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\chapter{Homology and abelianization of \texorpdfstring{$\oo$}{ω}-categories}
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\chaptermark{Homology of $\omega$-categories}
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\section{Homology via the nerve}
\begin{paragr}
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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:
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    \begin{itemize}
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     \item[-] the weak equivalences are the quasi-isomorphisms, i.e.\ morphisms of chain complexes that induce an isomorphism on homology groups,
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              \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}
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  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
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  \[
  \partial(x):=\sum_{i=0}^n(-1)^i\partial_i(x).
  \]
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  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.
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\end{paragr}
\begin{paragr}
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  Recall that an $n$-simplex $x$ of a simplicial set $X$ is \emph{degenerate} if there exists an epimorphism $\varphi : [n] \to [m]$ with $m<n$ and an $m$-simplex $y$ such that $X(\varphi)(y)=x$. We denote by $D_n(X)$ the subgroup of $K_n(X)$ generated by the degenerate $n$-simplices and by $\kappa_n(X)$ the abelian group of \emph{normalized $n$\nbd{}chains}:
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  \[
  \kappa_n(X)=K_n(X)/D_n(X).
  \]
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  Using the simplicial identities, it can be shown that $\partial(D_n(X)) \subseteq D_{n-1}(X)$ for every $n>0$. Hence, there is an induced differential which we still denote by $\partial$:
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  \[
  \partial : \kappa_n(X) \to \kappa_{n-1}(X).
  \]
This defines a chain complex $\kappa(X)$, which we call the \emph{normalized chain complex of $X$}. This yields a functor
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  \begin{align*}
    \kappa : \Psh{\Delta} &\to \Ch \\
    X &\mapsto \kappa(X).
  \end{align*}
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\end{paragr}
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\begin{lemma}\label{lemma:normcompquil}
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The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends the weak equivalences of simplicial sets to quasi-isomorphisms.  
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\end{lemma}
\begin{proof}
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  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
  \[
  \{\Lambda^i_n \hookrightarrow \Delta_n \vert n \in \mathbb{N}, 0 \leq i \leq n\}
  \]
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  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.
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  \end{proof}
\begin{remark}
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  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
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    \[
    U : \Ch \simeq \Ab(\Delta) \to \Psh{\Delta}
    \]
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    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}
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\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}
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\begin{definition}\label{def:hmlgycat}
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  The \emph{singular homology functor for $\oo$\nbd{}categories} $\sH^{\sing}$ is defined as the following composition
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  \[
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  \sH^{\sing} : \ho(\oo\Cat^{\Th}) \overset{\overline{N_{\omega}}}{\longrightarrow} \ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \ho(\Ch).
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    \]
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    For an $\oo$\nbd{}category $C$, $\sH^{\sing}(C)$ is the \emph{singular homology of $C$}.
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\end{definition}
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\begin{paragr}\label{paragr:singularhmlgygroup}
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  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
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  \[
  H_k^{\sing}(C):=H_k(\sH^{\sing}(C))=H_k(\kappa(N_{\oo}(C))),
  \]
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  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.
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\end{paragr}
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%% \begin{paragr}
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%%   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.
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%%   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}

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\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
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  that \emph{singular homology of $\oo$\nbd{}categories} ought to be simply called
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  \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}.
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\end{remark}
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\begin{remark}
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  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. 
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\end{remark}
\begin{paragr}
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  We will also denote by $\sH^{\sing}$ the morphism of op-prederivators defined as the following composition
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  \[
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  \sH^{\sing} : \Ho(\oo\Cat^{\Th}) \overset{\overline{N_{\omega}}}{\longrightarrow} \Ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \Ho(\Ch).
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  \]
\end{paragr}
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\begin{proposition}\label{prop:singhmlgycocontinuous}
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  The singular homology \[\sH^{\sing} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)\] is homotopy cocontinuous. 
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\end{proposition}
\begin{proof}
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  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}.
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\end{proof}
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\section{Abelianization}
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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 $G$.
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\begin{paragr}
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  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
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  \[
  x \comp_k y \sim x+y
  \]
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  for all $x,y \in C_n$ that are $k$\nbd{}composable for some $k<n$. For $n=0$,
  this means that $\lambda_0(C)=\mathbb{Z}C_0$. Now let $f : C \to D$ be an
  $\oo$\nbd{}functor. For every $n \geq 0$, the definition of $\oo$\nbd{}functor
  implies that the linear map
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  \begin{align*}
    \mathbb{Z}C_n &\to \mathbb{Z}D_{n}\\
    x \in C_n &\mapsto f(x)
  \end{align*}
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  induces a linear map
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  \[
  \lambda_n(f) : \lambda_n(C) \to \lambda_n(D).
  \]
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  This defines a functor $\lambda_n : \oo\Cat \to \Ab$.
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  For $n>0$, consider the linear map
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  \begin{align*}
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  \mathbb{Z}C_n &\to \mathbb{Z}C_{n-1}\\
  x \in C_n &\mapsto t(x)-s(x).
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  \end{align*}
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  The axioms of $\oo$\nbd{}categories imply that it induces a map
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  \[
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  \partial : \lambda_{n}(C) \to \lambda_{n-1}(C)
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  \]
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  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 $\lambda(C)$:
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  \[
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  \lambda_0(C) \overset{\partial}{\longleftarrow} \lambda_1(C) \overset{\partial}{\longleftarrow} \lambda_2(C) \overset{\partial}{\longleftarrow} \cdots
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  \]
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 and for every $f : C \to D$ a morphism of chain complexes
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  \[
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  \lambda(f) : \lambda(C) \to \lambda(D).
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  \]
  Altogether, this defines a functor
  \[
  \lambda : \omega\Cat \to \Ch,
  \]
  which we call the \emph{abelianization functor}.
\end{paragr}
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\begin{lemma}\label{lemma:adjlambda}
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  The functor $\lambda$ is a left adjoint.
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\end{lemma}
\begin{proof}
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  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 $\omega\Cat(\Ab) \to \omega\Cat$.
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\end{proof}
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As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain complex $\lambda(C)$ admits a nice expression.
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\begin{paragr}
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  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 $n\geq 0$ 
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  \begin{align*}
    B^n : \Ab &\to n\Cat \\
    G &\mapsto B^nG,
  \end{align*}
  where $\Ab$ is the category of abelian groups.
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  Besides, let us write $\lambda_n$ again for the functor
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  \begin{align*}
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    \lambda_n : n\Cat &\to \Ab\\
    C&\mapsto \lambda_n(C).
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  \end{align*}
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  (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 $B^n : \Ab \to n\Cat$.
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\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.

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  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
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  \begin{align*}
    \Hom_{n\Cat}(C,B^nG) &\to \Hom_{\Set}(C_n,\vert G \vert)\\
    F &\mapsto F_n,
  \end{align*}
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 is injective and its image consists of those functions $f : C_n \to \vert G \vert$ such that:
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  \begin{enumerate}[label=(\roman*)]
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  \item\label{cond:comp} for every $0 \leq k <n $ and every pair $(x,y)$ of $k$\nbd{}composable $n$\nbd{}cells of $C$, we have
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    \[
    f(x\comp_ky) = f(x)+f(y),
    \]
  \item\label{cond:unit} for every $x \in C_{n-1}$, we have
    \[
    f(1_x)=0.
    \]
  \end{enumerate}
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  Let us see that because $G$ is an abelian group (recall that Lemma \ref{lemma:nfunctortomonoid} was stated for the general case of commutative monoids), condition \ref{cond:comp} imply condition \ref{cond:unit}. Let $f : C_n \to \vert G \vert$ be a function that satisfies condition \ref{cond:comp} and let $x \in C_{n-1}$. We have $1_x\comp_{n-1} 1_x = 1_x$, hence
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  \[
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  f(1_x)=f(1_x \comp_{n-1} 1_x)=f(1_x)+f(1_x),
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  \]
  and then
  \[
  f(1_x)=0
  \]
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  because every element of an (abelian) group has an inverse. Now, because of the adjunction morphism
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  \[
  \Hom_{\Set}(C_n,\vert G \vert) \simeq \Hom_{\Ab}(\mathbb{Z}C_n,G),
  \]
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  we have that $\Hom_{n\Cat}(C,B^nG)$ is naturally isomorphic to the set of morphisms of abelian groups $g : \mathbb{Z}C_n \to G$ such that for every pair $(x,y)$ of $k$\nbd{}composable elements of $C_n$ for some $k<n$, we have
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  \[
  g(x\comp_ky)=g(x)+g(y).
  \]
  By definition, this set is naturally isomorphic to the set of morphisms of abelian groups from $\lambda_n(C)$ to $G$. In other words, we have
  \[
  \Hom_{n\Cat}(C,B^nG)\simeq \Hom_{\Ab}(\lambda_n(C),G).\qedhere
  \]
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\end{proof}
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\begin{paragr}\label{paragr:abelpolmap}
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 Let $C$ be an $\oo$\nbd{}category, $n \in \mathbb{N}$ and $E \subseteq C_n$ a subset of the $n$-cells. We obtain a map $\mathbb{Z}E \to \lambda_n(C)$ defined as the composition
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  \[
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  \mathbb{Z}E \to \mathbb{Z}C_n \to \lambda_n(C),
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  \]
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  where the map on the left is induced by the canonical inclusion of $E$ in $C_n$ and the map on the right is the quotient map from the definition of $\lambda_n(C)$. 
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\end{paragr}
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\begin{lemma}\label{lemma:abelpol}
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  Let $C$ be a \emph{free} $\oo$\nbd{}category and let $\Sigma=(\Sigma_n)_{n \in \mathbb{N}}$ be its basis. For every $n \in \mathbb{N}$, the map
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  \[
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  \mathbb{Z}\Sigma_n \to \lambda_n(C)
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  \]
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  from the previous paragraph, is an isomorphism.
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\end{lemma}
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\begin{proof}
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  %% Let $G$ be an abelian group. For any $n \in \mathbb{N}$, we define an $n$-category $B^nG$ with:
  %%   \begin{itemize}
  %%   \item[-] $(B^nG)_{k}$ is a singleton set for every $k < n$,
  %%   \item[-] $(B^nG)_n = G$
  %%   \item[-] for all $x$ and $y$ in $G$ and $i<n$,
  %%     \[x \ast_i y := x +y.\]
  %%   \end{itemize}
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  %%   It is straightforward to check that this defines an $n$-category. Note that the previous definition would still make sense with $G$ an abelian \emph{monoid}. Moreover, when $n=1$, we didn't even need it to be abelian, but for $n\geq 2$ this hypothesis is necessary because of the Eckmann-Hilton argument. For $n=0$, we only needed that $G$ was a set.
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  %%   This defines a functor
  %%   \[
  %%   \begin{aligned}
  %%     B^n : \Ab &\to n\Cat\\
  %%     G &\mapsto B^nG,
  %%   \end{aligned}
  %%   \]
  %%   which is easily seen to be right adjoint to the functor
  %%   \[
  %%   \begin{aligned}
  %%     n\Cat &\to \Ab\\
  %%     X &\mapsto \lambda_n(X).
  %%   \end{aligned}
  %%   \]
  %% 
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    Notice first that for every $\oo$\nbd{}category $C$, we have $\lambda_n(\tau_{\leq n}^s(C))=\lambda_n(C)$. Suppose now that $C$ is free with basis $\Sigma=(\Sigma_n)_{n \in \mathbb{N}}$. Using Lemma \ref{lemma:adjlambdasusp} and Lemma \ref{lemma:freencattomonoid}, we obtain that for every abelian group $G$, we have
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        \begin{align*}
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      \Hom_{\Ab}(\lambda_n(C),G) &\simeq \Hom_{\Ab}(\lambda_n(\tau_{\leq n}^s(C)),G)\\
      &\simeq \Hom_{n\Cat}(\tau_{\leq n}^s(C),B^nG)\\
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      &\simeq \Hom_{\Set}(\Sigma_n,\vert G \vert)\\
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      &\simeq \Hom_{\Ab}(\mathbb{Z}\Sigma_n,G),
        \end{align*}
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        and it is easily checked that this isomorphism is induced by
        precomposition with the map
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        $\mathbb{Z}\Sigma_n \to \lambda_n(C)$ from the previous paragraph. The
        result follows then from the Yoneda Lemma.
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\end{proof}
\begin{paragr}
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  Let $C$ be a \emph{free} $\oo$\nbd{}category and write $\Sigma=(\Sigma_n)_{n \in \mathbb{N}}$ for its basis. For every $n \geq 0$ and every $\alpha \in \Sigma_n$, recall that we have proved in Proposition \ref{prop:countingfunction} the existence of a unique function $w_{\alpha} : C_n \to \mathbb{N}$ such that:
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  \begin{enumerate}[label=(\alph*)]
  \item\label{cond:countingfunctionfirst} $w_{\alpha}(\alpha)=1$,
  \item\label{cond:countingfunctionsecond} $w_{\alpha}(\beta)=0$ for every $\beta \in \Sigma_n$ such that $\beta\neq \alpha$,
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  \item\label{cond:countingfunctionthird} for every pair of $k$\nbd{}composable $n$\nbd{}cells of $C$ for some $k<n$, we have
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    \[
    w_{\alpha}(x\comp_k y)=w_{\alpha}(x) + w_{\alpha}(y).
    \]
  \end{enumerate}
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  We can then define for each $n \geq 0$, a map $w_n : C_n \to \mathbb{Z}\Sigma_n$ with the formula
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  \[w_n(x)=\sum_{\alpha \in \Sigma_n}w_{\alpha}(x)\cdot \alpha\]
  for every $x \in C_n$.

  
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  Condition \ref{cond:countingfunctionthird} implies that
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  \[
  w_n(x\comp_k y)=w_n(x)+w_n(y)
  \]
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  for every pair $(x,y)$ of $k$\nbd{}composable $n$\nbd{}cells of $C$, and conditions \ref{cond:countingfunctionfirst} and \ref{cond:countingfunctionsecond} imply that
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  \[
  w_n(\alpha)=\alpha
  \]
  for every $\alpha \in \Sigma_n$. 
\end{paragr}
\begin{proposition}\label{prop:abelianizationfreeoocat}
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  Let $C$ be a free $\oo$\nbd{}category and let $(\Sigma_n)_{n \in \mathbb{N}}$ be its basis. The chain complex $\lambda(C)$ is canonically isomorphic to the chain complex
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  \[
  \mathbb{Z}\Sigma_0 \overset{\partial}{\longleftarrow} \mathbb{Z}\Sigma_1 \overset{\partial}{\longleftarrow} \mathbb{Z}\Sigma_2 \overset{\partial}{\longleftarrow} \cdots
  \]
  where $\partial : \mathbb{Z}\Sigma_n \to \mathbb{Z}\Sigma_{n-1}$ is the linear map defined by the formula
  \[
  \partial(x)=w_{n-1}(\trgt(x))-w_{n-1}(\src(x))
  \]
 for every $x \in \Sigma_n$.

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 With this identification, if $C'$ is another free $\oo$\nbd{}category and if $F : C \to C'$ is an $\oo$\nbd{}functor (not necessarily rigid), then the map $\lambda_n(F) : \lambda_n(C) \to \lambda_{n}(C')$ reads
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 \[
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 \lambda_n(F)(x)=w'_n(F(x))
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 \]
 for every $x \in \Sigma_n$.
\end{proposition}
\begin{proof}
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  For $n \geq 0$, write $\phi_n : \mathbb{Z}\Sigma_n \to \lambda_n(C)$ for the map defined in \ref{paragr:abelpolmap} (which we know is an isomorphism from Lemma \ref{lemma:abelpol}).

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  The map $w_n: C_n \to \mathbb{Z}\Sigma_n$ induces a map $\mathbb{Z}C_n \to \mathbb{Z}\Sigma_n$ by linearity, which in turn induces a map $\lambda_n(C) \to \mathbb{Z}\Sigma_n$ (because $w_n(x \comp_k y) = w_n(x)+w_n(y)$ for every pair $(x,y)$ of $k$\nbd{}composable $n$\nbd{}cells). Write $\psi_n$ for this last map. It is immediate to check that the composition
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  \[
  \mathbb{Z}\Sigma_n \overset{\phi_n}{\longrightarrow} \lambda_n(C) \overset{\psi_n}{\longrightarrow} \mathbb{Z}\Sigma_n
  \]
  gives the identity on $\mathbb{Z}\Sigma_n$. Hence, $\psi_n$ is the inverse of $\phi_n$.

  Now, for $n>0$, 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
  \[
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  \mathbb{Z}\Sigma_n \overset{\phi_n}{\longrightarrow} \lambda_n(C) \overset{\partial}{\longrightarrow} \lambda_{n-1}(C) \overset{\psi_{n-1}}{\longrightarrow} \mathbb{Z}\Sigma_{n-1}.
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  \]
  The first part of the proposition follows then from Lemma \ref{lemma:abelpol}.

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  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
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  \[
  \mathbb{Z}\Sigma_n \overset{\phi_n}{\longrightarrow} \lambda_n(C) \overset{\lambda_n(F)}{\longrightarrow} \lambda_n(C') \overset{\psi_n}{\longrightarrow} \mathbb{Z}\Sigma'_n.\qedhere
  \]
 \end{proof}
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 \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
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   $g$, which means that \[\gamma^{\Ch}(f)=\gamma^{\Ch}(g).\]
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   \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}).
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   \begin{lemma}\label{lemma:abeloplax}
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    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 $\lambda(v)$.
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   \end{lemma}
   \begin{proof}
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     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.\ $\lambda_n(D)$).
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  Let $h_n$ be the map
     \[
     \begin{aligned}
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       h_n : \lambda_n(C) &\to \lambda_{n+1}(D)\\
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       [x] & \mapsto [\alpha_x].
       \end{aligned}
     \]
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     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
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     \[
     \partial (h_n(x)) + h_{n-1}(\partial(x)) = [v(x)] - [u(x)].
     \]
     Details are left to the reader. 
   \end{proof}
   \begin{proposition}
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     The abelianization functor $\lambda : \oo\Cat \to \Ch$ is left Quillen with respect to the folk model structure on $\oo\Cat$.
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   \end{proposition}
   \begin{proof}
          The fact that $\lambda$ is a left adjoint is Lemma \ref{lemma:adjlambda}.

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     A simple computation using Lemma \ref{prop:abelianizationfreeoocat} shows that for every $n\in \mathbb{N}$,
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     \[
     \lambda(i_n) : \lambda(\sS_{n-1}) \to \lambda(\sD_{n})
     \]
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   is a monomorphism with projective cokernel. Hence $\lambda$ sends folk cofibrations to cofibrations of chain complexes.
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     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}).
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     From Lemma \ref{lemma:abeloplax}, we conclude that $\lambda$ sends folk trivial cofibrations to trivial cofibrations of chain complexes.
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   \end{proof}
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   In particular, $\lambda$ is totally left derivable (when $\oo\Cat$ is equipped with folk weak equivalences). This motivates the following definition.
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   \begin{definition}\label{de:polhom}
     The \emph{polygraphic homology functor}
     \[
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     \sH^{\pol} : \ho(\oo\Cat^{\folk}) \to \ho(\Ch)
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     \]
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     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 $C$}. 
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   \end{definition}
   \begin{paragr}
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     Similarly to singular homology groups, for $k\geq0$ the $k$\nbd{}th polygraphic homology group of an $\oo$\nbd{}category $C$ is defined as
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     \[
     H^{\pol}_k(C):=H_k(\sH^{\pol}(C))
     \]
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     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
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     \[
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     P \to C,
     \]
     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})
         \[
     \begin{tikzcd}
        \oo\Cat \ar[d,"\gamma^{\folk}"'] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
      \ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\pol}"'] & \ho(\Ch).
       \ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >= 1em, Rightarrow]
     \end{tikzcd}
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     \]
   \end{paragr}
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   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.
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   \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 $\epsilon\in\{0,1\}$.

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      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
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      \[
      v \star \alpha' = \alpha \star u.
      \]
\end{lemma}
   \begin{proof}
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     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
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      \[
      \begin{tikzcd}
            (\sD_0\amalg \sD_0) \otimes C \simeq  C \amalg C \ar[d,"i_1 \otimes C"'] \ar[dr,"{\langle u, v \rangle}"] &\\
        \sD_1 \otimes C \ar[r,"\alpha"'] & D
      \end{tikzcd}
      \]
      (where $i_1 : \sD_0 \amalg \sD_0 \simeq \sS_0 \to \sD_1$ is the morphism introduced in \ref{paragr:inclusionsphereglobe}) is commutative.

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      Now, the hypotheses of the lemma yield the following commutative square
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      \[
      \begin{tikzcd}
        (\sD_0 \amalg \sD_0)\otimes C' \ar[d,"{i_1\otimes C'}"'] \ar[rr,"{\langle f'_0, f_1' \rangle}"] && D' \ar[d,"v"] \\
        \sD_1\otimes C'\ar[r,"\sD_1 \otimes u"'] & \sD_1\otimes C \ar[r,"\alpha"] & D
        \end{tikzcd}
      \]
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      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
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      \[
      \alpha' : \sD_1\otimes C' \to D'.
      \]
      The commutativity of the two induced triangles shows what we needed to prove.
   \end{proof}
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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.
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\begin{lemma}\label{lemma:oplaxpolhmlgy}
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  Let $u,v : C \to D$ be two $\oo$\nbd{}functors. If there exists an oplax transformation $u\Rightarrow v$, then
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  \[
  \sH^{\pol}(u)=\sH^{\pol}(v).
  \]
\end{lemma}
\begin{proof}
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  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}).
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  In the general case, let
  \[
  p : C' \to C
  \]
  and
  \[
  q : D' \to D
  \]
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  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
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  \[
  \begin{tikzcd}
    C' \ar[d,"p"] \ar[r,"u'"] & D' \ar[d,"q"] \\
    C \ar[r,"u"] & D
  \end{tikzcd}
  \text{ and }
  \begin{tikzcd}
    C' \ar[d,"p"] \ar[r,"v'"] & D' \ar[d,"q"] \\
    C \ar[r,"v"] & D
  \end{tikzcd}  
  \]
  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.
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\begin{proposition}\label{prop:oplaxhmtpypolhmlgy}
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  Let $u : C \to D$ be an $\oo$\nbd{}functor. If $u$ is an oplax homotopy equivalence, then the induced morphism
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  \[
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  \sH^{\pol}(u) : \sH^{\pol}(C) \to \sH^{\pol}(D)
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  \]
  is an isomorphism.
\end{proposition}
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\begin{paragr}\label{paragr:polhmlgythomeq}
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  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.
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\end{paragr}
\begin{remark}
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  Lemma \ref{lemma:liftingoplax}, Lemma \ref{lemma:oplaxpolhmlgy} and Proposition \ref{prop:oplaxhmtpypolhmlgy} are also true if we replace ``oplax'' by ``lax'' everywhere. 
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\end{remark}

\begin{paragr}
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     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 $\sH^{\pol}$:
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     \[
     \sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch).
     \]
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     Moreover, we also have a universal $2$-morphism which we again denote by $\alpha^{\pol}$:
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              \[
     \begin{tikzcd}
        \oo\Cat \ar[d,"\gamma^{\folk}"'] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
      \Ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\pol}"'] & \Ho(\Ch).
       \ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >= 1em, Rightarrow]
     \end{tikzcd}
     \]
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\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}
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   \section{Singular homology as derived abelianization}\label{section:singhmlgyderived}
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   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.
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   \begin{lemma}\label{lemma:nuhomotopical}
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     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.
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   \end{lemma}
   \begin{proof}
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     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}).
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   \end{proof}
   \begin{remark}
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     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.
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   \end{remark}
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   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}).
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   \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).
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   \end{lemma}
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   \begin{proof}
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     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}.
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   \end{proof}
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   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.
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    \begin{lemma}\label{lemma:adjeq}
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     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 $kg$.
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   \end{lemma}
   We can now state and prove the promised result.
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   \begin{theorem}\label{thm:hmlgyderived}
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     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
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     \[
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     \LL \lambda^{\Th} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)
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     \]
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     is isomorphic to the singular homology
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     \[
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     \sH^{\sing} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch).
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     \]
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   \end{theorem}
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   \begin{proof}
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     Let $\nu$ be the right adjoint of the abelianization functor (see Lemma \ref{lemma:adjlambda}) and consider the following adjunctions
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     \[
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     \begin{tikzcd}
       \Psh{\Delta} \ar[r,shift left,"c_{\omega}"] &\oo\Cat \ar[l,shift left,"N_{\omega}"] \ar[r,shift left,"\lambda"]& \Ch \ar[l,shift left,"\nu"]
       \end{tikzcd}
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     \]
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     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
     \[
     \nu : (\Ch,\W_{\Ch}) \to (\oo\Cat,\W^{\Th}),
     \]
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      thanks to Lemma \ref{lemma:nuhomotopical}.
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   \item The functor $N_{\omega}$ induces a morphism of localizers
     \[
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     N_{\omega} : (\oo\Cat,\W^{\Th}) \to (\Psh{\Delta},\W_{\Delta}),
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     \]
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     by definition of Thomason equivalences.
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   \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:
     \[
     \begin{tikzcd}
       \overline{\kappa} \simeq \overline{\lambda c_{\omega}} :  \Ho(\Psh{\Delta}) \ar[r,shift left] &  \ar[l,shift left] \Ho(\Ch) :\overline{N_{\omega}}\overline{\nu}.
       \end{tikzcd}
     \]
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     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:
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     \[
     \begin{tikzcd}
       \overline{\kappa} \overline{N_{\oo}} : \Ho(\oo\Cat^{\Th}) \ar[r,shift left]& \ar[l,shift left] \Ho(\Ch) : M \overline{N_{\oo}} \overline{\nu} \simeq \overline{\nu}.
       \end{tikzcd}
     \]
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     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. 
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   \end{proof}
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   \begin{remark}
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     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.
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   \end{remark}
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   \begin{paragr}\label{paragr:univmor}
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     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
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     \[
     \begin{tikzcd}
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        \oo\Cat \ar[d,"\gamma^{\Th}"'] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
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      \Ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"'] & \Ho(\Ch).
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       \ar[from=2-1,to=1-2,"\alpha^{\sing}",shorten <= 1em, shorten >= 1em, Rightarrow]
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     \end{tikzcd}
     \]
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     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
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     \[
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     \lambda c_{\oo} N_{\oo} \Rightarrow \lambda.
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     \]
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     Then $\alpha^{\sing}$ is nothing but the following composition of $2$\nbd{}morphisms
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     \[
     \begin{tikzcd}[column sep=huge]
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       \oo\Cat \ar[d,"\gamma^{\Th}"]\ar[r,bend left,"\lambda",""{name=A,below}] \ar[r,"\lambda c_{\oo} N_{\oo}"',""{name=B,above}] & \Ch \ar[d,"\gamma^{\Ch}"] \\
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        \Ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"'] & \Ho(\Ch),
        \ar[from=B,to=A,Rightarrow]\ar[from=1-1,to=2-2,phantom,"\simeq" description]
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     \end{tikzcd}
     \]
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     where the square is commutative (up to an isomorphism) because $\sH^{\sing}\simeq\overline{\lambda c_{\oo}} \overline{N_{\oo}}$.
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   \end{paragr}
   \section{Comparing homologies}
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   \begin{paragr}\label{paragr:cmparisonmap}
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     Recall from Proposition \ref{prop:folkisthom} that the identity functor on $\oo\Cat$ induces a morphism of localizers
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     \[(\oo\Cat,\W^{\folk}) \to (\oo\Cat,\W^{\Th}),\]
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     which in turn induces a functor
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     \[\J : \ho(\oo\Cat^{\folk}) \to \ho(\oo\Cat^{\Th}),\]
     such that
     \[
     \gamma^{\Th} = \J \circ \gamma^{\folk}.
     \]
     Now, consider the following triangle 
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     \begin{equation}\label{cmprisontrngle}
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     \begin{tikzcd}
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       \ho(\oo\Cat^{\folk}) \ar[r,"\J"] \ar[rd,"\sH^{\pol}"'] & \ho(\oo\Cat^{\Th}) \ar[d,"\sH^{\sing}"] \\
      & \ho(\Ch).
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     \end{tikzcd}
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     \end{equation}
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     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 $C$)
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     \[
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     \sH^{\pol}(C)\simeq \sH^{\sing}(C).
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     \]
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     As it happens, this is not possible as the following counter-example, due to Ara and Maltsiniotis, shows.
       \end{paragr}
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\begin{paragr}[Ara and Maltsiniotis' counter-example]\label{paragr:bubble}
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  Write $\mathbb{N}=(\mathbb{N},+,0)$ for the commutative monoid of non-negative integers  and let $C$ be the $2$\nbd{}category defined as
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  \[
  C:=B^2\mathbb{N}
  \]
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  (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 $\ho(\Ch)$)
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       \[
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     \begin{tikzcd}[column sep=small]
       \mathbb{Z} & 0 \ar[l] & \ar[l] \mathbb{Z} & \ar[l] 0 & \ar[l] 0 & \ar[l] \cdots
       \end{tikzcd}
     \]
     Hence, the polygraphic homology groups of $B$ are given by
     \[
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     H^{\pol}_k(C)=\begin{cases} \mathbb{Z} \text{ if } k=0,2\\ 0 \text{ in other cases.}\end{cases}  
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     \]
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     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). 
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\end{paragr}
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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 $\sH^{\pol}(\gamma^{\folk}(u))$.
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\begin{proposition}\label{prop:polhmlgynotinvariant}
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  There exists at least one Thomason equivalence \[u : C \to D\] such that the induced morphism
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  \[
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  \sH^{\pol}(u) : \sH^{\pol}(C) \to \sH^{\pol}(D)
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  \]
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  is not an isomorphism of $\ho(\Ch)$.
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\end{proposition}
\begin{proof}
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  Suppose the converse, which is that the functor
  \[
  \sH^{\pol} \circ \gamma^{\folk} : \oo\Cat \to \ho(\Ch)
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  \]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
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  \[
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  \J : \ho(\oo\Cat^{\folk}) \to \ho(\oo\Cat^{\Th}).
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  \]
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  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
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  \[
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  \overline{\sH^{\pol}} : \ho(\oo\Cat^{\Th}) \to \ho(\Ch)
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  \]
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  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
   \[
   \begin{tikzcd}
     \oo\Cat \ar[r,"\lambda"] \ar[d,"\gamma^{\Th}"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
     \ho(\oo\Cat^{\Th}) \ar[r,"\overline{\sH^{\pol}}"'] & \ho(\Ch).
     \ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >=1em,Rightarrow]
   \end{tikzcd}
   \]
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   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
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      \[
   \begin{tikzcd}
     \oo\Cat \ar[r,"\lambda"] \ar[d,"\gamma^{\Th}=\J\circ \gamma^{\folk}"'] & \Ch \ar[d,"\gamma^{\Ch}"] \\
     \ho(\oo\Cat^{\Th}) \ar[r,"G"'] & \ho(\Ch).
     \ar[from=2-1,to=1-2,"\beta",shorten <= 1em, shorten >=1em,Rightarrow]
   \end{tikzcd}
   \]
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   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
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   \[
      \begin{tikzcd}
     \oo\Cat \ar[r,"\lambda"] \ar[d,"\gamma^{\folk}"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
     \ho(\oo\Cat^{\folk}) \ar[d,"\J"] \ar[r,"\sH^{\pol}"'] & \ho(\Ch)\\
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     \ho(\oo\Cat^{\Th}) \ar[ru,"G"',bend right] &.
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     \ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >=1em,Rightarrow]\\
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     \ar[from=3-1,to=2-2,"\delta"',shorten <= 1em, shorten >= 1em,Rightarrow]
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   \end{tikzcd}
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      \]
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      But since $\J$ acts as a localization functor, $\delta$ also factorizes uniquely as
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      \[
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      \begin{tikzcd}[column sep=small] \ho(\oo\Cat^{\folk}) \ar[r,"\J"] & \ho(\oo\Cat^{\Th}) \ar[r,bend left,"\overline{\sH^{\pol}}",""{name=A,below}] \ar[r,bend right, "G"',pos=16/30,""{name=B,above}] & \ho(\Ch). \ar[from=B,to=A,Rightarrow,"\delta'"]\end{tikzcd}
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      \]
      Altogether we have that $\beta$ factorizes as
      \[
   \begin{tikzcd}
     \oo\Cat \ar[r,"\lambda"] \ar[d,"\gamma^{\Th}"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
     \ho(\oo\Cat^{\Th}) \ar[r,"\overline{\sH^{\pol}}",""{name=B,below}] & \ho(\Ch).
     \ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >=1em,Rightarrow]
     \ar[from=2-1,to=2-2,"G"',pos=16/30,bend right,""{name=A,above}]
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     \ar[from=A,to=B,Rightarrow,"\delta'"]
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   \end{tikzcd}
   \]
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   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
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   \[
   \sH^{\sing}\simeq \overline{\sH^{\pol}}.
   \]
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   But since $\J$ is the identity on objects, this implies that for every $\oo$\nbd{}category $C$ we have
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   \[
   \sH^{\sing}(C)\simeq \overline{\sH^{\pol}}(C)=\sH^{\pol}(C),
   \]
   which we know is impossible.
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 \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
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   homology of an $\oo$\nbd{}category is \emph{not} a well defined invariant. This
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   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}
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\begin{paragr}\label{paragr:defcancompmap}
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  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
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     \[
     \begin{tikzcd}
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       \oo\Cat \ar[d,"\gamma^{\Th}"] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
       \ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"] & \ho(\Ch),
       \ar[from=2-1,to=1-2,"\alpha^{\sing}",shorten <= 1em, shorten >= 1em, Rightarrow]
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     \end{tikzcd}
     \]
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     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
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     \begin{equation}\label{cmparisonmapdiag}
     \begin{tikzcd}
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       \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]
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     \end{tikzcd}
     \end{equation}
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     such that $\alpha^{\sing}$ factorizes as 
          \[
     \begin{tikzcd}
       \oo\Cat \ar[d,"\gamma^{\folk}"] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\
       \ho(\oo\Cat^{\folk})\ar[d,"\J"] \ar[r,"\sH^{\pol}",""{name=B,below}] & \ho(\Ch)\\
       \ho(\oo\Cat^{\Th}) \ar[ru,"\sH^{\sing}"',bend right=15] &
       \ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >= 1em, Rightarrow]
       \ar[from=3-1,to=B,Rightarrow,"\pi",shorten <= 1em, shorten >= 1em]
     \end{tikzcd}
     \]
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    %% \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.
   
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     Since $\J$ is nothing but the identity on objects, for every $\oo$\nbd{}category $C$, the natural transformation $\pi$ yields a map 
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     \[
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     \pi_C : \sH^{\sing}(C) \to \sH^{\pol}(C),
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     \]
     which we shall refer to as the \emph{canonical comparison map.}
\end{paragr}
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   \begin{remark}
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     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 $\Ch$
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     \[
    \lambda c_{\oo}N_{\oo}(C) \to \lambda(C)
     \]
     induced by the co-unit of $c_{\oo} \dashv N_{\oo}$.
   \end{remark}
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  % This motivates the following definition.
   \begin{definition}
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     An $\oo$\nbd{}category $C$ is said to be \emph{\good{}} when the canonical comparison map
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     \[
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     \pi_C : \sH^{\sing}(C) \to \sH^{\pol}(C)
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     \]
     is an isomorphism of $\ho(\Ch)$.
     \end{definition}
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  \begin{paragr}
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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. 
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     \end{paragr}
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%% \section{A criterion to detect \good{} $\oo$\nbd{}categories}
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%% 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.
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\begin{paragr}\label{paragr:prelimcriteriongoodcat}
  Similarly to \ref{paragr:cmparisonmap}, the morphism of localizers
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  \[
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 (\oo\Cat,\W^{\folk}) \to (\oo\Cat,\W^{\Th})
 \]
  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[