left,"\lambda"]& \Ch \ar[l,shift left,"\nu"]
\end{tikzcd}
\]
where the functors from left to right are the left adjoints.
We know that:
\begin{itemize}[label=-]
\item The functor $\nu$ induces a morphism of localizers
\[
\nu : (\Ch,\W_{\Ch}) \to (\oo\Cat,\W^{\Th}),
\]
thanks to Lemma \ref{lemma:nuhomotopical}.
\item The functor $N_{\omega}$ induces a morphism of localizers
\[
N_{\omega} : (\oo\Cat,\W^{\Th}) \to (\Psh{\Delta},\W_{\Delta}),
\]
by definition of Thomason equivalences.
\item There is an isomorphism of functors $\lambda c_{\omega} \simeq \kappa$ (Lemma \ref{lemma:abelor}), hence an induced morphism of localizers
\[
(\lambda c_{\omega})\simeq \kappa : (\Psh{\Delta},\W_{\Delta}) \to (\Ch,\W_{\Ch}),
\]
thanks to Lemma \ref{lemma:normcompquil}.
\end{itemize}
It follows that there is an induced adjunction at the level of op-prederivators:
\[
\begin{tikzcd}
\overline{\kappa} \simeq \overline{\lambda c_{\omega}} : \Ho(\Psh{\Delta}) \ar[r,shift left] & \ar[l,shift left] \Ho(\Ch) :\overline{N_{\omega}}\overline{\nu}.
\end{tikzcd}
\]
Now, we know from Theorem \ref{thm:gagna} that $\overline{N_{\omega}}$ is an equivalence of op\nbd{}prederivators, and thus admits a quasi-inverse. Let $ M : \Ho(\Psh{\Delta}) \to \Ho(\oo\Cat)$ be such a quasi-inverse. From Lemma \ref{lemma:adjeq}, we deduce that we have an adjunction:
\[
\begin{tikzcd}
\overline{\kappa} \overline{N_{\oo}} : \Ho(\oo\Cat^{\Th}) \ar[r,shift left]& \ar[l,shift left] \Ho(\Ch) : M \overline{N_{\oo}} \overline{\nu} \simeq \overline{\nu}.
\end{tikzcd}
\]
From Proposition \ref{prop:gonzalezcritder}, we conclude that $\lambda : \oo\Cat \to \Ch$ is strongly left derivable and that $ \LL\lambda^{\Th} \simeq \overline{\kappa} \overline{N_{\oo}}$, which is, by definition, the singular homology.
\end{proof}
\begin{remark}
Beware that neither $c_{\oo} : \Psh{\Delta} \to \oo\Cat$ sends all weak equivalences of simplicial sets to Thomason equivalences nor $\lambda : \oo\Cat \to \Ch$ sends all Thomason equivalences to quasi-isomorphisms. But this does not contradict the fact that $\lambda c_{\oo} : \Psh{\Delta} \to \Ch$ does send all weak equivalences of simplicial sets to quasi-isomorphisms.
\end{remark}
\begin{paragr}\label{paragr:univmor}
Since $\sH^{\sing} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)$ is the left derived
morphisms of op-prederivators of the abelianization functor, it comes with a universal $2$-morphism
\[
\begin{tikzcd}
\oo\Cat \ar[d,"\gamma^{\Th}"'] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
\Ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"'] & \Ho(\Ch).
\ar[from=2-1,to=1-2,"\alpha^{\sing}",shorten <= 1em, shorten >= 1em, Rightarrow]
\end{tikzcd}
\]
A thorough reading of the proofs of Proposition \ref{prop:gonzalezcritder} and Theorem \ref{thm:hmlgyderived} enables us to give the following description of $\alpha^{\sing}$. By post-composing the co-unit of the adjunction $c_{\oo} \dashv N_{\oo}$ with the abelianization functor, we obtain $2$-morphism
\[
\lambda c_{\oo} N_{\oo} \Rightarrow \lambda.
\]
Then $\alpha^{\sing}$ is nothing but the following composition of $2$\nbd{}morphisms
\[
\begin{tikzcd}[column sep=huge]
\oo\Cat \ar[d,"\gamma^{\Th}"]\ar[r,bend left,"\lambda",""{name=A,below}] \ar[r,"\lambda c_{\oo} N_{\oo}"',""{name=B,above}] & \Ch \ar[d,"\gamma^{\Ch}"] \\
\Ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"'] & \Ho(\Ch),
\ar[from=B,to=A,Rightarrow]\ar[from=1-1,to=2-2,phantom,"\simeq" description]
\end{tikzcd}
\]
where the square is commutative (up to an isomorphism) because $\sH^{\sing}\simeq\overline{\lambda c_{\oo}} \overline{N_{\oo}}$.
\end{paragr}
\section{Comparing homologies}
\begin{paragr}\label{paragr:cmparisonmap}
Recall from Proposition \ref{prop:folkisthom} that the identity functor on $\oo\Cat$ induces a morphism of localizers
\[(\oo\Cat,\W^{\folk}) \to (\oo\Cat,\W^{\Th}),\]
which in turn induces a functor
\[\J : \ho(\oo\Cat^{\folk}) \to \ho(\oo\Cat^{\Th}),\]
such that
\[
\gamma^{\Th} = \J \circ \gamma^{\folk}.
\]
Now, consider the following triangle
\begin{equation}\label{cmprisontrngle}
\begin{tikzcd}
\ho(\oo\Cat^{\folk}) \ar[r,"\J"] \ar[rd,"\sH^{\pol}"'] & \ho(\oo\Cat^{\Th}) \ar[d,"\sH^{\sing}"] \\
& \ho(\Ch).
\end{tikzcd}
\end{equation}
A natural question to ask is whether this triangle is commutative (up to an isomorphism). Since $\J$ is the identity on objects, this amounts to ask whether for every $\oo$\nbd{}category $C$ we have an isomorphism (natural in $C$)
\[
\sH^{\pol}(C)\simeq \sH^{\sing}(C).
\]
As it happens, this is not possible as the following counter-example, due to Ara and Maltsiniotis, shows.
\end{paragr}
\begin{paragr}[Ara and Maltsiniotis' counter-example]\label{paragr:bubble}
Write $\mathbb{N}=(\mathbb{N},+,0)$ for the commutative monoid of non-negative integers and let $C$ be the $2$\nbd{}category defined as
\[
C:=B^2\mathbb{N}
\]
(see \ref{paragr:suspmonoid}). As usual, we consider $C$ as an $\oo$\nbd{}category with only unit cells strictly above dimension $2$. This $\oo$\nbd{}category is free; namely its $k$\nbd{}basis is a singleton for $k=0$ and $k=2$, and the empty set otherwise. In particular $C$ is cofibrant for the folk model structure (Proposition \ref{prop:freeiscofibrant}) and it follows from Proposition \ref{prop:abelianizationfreeoocat} that $\sH^{\pol}(C)$ is given by the chain complex (seen as an object of $\ho(\Ch)$)
\[
\begin{tikzcd}[column sep=small]
\mathbb{Z} & 0 \ar[l] & \ar[l] \mathbb{Z} & \ar[l] 0 & \ar[l] 0 & \ar[l] \cdots
\end{tikzcd}
\]
Hence, the polygraphic homology groups of $B$ are given by
\[
H^{\pol}_k(C)=\begin{cases} \mathbb{Z} \text{ if } k=0,2\\ 0 \text{ in other cases.}\end{cases}
\]
On the other hand, it is proven in \cite[Theorem 4.9 and Example 4.10]{ara2019quillen} that (the nerve of) $C$ is a $K(\mathbb{Z},2)$. In particular, it has non-trivial singular homology groups in every even dimension. This proves that $\sH^{\pol}(C)$ is \emph{not} isomorphic to $\sH^{\sing}(C)$; which means that triangle \eqref{cmprisontrngle} cannot be commutative (up to an isomorphism).
\end{paragr}
Another consequence of the above counter-example is the following result, which we claimed in \ref{paragr:polhmlgythomeq}. Recall that given a morphism $u : C \to D$ of $\oo\Cat$, we write $\sH^{\pol}(u)$ instead of $\sH^{\pol}(\gamma^{\folk}(u))$.
\begin{proposition}\label{prop:polhmlgynotinvariant}
There exists at least one Thomason equivalence \[u : C \to D\] such that the induced morphism
\[
\sH^{\pol}(u) : \sH^{\pol}(C) \to \sH^{\pol}(D)
\]
is not an isomorphism of $\ho(\Ch)$.
\end{proposition}
\begin{proof}
Suppose the converse, which is that the functor
\[
\sH^{\pol} \circ \gamma^{\folk} : \oo\Cat \to \ho(\Ch)
\]sends Thomason equivalences to isomorphisms of $\ho(\Ch)$. Because of the inclusion $\W^{\folk} \subseteq \W^{\Th}_{\oo}$, the category $\ho(\oo\Cat^{\Th})$ may be identified with the localization of $\ho(\oo\Cat^{\folk})$ with respect to $\gamma^{\folk}(\W^{\Th}_{\oo})$ and then the localization functor is nothing but
\[
\J : \ho(\oo\Cat^{\folk}) \to \ho(\oo\Cat^{\Th}).
\]
From this observation and because of the hypothesis we made on Thomason equivalences inducing isomorphisms in polygraphic homology, we deduce the existence of a functor
\[
\overline{\sH^{\pol}} : \ho(\oo\Cat^{\Th}) \to \ho(\Ch)
\]
such that we have
\[
\overline{\sH^{\pol}}\circ \J = \sH^{\pol},
\]
and because of the equality $\gamma^{\Th} = \J \circ \gamma^{\folk}$, the universal natural transformation $\alpha^{\pol}$ now reads
\[
\begin{tikzcd}
\oo\Cat \ar[r,"\lambda"] \ar[d,"\gamma^{\Th}"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
\ho(\oo\Cat^{\Th}) \ar[r,"\overline{\sH^{\pol}}"'] & \ho(\Ch).
\ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >=1em,Rightarrow]
\end{tikzcd}
\]
Let us show that $(\overline{\sH^{\pol}},\alpha^{\pol})$ is the left derived functor of $\lambda$ when $\oo\Cat$ is equipped with the Thomason equivalences. Let $G$ and $\beta$ be as in the following $2$\nbd{}diagram
\[
\begin{tikzcd}
\oo\Cat \ar[r,"\lambda"] \ar[d,"\gamma^{\Th}=\J\circ \gamma^{\folk}"'] & \Ch \ar[d,"\gamma^{\Ch}"] \\
\ho(\oo\Cat^{\Th}) \ar[r,"G"'] & \ho(\Ch).
\ar[from=2-1,to=1-2,"\beta",shorten <= 1em, shorten >=1em,Rightarrow]
\end{tikzcd}
\]
Since $\sH^{\pol}$ is the left derived functor of $\lambda$ when $\oo\Cat$ is equipped with the folk weak equivalences, there exists a unique $\delta : G \circ \J \Rightarrow \sH^{\pol}$ that factorizes $\beta$ as
\[
\begin{tikzcd}
\oo\Cat \ar[r,"\lambda"] \ar[d,"\gamma^{\folk}"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
\ho(\oo\Cat^{\folk}) \ar[d,"\J"] \ar[r,"\sH^{\pol}"'] & \ho(\Ch)\\
\ho(\oo\Cat^{\Th}) \ar[ru,"G"',bend right] &.
\ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >=1em,Rightarrow]\\
\ar[from=3-1,to=2-2,"\delta"',shorten <= 1em, shorten >= 1em,Rightarrow]
\end{tikzcd}
\]
But since $\J$ acts as a localization functor, $\delta$ also factorizes uniquely as
\[
\begin{tikzcd}[column sep=small] \ho(\oo\Cat^{\folk}) \ar[r,"\J"] & \ho(\oo\Cat^{\Th}) \ar[r,bend left,"\overline{\sH^{\pol}}",""{name=A,below}] \ar[r,bend right, "G"',pos=16/30,""{name=B,above}] & \ho(\Ch). \ar[from=B,to=A,Rightarrow,"\delta'"]\end{tikzcd}
\]
Altogether we have that $\beta$ factorizes as
\[
\begin{tikzcd}
\oo\Cat \ar[r,"\lambda"] \ar[d,"\gamma^{\Th}"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
\ho(\oo\Cat^{\Th}) \ar[r,"\overline{\sH^{\pol}}",""{name=B,below}] & \ho(\Ch).
\ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >=1em,Rightarrow]
\ar[from=2-1,to=2-2,"G"',pos=16/30,bend right,""{name=A,above}]
\ar[from=A,to=B,Rightarrow,"\delta'"]
\end{tikzcd}
\]
The uniqueness of such a factorization follows from a similar argument which is left to the reader. This proves that $\overline{\sH^{\pol}}$ is the left derived functor of $\lambda$ when $\oo\Cat$ is equipped with the Thomason equivalences and in particular we have
\[
\sH^{\sing}\simeq \overline{\sH^{\pol}}.
\]
But since $\J$ is the identity on objects, this implies that for every $\oo$\nbd{}category $C$ we have
\[
\sH^{\sing}(C)\simeq \overline{\sH^{\pol}}(C)=\sH^{\pol}(C),
\]
which we know is impossible.
\end{proof}
\begin{remark}\label{remark:polhmlgyisnotinvariant}
It follows from the previous result that if we think of $\oo$\nbd{}categories as a
model for homotopy types (see Theorem \ref{thm:gagna}), then the polygraphic
homology of an $\oo$\nbd{}category is \emph{not} a well defined invariant. This
justifies what we said in remark \ref{remark:singularhmlgyishmlgy}, which is
that \emph{singular homology} is the only ``correct'' homology of $\oo$\nbd{}categories.
\end{remark}
\begin{paragr}\label{paragr:defcancompmap}
Even though triangle \eqref{cmprisontrngle} is not commutative (even up to an isomorphism), it can be filled up with a $2$-morphism. Indeed, consider the following $2$\nbd{}square
\[
\begin{tikzcd}
\oo\Cat \ar[d,"\gamma^{\Th}"] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
\ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"] & \ho(\Ch),
\ar[from=2-1,to=1-2,"\alpha^{\sing}",shorten <= 1em, shorten >= 1em, Rightarrow]
\end{tikzcd}
\]
Since $\gamma^{\Th}=\J\circ \gamma^{\folk}$ and the polygraphic homology is the total left derived functor of the abelianization functor when $\oo\Cat$ is equipped with folk weak equivalences, we obtain by universal property (see \ref{paragr:defleftderived}) a unique natural transformation
\begin{equation}\label{cmparisonmapdiag}
\begin{tikzcd}
\ho(\oo\Cat^{\folk}) \ar[d,"\J"] \ar[rd,"\sH^{\pol}",""{name=A,below}] & \\
\ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"'] & \ho(\Ch)\ar[from=2-1,to=A,"\pi",Rightarrow]
\end{tikzcd}
\end{equation}
such that $\alpha^{\sing}$ factorizes as
\[
\begin{tikzcd}
\oo\Cat \ar[d,"\gamma^{\folk}"] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
\ho(\oo\Cat^{\folk})\ar[d,"\J"] \ar[r,"\sH^{\pol}",""{name=B,below}] & \ho(\Ch)\\
\ho(\oo\Cat^{\Th}) \ar[ru,"\sH^{\sing}"',bend right=15] &
\ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >= 1em, Rightarrow]
\ar[from=3-1,to=B,Rightarrow,"\pi",shorten <= 1em, shorten >= 1em]
\end{tikzcd}
\]
%% \begin{equation}\label{trianglecomparisonmap}
%% \begin{tikzcd}
%% \sH^{\pol}\circ \gamma^{\folk} \ar[r,"\pi\ast\gamma^{\folk}",Rightarrow] \ar[rd,"\alpha^{\folk}\circ (\pi \ast \gamma^{\folk})"',Rightarrow] & \sH^{\sing}\circ \J \circ \gamma^{\folk} \ar[d,"\alpha^{\sing}\ast (\J \circ \gamma^{\folk})",Rightarrow]\\
%% &\gamma^{\Ch}\circ \lambda
%% \end{tikzcd}
%% \end{equation}
%% is commutative.
Since $\J$ is nothing but the identity on objects, for every $\oo$\nbd{}category $C$, the natural transformation $\pi$ yields a map
\[
\pi_C : \sH^{\sing}(C) \to \sH^{\pol}(C),
\]
which we shall refer to as the \emph{canonical comparison map.}
\end{paragr}
\begin{remark}
When $C$ is free, it follows from the considerations in \ref{paragr:univmor} that the canonical comparison map $\pi_C$ can be identified with the image by $\gamma^{\Ch}$ of the morphism of $\Ch$
\[
\lambda c_{\oo}N_{\oo}(C) \to \lambda(C)
\]
induced by the co-unit of $c_{\oo} \dashv N_{\oo}$.
\end{remark}
% This motivates the following definition.
\begin{definition}
An $\oo$\nbd{}category $C$ is said to be \emph{\good{}} when the canonical comparison map
\[
\pi_C : \sH^{\sing}(C) \to \sH^{\pol}(C)
\]
is an isomorphism of $\ho(\Ch)$.
\end{definition}
\begin{paragr}
The rest of this dissertation is devoted to the study of \good{} $\oo$\nbd{}categories. Examples of such $\oo$\nbd{}categories will be presented later. Following the perspective of Remark \ref{remark:polhmlgyisnotinvariant}, polygraphic homology can be thought of as a way to compute singular homology of \good{} $\oo$\nbd{}categories.
\end{paragr}
%% \section{A criterion to detect \good{} $\oo$\nbd{}categories}
%% We shall now proceed to give an abstract criterion to find \good{} $\oo$\nbd{}categories. %In the rest of this dissertation, we will exploit this criterion to exhibit the largest classes possible of \good{} $\oo$\nbd{}categories.
\begin{paragr}\label{paragr:prelimcriteriongoodcat}
Similarly to \ref{paragr:cmparisonmap}, the morphism of localizers
\[
(\oo\Cat,\W^{\folk}) \to (\oo\Cat,\W^{\Th})
\]
induces a morphism of op-prederivators
\[
\J : \Ho(\oo\Cat^{\folk}) \to \Ho(\oo\Cat^{\Th})
\]
such that the triangle in the category of op-prederivators
\[
\begin{tikzcd}
\oo\Cat \ar[d,"\gamma^{\folk}"'] \ar[rd,"\gamma^{\Th}"] &\\
\Ho(\oo\Cat^{\folk}) \ar[r,"\J"'] & \Ho(\oo\Cat^{\Th})
\end{tikzcd}
\]
is commutative.
It follows from what we said in \ref{paragr:compweakeq} that the morphism $\J$ cannot be an equivalence of op-prederivators. As we shall see later, $\J$ is not even homotopy cocontinuous. In particular, this implies that given a diagram $d : I \to \oo\Cat$, the canonical arrow of $\ho(\oo\Cat^{\Th})$
\begin{equation}\label{equation:Jhocolim}
\hocolim^{\Th}_{I}(\J_I(d)) \to \J_e(\hocolim_{I}^{\folk}(d))
\end{equation}
induced by $\J$ (see \ref{paragr:canmorphismcolimit}) is generally \emph{not} an isomorphism. Note that since
\[\J_I : \Ho(\oo\Cat^{\folk})(I) \to \Ho(\oo\Cat^{\Th})(I)\]
is the identity on objects for every small category $I$, morphism \eqref{equation:Jhocolim} simply reads
\[
\hocolim_I^{\Th}(d) \to \hocolim_I^{\folk}(d).
\]
Even if this is not always true, there are some particular diagrams $d$ for which the above morphism is indeed an isomorphism. The criterion to find \good{} $\oo$\nbd{}categories given in the proposition below is based on this observation.
\end{paragr}
%% \begin{paragr}\label{paragr:compcriterion}
%% Both the polygraphic homology
%% \[
%% \sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch)
%% \]
%% and the Street homology
%% \[
%% \sH : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)
%% \]
%% are homotopy cocontinuous (Definition \ref{def:cocontinuous}). In the first case, this follows from Theorem \ref{thm:cisinskiII} and the fact that $\lambda : \oo\Cat \to \Ch$ is left Quillen with respect to the folk model structure on $\oo\Cat$. In the second case, this follows from the fact that $\overline{N_{\oo}} : \Ho(\oo\Cat^{\Th} \to \Ho(\Ch)$ induces an equivalence of op-prederivators and that $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen. Besides, the construction of the canonical comparison map from Paragraph \ref{paragr:cmparisonmap} can be reproduced \emph{mutatis mutandis} in the $2$\nbd{}category of op-prederivators, yielding a $2$-morphism of op-prederivators
%% \[
%% \begin{tikzcd}
%% \Ho(\oo\Cat^{\folk}) \ar[r,"\J"] \ar[rd,"\sH^{\pol}"',""{name=A,above}] & \Ho(\oo\Cat^{\Th}) \ar[d,"\sH"] \\
%% & \Ho(\Ch).\ar[from=A,to=1-2,"\pi",Rightarrow]
%% \end{tikzcd}
%% \]
%% We will see later that $\J$ cannot be homotopy cocontinuous as if it were, every $\oo$\nbd{}category would be \good{}. \todo{Mettre ref interne de où il sera montré que le morphisme n'est pas cocontinu.}
%% \end{paragr}
\begin{proposition}\label{prop:criteriongoodcat}
Let $C$ be an $\oo$\nbd{}category. Suppose that there exists a diagram
\[
d : I \to \oo\Cat
\]
and a co-cone
\[
(\varphi_i : d(i) \to C)_{i \in \Ob(I)}
\]
such that:
\begin{enumerate}[label=(\roman*)]
\item For every $i \in \Ob(I)$, the $\oo$\nbd{}category $d(i)$ is \good{}.
\item The canonical morphism
\[
\hocolim^{\folk}d \to C
\]
is an isomorphism of $\ho(\oo\Cat^{\folk})$.
\item The canonical morphism
\[
\hocolim^{\Th}d \to C
\]
is an isomorphism of $\ho(\oo\Cat^{\Th})$.
\end{enumerate}
Then the $\oo$\nbd{}category $C$ is \good{}.
\end{proposition}
\begin{proof}
Notice first that all the constructions from \ref{paragr:defcancompmap} may be reproduced \emph{mutatis mutandis} at the level of op-prederivators. In particular, we obtain a $2$\nbd{}morphism of op-prederivators
\[
\begin{tikzcd}
\Ho(\oo\Cat^{\folk}) \ar[d,"\J"] \ar[rd,"\sH^{\pol}",""{name=A,below}] & \\
\Ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"'] & \Ho(\Ch)\ar[from=2-1,to=A,"\pi",Rightarrow].
\end{tikzcd}
\]
Then, by naturality, we have a commutative diagram in $\ho(\Ch)$:
\[
\begin{tikzcd}
\displaystyle\hocolim_{i\in I}\sH^{\sing}(d_i) \ar[d] \ar[r] & \displaystyle\hocolim_{i \in I}\sH^{\pol}(d_i) \ar[d] \\
\displaystyle\sH^{\sing}(\hocolim_{i \in I}^{\Th}(d_i)) \ar[r] \ar[d] & \displaystyle\sH^{\pol}(\hocolim^{\folk}_{i \in I}(d_i))\ar[d] \\
\sH^{\sing}(C) \ar[r] & \sH^{\pol}(C),
\end{tikzcd}
\]
where:
\begin{itemize}[label=-]
\item the top and bottom horizontal arrows are induced by $\pi$,
\item the middle horizontal arrow is induced by $\pi$ and the canonical morphism \[\hocolim_{i \in I}^{\Th}(d_i)\to \hocolim_{i \in I}^{\folk}(d_i)\] from \ref{paragr:prelimcriteriongoodcat},
\item the top vertical arrows are the canonical morphisms induced by every morphism of op-prederivators (see \ref{paragr:canmorphismcolimit}),
\item the bottom vertical arrows are induced by the co-cone \[(\varphi_i : d(i) \to C)_{i \in \Ob(I)}.\]
\end{itemize}
Since $\sH^{\pol}$ and $\sH^{\sing}$ are both homotopy cocontinuous (Proposition \ref{prop:singhmlgycocontinuous} and Proposition \ref{prop:polhmlgycocontinuous} respectively), both top vertical arrows are isomorphisms. Hypotheses $(ii)$ and $(iii)$ imply that the bottom vertical arrows are isomorphisms and hypothesis $(i)$ imply that the top horizontal arrow is an isomorphism. By a 2-out-of-3 property, the bottom horizontal arrow is an isomorphism, which means exactly that $C$ is \good{}.
\end{proof}
The previous proposition admits the following corollary, which will be of great use in later chapters.
\begin{corollary}\label{cor:usefulcriterion}
Let
\[
\begin{tikzcd}
A \ar[r,"u"] \ar[d,"f"] & B \ar[d,"g"] \\
C \ar[r,"v"] & D
\ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
\end{tikzcd}
\]
be a cocartesian square of $\oo\Cat$ such that:
\begin{enumerate}[label=(\alph*)]
\item the $\oo$\nbd{}categories $A$,$B$ and $C$ are free and \good{},
\item at least one of the morphisms $u : A \to B$ or $f : A \to C$ is a folk cofibration,
\item the square is Thomason homotopy cocartesian.
\end{enumerate}
Then, the $\oo$\nbd{}category $D$ is \good{}.
\end{corollary}
\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 (Lemma \ref{lemma:hmtpycocartesianreedy}). The conclusion follows then from Proposition \ref{prop:criteriongoodcat}.
\end{proof}
\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.
We have already seen that $\iota_n$ 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 $kn$
are units, we trivially have that $f(x)=f(y)$ and $\beta$ is the unit on
$f(x)$. Now let $x'$ and $y'$ be parallel $n$\nbd{}cells of $C$ such that
$\eta_C(x')=x$ and $\eta_C(y')=y$ (this is always possible by definition
of $T(C)$). We have $\eta_{D}(f(x'))=f(x)=f(y)=\eta_{D}(f(y'))$. By
definition of the functor $\tau^{i}_{\leq n}$, this means that there
exists a zigzag of $(n+1)$\nbd{}cells of $D$ from $f(x')$ to $f(y')$.
More precisely, this means that there exists a sequence
\[
(z_0,\beta_1,z_1,\cdots,z_{p-1},\beta_p,z_p)
\]
where the $z_i$ are all parallel $n$\nbd{}cells of $C$ with $z_0=f(x')$
and $z_p=f(y')$, and each $\beta_i$ is $(n+1)$\nbd{}cell of $C$ either
from $z_{i-1}$ to $z_i$ or from $z_{i}$ to $z_{i-1}$. Using the fact that
$f$ is a folk trivial fibration, it is easy to prove the existence of a
zigzag from $x'$to $y'$, which implies in particular that $x=\eta_C(x')=\eta_C(y')=y$.
\item[Case $k>n$:] Since all $k$\nbd{}cells of $T(C)$ and $T(D)$ with $k>n$
are units, we trivially have $f(x)=f(y)$ and $x=y$.
\end{description}
Altogether, this proves that $T(f)$ is a folk trivial fibration, hence a folk
weak equivalence.
\end{proof}
For later reference, we put here the following lemma.
\begin{lemma}\label{lemma:cofncatfolk}
If an $n$\nbd{}category $C$ has a $k$\nbd{}basis for every $0 \leq k \leq n-1$, then it is cofibrant for the folk model structure on $n\Cat$.
\end{lemma}
\begin{proof}
Since $\tau^{i}_{\leq n}$ is a left Quillen functor, it suffices in virtue of
Proposition \ref{prop:freeiscofibrant} to show that
there exists a free $\oo$\nbd{}category $C'$ such that $\tau^{i}_{\leq n}(C')=C$.
Let us write $\Sigma_k$ for the $k$\nbd{}base of $C$ with $0\leq k\leq n-1$ and
let $C'$ be the free $\oo$\nbd{}category such that:
\begin{itemize}[label=-]
\item the $k$\nbd{}base of $C'$ is $\Sigma_k$ for every $0 \leq k \leq n-1$,
\item the $n$\nbd{}base of $C'$ is the set $C_n$,
\item the $(n+1)$\nbd{}base of $C'$ is the set
\[
\{(x,y)\,\vert\, x \text{ and } y \text{ are parallel } n \text{-cells of } C\},
\]
the source (resp.\ target) of $(x,y)$ being $x$ (resp.\ $y$),
\item the $k$\nbd{}base of $C'$ is empty for $k > n+1$ (i.e.\ $C'$ is an $(n+1)$\nbd{}category).
\end{itemize}
(For such recursive constructions of free $\oo$\nbd{}categories, see Section
\ref{section:freeoocataspolygraph}, and in particular Proposition
\ref{prop:freeonpolygraph}.)
We invite the reader to verify for himself that indeed $\tau^{i}_{\leq n}(C')=C$.
\end{proof}
\begin{example}
Every (small) category is cofibrant for the folk model structure on $\Cat$.
\end{example}
We now turn to truncations of chain complexes.
\begin{paragr}
Let $\Ch^{\leq n}$ be the category of chain complexes concentrated in degrees between $0$ and $n$. This means that an object $K$ of $\Ch^{\leq n}$ is a diagram of abelian groups of the form
\[
K_0 \overset{\partial}{\longleftarrow} K_1 \overset{\partial}{\longleftarrow} K_2 \overset{\partial}{\longleftarrow} \cdots \overset{\partial}{\longleftarrow} K_n,
\]
where $\partial \circ \partial =0$, and morphisms of $\Ch^{\leq n}$ are defined the expected way. Write $\iota_n : \Ch^{\leq n} \to \Ch$ for the canonical functor that sends an object $K$ of $\Ch^{\leq n}$ to the chain complex
\[
K_0 \overset{\partial}{\longleftarrow} K_1 \overset{\partial}{\longleftarrow} K_2 \overset{\partial}{\longleftarrow} \cdots \overset{\partial}{\longleftarrow} K_n \longleftarrow 0 \longleftarrow 0 \longleftarrow \cdots.
\]
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 $k >n$.
% 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 to the case of $n$\nbd{}categories, the functor $\iota_n : \Ch^{\leq n} \to \Ch$ has a left adjoint $\tau^{i}_{\leq n} : \Ch \to \Ch^{\leq}$, where for a chain complex $K$, $\tau^{i}_{\leq n}(K)$ is the object of $\Ch^{\leq n}$ defined as
\[
K_0 \overset{\partial}{\longleftarrow} K_1 \overset{\partial}{\longleftarrow} \cdots \overset{\partial}{\longleftarrow} K_{n-1} \overset{\partial}{\longleftarrow} K_{n}/{\partial(K_{n+1})}.
\]
Again, as with $n$\nbd{}categories, we can use the adjunction
\[
\begin{tikzcd}
\tau^{i}_{\leq n} : \Ch\ar[r,shift left] & \ar[l,shift left]\Ch^{\leq n} : \iota_n
\end{tikzcd}
\]
to create a model structure on $\Ch^{\leq n}$.
\end{paragr}
\begin{proposition}
There exists a model structure on $\Ch^{\leq n}$ such that:
\begin{itemize}[label=-]
\item the weak equivalences are exactly those morphisms $f : K \to K'$ such that $\iota_n(f)$ is a weak equivalence for the projective model structure on $\Ch$,
\item the fibrations are exactly those morphisms $f : K \to K'$ such that $\iota_n(f)$ is a fibration for the projective model structure on $\Ch$.
\end{itemize}
\end{proposition}
\begin{proof}
This is a typical example of a transfer of a cofibrantly generated model
structure along a right adjoint as in \cite[Proposition
2.3]{beke2001sheafifiableII}. Since the weak equivalences of the projective model
structure on $\Ch$ are closed under filtered colimits \cite[Theorem 2.6.15]{weibel1995introduction}, the only \emph{a priori} non-trivial hypothesis to check is that there exists a set $J$ of generating trivial cofibrations of the projective model structure on $\Ch$ such that for every $j \colon A \to B$ in $J$ and every cocartesian square
\[
\begin{tikzcd}
\tau^{i}_{\leq n}(A) \ar[r] \ar[d,"\tau^{i}_{\leq n}(j)"'] & X \ar[d,"g"] \\
\tau^{i}_{\leq n}(B) \ar[r] & Y,
\ar[from=1-1, to=2-2, phantom, "\ulcorner",very near end]
\end{tikzcd}
\]
the morphism $\iota_n(g)$ is a weak equivalence of $\Ch$. As explained in \cite[Proposition 7.19]{dwyer1995homotopy}, there exists a set of generating trivial cofibrations of the projective model structure on $\Ch$ consisting of the maps
\[
0 \to D_k
\]
for each $k >0$, where $D_k$ is the following chain complex concentrated in degree $k$ and $k-1$
\[
0 \leftarrow \cdots \leftarrow 0 \leftarrow \mathbb{Z} \overset{\mathrm{id}}{\leftarrow} \mathbb{Z} \leftarrow 0 \leftarrow \cdots
\]
What is left to show then is that for every $k > 0$ and every object $X$ of $\Ch^{\leq n}$, the canonical inclusion map
\[
X \to X \oplus \tau^{i}_{\leq n}(D_k)
\]is sent by $\iota_n$ to a weak equivalence of $\Ch$. This follows immediately from the fact that homology groups commute with direct sums.
\end{proof}
\begin{paragr}
We refer to the model structure of the previous proposition as the \emph{projective model structure on $\Ch^{\leq n}$}.
\end{paragr}
\begin{lemma}\label{lemma:unitajdcomp}
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}
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}
As a consequence of this lemma, we have the analogous of Proposition \ref{prop:truncationhomotopical}.
\begin{proposition}
The functor $\tau^{i}_{\leq n} : \Ch \to \Ch^{\leq n}$ sends the weak equivalences of the projective model structure on $\Ch$ to weak equivalences of the projective model structure on $\Ch^{\leq n}$.
\end{proposition}
\begin{proof}
Let $f : K \to K'$ be a weak equivalence for the projective model structure on $\Ch$ and consider the naturality square
\[
\begin{tikzcd}[column sep=huge]
K \ar[d,"\eta_K"] \ar[r,"f"] & K' \ar[d,"\eta_K'"] \\
\iota_n\tau^{i}_{\leq n}(K) \ar[r,"\iota_n\tau^{i}_{\leq n}(f)"] & \iota_n\tau^{i}_{\leq n}(K'),
\end{tikzcd}
\]
where $\eta$ is the unit map of the adjunction $\tau^{i}_{\leq n} \dashv \iota_n$. It follows from Lemma \ref{lemma:unitajdcomp} that
\[
H_k(\iota_n\tau^{i}_{\leq n}(f)) : H_k(\iota_n\tau^{i}_{\leq n}(K)) \to \iota_n\tau^{i}_{\leq n}(K'))
\]
is an isomorphism for every $k \leq n$. Since obviously $H_k(\iota_n\tau^{i}_{\leq n}(f))$ is also an isomorphism for $k > n$, this proves the result.
\end{proof}
We now investigate the relation between truncation and abelianization.
\begin{paragr}
Let $C$ be $n$\nbd{}category. A straightforward computation shows that the chain complex $\lambda(\iota_n(C))$ is such that
\[
\lambda_k(\iota_n(C))=0
\]
for every $k > n$ and thus $\lambda(\iota_n(C))$ can be seen as an object of $\Ch^{\leq n}$. Hence, 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*}
and 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}\label{lemma:abelianizationtruncation}
The square
\[
\begin{tikzcd}
\oo\Cat \ar[d,"\tau_{\leq n}^{i}"] \ar[r,"\lambda"] & \Ch \ar[d,"\tau^{i}_{\leq n}"] \\
n\Cat \ar[r,"\lambda_{\leq n}"] & \Ch^{\leq n}
\end{tikzcd}
\]
is commutative (up to a canonical isomorphism).
\end{lemma}
\begin{proof}
Notice first that we have a natural transformation
\[
\beta : \tau^{i}_{\leq n}\circ \lambda \Rightarrow \lambda_{\leq n} \circ \tau^{i}_{\leq n}
\]
defined as
\[
\begin{tikzcd}
\oo\Cat \ar[r,"\tau_{\leq n}^{i}"] \ar[rd,"\mathrm{id}"',""{name=A,right}] & n\Cat \ar[d,"\iota_n"] \ar[r,"\lambda_{\leq n}"] & \Ch^{\leq n} \ar[d,"\iota_n"'] \ar[dr,"\mathrm{id}",""{name=B,left}] & \\
&\oo\Cat \ar[r,"\lambda"] & \Ch \ar[r,"\tau^{i}_{\leq n}"'] & \Ch^{\leq n}.
\ar[from=A, to=1-2,Rightarrow,"\eta"]
\ar[from=2-3,to=B,Rightarrow,"\epsilon"]
\end{tikzcd}
\]
Since for every $\oo$\nbd{}category $C$ and every $k n$, we would have $H_k^{\pol}(\iota_n(C))=0$ for every $k >n$, which is not even true for the case $n=1$ as we shall see in the following chapter.
\end{remark}
A useful consequence of Proposition \ref{prop:polhmlgytruncation} is the following corollary.
\begin{corollary}\label{cor:polhmlgycofibrant}
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
\[
H_k^{\pol}(C) \simeq H_k(\lambda(C))
\]
for every $0 \leq k \leq n$.
\end{corollary}
\begin{proof}
From Lemma \ref{lemma:abelianizationtruncation} and Proposition \ref{prop:polhmlgytruncation}, we deduce that the morphism $\overline{\tau^{i}_{\leq n}}(\alpha^{\pol})$ of $\ho(\Ch^{\leq n})$ can be identified with the canonical morphism
\[
\LL \lambda_{\leq n}(\tau^{i}_{\leq n}(C)) \to \lambda_{\leq n}(\tau^{i}_{\leq n}(C)).
\]
From Lemma \ref{lemma:cofncatfolk}, we have that $\tau^{i}_{\leq n}(C)$ is cofibrant for the folk model structure on $n\Cat$, and the result follows immediately from the fact that $\lambda_{\leq n}$ is left Quillen.
\end{proof}
\begin{paragr}\label{paragr:polhmlgylowdimension}
Since every $\oo$\nbd{}category trivially admits its set of $0$\nbd{}cells as a $0$\nbd{}base, it follows from the previous proposition that for every $\oo$\nbd{}category $C$ we have
\[
\sH^{\pol}_0(C)\simeq H_0(\lambda(C))
\]
and
\[
\sH^{\pol}_1(C) \simeq H_1(\lambda(C)).
\]
Intuitively speaking, 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
%% \[
%% \sH^{\pol}(C) \to \sH^{\pol}(\iota_n\tau^{i}_{\leq n}(C))
%% \]
%% induces isomorphisms
%% \[
%% H^{\pol}_k(C) \simeq H^{\pol}_k(\iota_n\tau^{i}_{\leq n}(C))
%% \]
%% for every $0 \leq k \leq n$.
%% \end{proposition}
%% \begin{proof}
%% Let $f : P \to C$ be a cofibrant replacement for $C$. \todo{À finir}.
%% \end{proof}
We now turn to the relation between truncation and singular homology of $\oo$\nbd{}categories. Recall that for every $n \geq 0$, the nerve functor $N_n : n\Cat \to \Psh{\Delta}$ is defined as the following composition
\[
N_n : n\Cat \overset{\iota_n}{\longrightarrow} \oo\Cat \overset{N_{\oo}}{\longrightarrow} \Psh{\Delta},
\]
and for $n \in \mathbb{N}\cup \{\oo\}$ we write $c_n : \Psh{\Delta} \to n\Cat$ for the left adjoint of $N_n$.
\begin{lemma}
For every $n \in \mathbb{N}$, the following triangle of functors
\[
\begin{tikzcd}
\Psh{\Delta} \ar[r,"c_{\oo}"] \ar[d,"c_n"] & \oo\Cat \ar[dl,"\tau^{i}_{\leq n}"] \\
n\Cat&
\end{tikzcd}
\]
is commutative (up to an isomorphism).
\end{lemma}
\begin{proof}
Straightforward consequence of the fact that $N_n = N_{\oo} \circ \iota_n$ and the fact that the composition of left adjoints is the left adjoint of the composition.
\end{proof}
\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}\label{lemma:truncationcounit}
For every $\oo$\nbd{}category $C$, the canonical morphism of $\Cat$
\[
c_1N_{\oo}(C) \to \tau^{i}_{\leq 1}(C)
\]
is an isomorphism.
\end{lemma}
\begin{proof}
Let $C$ be an $\oo$\nbd{}category and $D$ be a (small) category. By adjunction, we have
\begin{equation}
\Hom_{\Cat}(c_1N_{\oo}(C),D) \simeq \Hom_{\Psh{\Delta}}(N_{\oo}(C),N_1(D)).
\end{equation}
Now let $\Delta_{\leq 2}$ be the full subcategory of $\Delta$ spanned by
$[0]$, $[1]$ and $[2]$ and let $i : \Delta_{\leq 2} \to \Delta$ be the
canonical inclusion. This inclusion induces by pre-composition a functor $i^*
: \Psh{\Delta} \to \Psh{\Delta_{\leq 2}}$ which has a right-adjoint $i_* :
\Psh{\Delta_{\leq 2}} \to \Psh{\Delta}$. Recall that the nerve of a (small) category is $2$-coskeletal (see for example \cite[Theorem 5.2]{street1987algebra}), which means that for every category $D$, the unit morphism $ N_1(D) \to i_* i^* (N_1(D))$ is an isomorphism of simplicial sets. In particular, we have
\begin{align*}
\Hom_{\Psh{\Delta}}(N_{\oo}(C),N_1(D)) &\simeq \Hom_{\Psh{\Delta}}(N_{\oo}(C),i_* i^* (N_1(D)))\\
&\simeq \Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))).
\end{align*}
Using the description of $\Or_0$, $\Or_1$ and $\Or_2$ from \ref{paragr:orientals}, we deduce that a morphism $F : i^*(N_{\oo}(C)) \to i^*(N_1(D))$ of $\Psh{\Delta_{\leq 2}}$ consists of a function $F_0 : C_0 \to D_0$ and a function $F_1 : C_1 \to D_1$ such that
\begin{itemize}[label=-]
\item for every $x \in C_0$, we have $F_1(1_x)=1_{F_0(x)}$,
\item for every $x \in C_1$, we have
\[\src(F_1(x))=F_0(\src(x))) \text{ and }\trgt(F_1(x))=F_0(\trgt(x))),\]
\item for every $2$\nbd{}triangle
\[
\begin{tikzcd}
& Y \ar[rd,"g"] & \\
X \ar[ru,"f"] \ar[rr,"h"',""{name=A,above}] & & Z
\ar[from=A,to=1-2,Rightarrow,"\alpha"]
\end{tikzcd}
\]
in $C$, we have $F_1(g)\comp_0 F_1(f)=F_1(h)$.
\end{itemize}
In particular, it follows that $F_1$ is compatible with composition of $1$\nbd{}cells in an obvious sense and that for every $2$\nbd{}cell $\alpha : f \Rightarrow g$ of $C$, we have $F_1(f)=F_1(g)$. This means exactly that we have a natural isomorphism
\[
\Hom_{\Psh{\Delta_{\leq 2}}}(i^*(N_{\oo}(C)),i^*(N_1(D))) \simeq \Hom_{\Cat}(\tau_{\leq 1}^{i}(C),D).
\]
Altogether, we have
\[
\Hom_{\Cat}(c_1N_{\oo}(C),D) \simeq \Hom_{\Cat}(\tau_{\leq 1}^{i}(C),D),
\]
which proves that
\[
c_1N_{\oo}(C) \simeq \tau_{\leq 1}^{i}(C)
\]
and a thorough analysis of naturality shows that this isomorphism is nothing but the canonical morphism $c_1N_{\oo}(C) \to \tau_{\leq 1}^{i}(C)$.
\end{proof}
We can now prove the important following proposition.
\begin{proposition}\label{prop:singhmlgylowdimension}
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}
Let $C$ be an $\oo$\nbd{}category. Recall from \ref{paragr:univmor} that the canonical morphism $\alpha^{\sing} : \sH^{\sing}(C) \to \lambda(C)$ is nothing but the image by the localization functor $\Ch \to \ho(\Ch)$ of the morphism
\[
\lambda c_{\oo}N_{\oo}(C) \to \lambda(C)
\]
induced by the co-unit of the adjunction $c_{\oo} \dashv N_{\oo}$. From \ref{prop:polhmlgytruncation} we have that
\[
\tau^{i}_{\leq 1}\lambda c_{\oo}N_{\oo}(C) \simeq \lambda_{\leq 1} \tau_{\leq 1}^{i} c_{\oo} N_{\oo}(C)=\lambda_{\leq 1} c_1 N_{\oo}(C),
\]
and from Lemma \ref{lemma:truncationcounit} we obtain
\[
\tau_{\leq 1 }^{i} \lambda c_{\oo} N_{\oo}(C) \simeq \lambda_{\leq 1} \tau^{i}_{\leq 1}(C) \simeq \tau^{i}_{\leq 1}\lambda(C).
\]
This means exactly that the image by $\overline{\tau^{i}_{\leq 1}}$ of $\alpha^{\sing}$ is an isomorphism, which is what we wanted to prove.
\end{proof}
Finally, we obtain the result we were aiming for.
\begin{proposition}\label{prop:comphmlgylowdimension}
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}
\begin{proof}
Let $C$ be an $\oo$\nbd{}category and consider the following commutative triangle of $\ho(\Ch)$
\[
\begin{tikzcd}[column sep=tiny]
\sH^{\sing}(C) \ar[rd,"\alpha^{\sing}"'] \ar[rr,"\pi_C"] & & \sH^{\pol}(C) \ar[dl,"\alpha^{\pol}"] \\
&\lambda(C)&.
\end{tikzcd}
\]
From Proposition \ref{prop:singhmlgylowdimension}, we know that $\alpha^{\sing}$ induces isomorphisms \[H_k^{\sing}(C) \simeq H_k(\lambda(C))\] for $k \in \{0,1\}$ and from Corollary \ref{cor:polhmlgycofibrant} and Paragraph \ref{paragr:polhmlgylowdimension} we know that $\alpha^{\pol}$ induces isomorphisms $H_k^{\pol}(C) \simeq H_k(\lambda(C))$ for $k \in \{0,1\}$. The result follows then from an immediate 2-out-of-3 property.
\end{proof}
\begin{paragr}\label{paragr:conjectureH2}
A natural question following the above proposition is:
\begin{center}
For which $k \geq 0$ do we have $H_k^{\sing}(C) \simeq H_k^{\pol}(C)$ for every $\oo$\nbd{}category $C$ ?
\end{center}
We have already seen in \ref{paragr:bubble} that when $C = B^2\mathbb{N}$ we have
\[
H_{2p}^{\sing}(B^2\mathbb{N}) \not\simeq H^{\pol}_{2p}(B^2\mathbb{N})
\]
for every $p \geq 2$. Furthermore, with a similar argument to the one given in \ref{paragr:bubble}, we have that for every $k \geq 3$, the (nerve of the) $\oo$\nbd{}category $B^k\mathbb{N}$ is a $K(\mathbb{Z},k)$. In particular, we have
\[
H_{2p+3}^{\sing}(B^{2p +1}\mathbb{N})\simeq \mathbb{Z}/{2\mathbb{Z}}
\]
for every $p \geq 1$ (see \cite[Theorem 23.1]{eilenberg1954groups}). On the other hand, since $B^k\mathbb{N}$ is a free $k$\nbd{}category, we have $H_n^{\pol}(B^k\mathbb{N})=0$ for all $n \geq k$. All in all, we have proved that for every $k \geq 4$, there exists at least one $\oo$\nbd{}category $C$ such that
\[
H_k^{\sing}(C) \not\simeq H_k^{\pol}(C).
\]
However, it is still an open question to know whether for $k \in \{2,3\}$ we have
\[
H^{\sing}_k(C) \simeq H^{\pol}_k(C)
\]
for every $\oo$\nbd{}category $C$. The only missing part to adapt the proof of Proposition \ref{prop:comphmlgylowdimension} for these values of $k$ is the analogue of Lemma \ref{lemma:truncationcounit}. But contrary to the case $k=1$, it is not generally true that the canonical morphism $c_k N_{\oo}(C) \to \tau^{i}_{\leq k}(C)$ is an isomorphism when $k \geq 2$. However, what we really need is that the image by $\lambda$ of this morphism be a quasi-isomorphism. In the case $k=2$, it seems that this canonical morphism admits an oplax $2$\nbd{}functor as an inverse up to oplax transformation which could be an hint towards the conjecture that $H^{\sing}_2(C) \simeq H^{\pol}_2(C)$ for every $\oo$\nbd{}category $C$.
\end{paragr}
%% 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 $\oo$\nbd{}categories in low dimension}
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