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 Leonard Guetta committed Oct 22, 2020 1 \chapter{Homology and abelianization of \texorpdfstring{$\oo$}{ω}-categories}  Leonard Guetta committed Jan 20, 2021 2 \chaptermark{Homology of $\omega$-categories}  Leonard Guetta committed Apr 29, 2020 3 4 \section{Homology via the nerve} \begin{paragr}  Leonard Guetta committed Nov 04, 2020 5 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:  Leonard Guetta committed Apr 29, 2020 6  \begin{itemize}  Leonard Guetta committed Oct 22, 2020 7  \item[-] the weak equivalences are the quasi-isomorphisms, i.e.\ morphisms of chain complexes that induce an isomorphism on homology groups,  Leonard Guetta committed Apr 29, 2020 8 9 10 11 12 13 14  \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}  Leonard Guetta committed Jan 20, 2021 15  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  Leonard Guetta committed Apr 29, 2020 16 17 18  $\partial(x):=\sum_{i=0}^n(-1)^i\partial_i(x).$  Leonard Guetta committed Sep 04, 2020 19 20 21 22 23 24  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.  Leonard Guetta committed Apr 29, 2020 25 26 \end{paragr} \begin{paragr}  Leonard Guetta committed Dec 27, 2020 27  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 $\partial$:  Leonard Guetta committed Apr 29, 2020 32 33 34 35  $\partial : \kappa_n(X) \to \kappa_{n-1}(X).$ This defines a chain complex $\kappa(X)$, which we call the \emph{normalized chain complex of $X$}. This yields a functor  Leonard Guetta committed Sep 04, 2020 36 37 38 39  \begin{align*} \kappa : \Psh{\Delta} &\to \Ch \\ X &\mapsto \kappa(X). \end{align*}  Leonard Guetta committed Apr 29, 2020 40 \end{paragr}  Leonard Guetta committed May 25, 2020 41 \begin{lemma}\label{lemma:normcompquil}  Leonard Guetta committed Oct 27, 2020 42 The functor $\kappa : \Psh{\Delta} \to \Ch$ is left Quillen and sends the weak equivalences of simplicial sets to quasi-isomorphisms.  Leonard Guetta committed Apr 29, 2020 43 44 \end{lemma} \begin{proof}  Leonard Guetta committed Oct 12, 2020 45 46 47 48 49 50 51 52  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 $\{\Lambda^i_n \hookrightarrow \Delta_n \vert n \in \mathbb{N}, 0 \leq i \leq n\}$  Leonard Guetta committed Dec 27, 2020 53  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.  Leonard Guetta committed Oct 12, 2020 54 55  \end{proof} \begin{remark}  Leonard Guetta committed Dec 27, 2020 56  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  Leonard Guetta committed Apr 29, 2020 57 58 59  $U : \Ch \simeq \Ab(\Delta) \to \Psh{\Delta}$  Leonard Guetta committed Oct 12, 2020 60 61  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}  Leonard Guetta committed Sep 04, 2020 62 63 64 65 \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}  Leonard Guetta committed Apr 29, 2020 66 \begin{definition}\label{def:hmlgycat}  Leonard Guetta committed Oct 25, 2020 67  The \emph{singular homology functor for $\oo$\nbd{}categories} $\sH^{\sing}$ is defined as the following composition  Leonard Guetta committed Apr 29, 2020 68  $ Leonard Guetta committed Sep 04, 2020 69  \sH^{\sing} : \ho(\oo\Cat^{\Th}) \overset{\overline{N_{\omega}}}{\longrightarrow} \ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \ho(\Ch).  Leonard Guetta committed Apr 29, 2020 70 $  Leonard Guetta committed Oct 01, 2020 71  For an $\oo$\nbd{}category $C$, $\sH^{\sing}(C)$ is the \emph{singular homology of $C$}.  Leonard Guetta committed Apr 29, 2020 72 \end{definition}  73 \begin{paragr}\label{paragr:singularhmlgygroup}  Leonard Guetta committed Oct 01, 2020 74  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  Leonard Guetta committed Sep 05, 2020 75 76 77  $H_k^{\sing}(C):=H_k(\sH^{\sing}(C))=H_k(\kappa(N_{\oo}(C))),$  Leonard Guetta committed Oct 27, 2020 78  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.  Leonard Guetta committed Sep 04, 2020 79 \end{paragr}  Leonard Guetta committed Sep 16, 2020 80 81  %% \begin{paragr}  Leonard Guetta committed Oct 25, 2020 82 %% 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.  Leonard Guetta committed Sep 16, 2020 83 84 85 %% 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}  Leonard Guetta committed Oct 13, 2020 86 87 88 89 \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  Leonard Guetta committed Oct 25, 2020 90  that \emph{singular homology of $\oo$\nbd{}categories} ought to be simply called  Leonard Guetta committed Oct 13, 2020 91 92  \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}.  Leonard Guetta committed Sep 05, 2020 93 \end{remark}  Leonard Guetta committed Apr 29, 2020 94 \begin{remark}  Leonard Guetta committed Dec 27, 2020 95  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.  Leonard Guetta committed Apr 29, 2020 96 97 \end{remark} \begin{paragr}  Leonard Guetta committed Sep 05, 2020 98  We will also denote by $\sH^{\sing}$ the morphism of op-prederivators defined as the following composition  Leonard Guetta committed Apr 29, 2020 99  $ Leonard Guetta committed Sep 05, 2020 100  \sH^{\sing} : \Ho(\oo\Cat^{\Th}) \overset{\overline{N_{\omega}}}{\longrightarrow} \Ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \Ho(\Ch).  Leonard Guetta committed Apr 29, 2020 101 102 $ \end{paragr}  Leonard Guetta committed Sep 17, 2020 103 \begin{proposition}\label{prop:singhmlgycocontinuous}  Leonard Guetta committed Oct 25, 2020 104  The singular homology $\sH^{\sing} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)$ is homotopy cocontinuous.  Leonard Guetta committed Sep 05, 2020 105 106 \end{proposition} \begin{proof}  Leonard Guetta committed Oct 25, 2020 107  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}.  Leonard Guetta committed Sep 05, 2020 108 \end{proof}  Leonard Guetta committed Apr 29, 2020 109 \section{Abelianization}  Leonard Guetta committed Dec 27, 2020 110 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 $G$.  Leonard Guetta committed Apr 29, 2020 111 \begin{paragr}  Leonard Guetta committed Oct 25, 2020 112  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  Leonard Guetta committed May 02, 2020 113 114 115  $x \comp_k y \sim x+y$  Leonard Guetta committed Feb 04, 2021 116 117 118 119  for all $x,y \in C_n$ that are $k$\nbd{}composable for some $k0$, consider the linear map  Leonard Guetta committed May 02, 2020 131  \begin{align*}  Leonard Guetta committed Sep 05, 2020 132 133  \mathbb{Z}C_n &\to \mathbb{Z}C_{n-1}\\ x \in C_n &\mapsto t(x)-s(x).  Leonard Guetta committed May 02, 2020 134  \end{align*}  Leonard Guetta committed Oct 25, 2020 135  The axioms of $\oo$\nbd{}categories imply that it induces a map  Leonard Guetta committed May 02, 2020 136  $ Leonard Guetta committed Sep 05, 2020 137  \partial : \lambda_{n}(C) \to \lambda_{n-1}(C)  Leonard Guetta committed May 02, 2020 138 $  Leonard Guetta committed Oct 01, 2020 139  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 $\lambda(C)$:  Leonard Guetta committed May 02, 2020 140  $ Leonard Guetta committed Sep 05, 2020 141  \lambda_0(C) \overset{\partial}{\longleftarrow} \lambda_1(C) \overset{\partial}{\longleftarrow} \lambda_2(C) \overset{\partial}{\longleftarrow} \cdots  Leonard Guetta committed May 02, 2020 142 $  Leonard Guetta committed Sep 08, 2020 143  and for every $f : C \to D$ a morphism of chain complexes  Leonard Guetta committed May 02, 2020 144  $ Leonard Guetta committed Sep 05, 2020 145  \lambda(f) : \lambda(C) \to \lambda(D).  Leonard Guetta committed May 02, 2020 146 147 148 149 150 151 152 $ Altogether, this defines a functor $\lambda : \omega\Cat \to \Ch,$ which we call the \emph{abelianization functor}. \end{paragr}  Leonard Guetta committed Sep 05, 2020 153 \begin{lemma}\label{lemma:adjlambda}  Leonard Guetta committed Oct 09, 2020 154  The functor $\lambda$ is a left adjoint.  Leonard Guetta committed Sep 05, 2020 155 156 \end{lemma} \begin{proof}  Leonard Guetta committed Oct 22, 2020 157  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 $\omega\Cat(\Ab) \to \omega\Cat$.  Leonard Guetta committed Sep 05, 2020 158 \end{proof}  Leonard Guetta committed Oct 01, 2020 159 As we shall now see, when the $\oo$\nbd{}category $C$ is \emph{free} the chain complex $\lambda(C)$ admits a nice expression.  Leonard Guetta committed Sep 06, 2020 160 \begin{paragr}  Leonard Guetta committed Oct 01, 2020 161  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 $n\geq 0$  Leonard Guetta committed Sep 07, 2020 162 163 164 165 166  \begin{align*} B^n : \Ab &\to n\Cat \\ G &\mapsto B^nG, \end{align*} where $\Ab$ is the category of abelian groups.  Leonard Guetta committed Sep 08, 2020 167   Leonard Guetta committed Oct 25, 2020 168  Besides, let us write $\lambda_n$ again for the functor  Leonard Guetta committed Sep 07, 2020 169  \begin{align*}  Leonard Guetta committed Sep 08, 2020 170 171  \lambda_n : n\Cat &\to \Ab\\ C&\mapsto \lambda_n(C).  Leonard Guetta committed Sep 07, 2020 172  \end{align*}  Leonard Guetta committed Sep 08, 2020 173 174 175 176  (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 $B^n : \Ab \to n\Cat$.  Leonard Guetta committed Sep 07, 2020 177 178 179 180 \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.  Leonard Guetta committed Oct 25, 2020 181  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  Leonard Guetta committed Sep 07, 2020 182 183 184 185  \begin{align*} \Hom_{n\Cat}(C,B^nG) &\to \Hom_{\Set}(C_n,\vert G \vert)\\ F &\mapsto F_n, \end{align*}  Leonard Guetta committed Feb 04, 2021 186  is injective and its image consists of those functions $f : C_n \to \vert G \vert$ such that:  Leonard Guetta committed Sep 07, 2020 187  \begin{enumerate}[label=(\roman*)]  Leonard Guetta committed Oct 01, 2020 188  \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 $ Leonard Guetta committed Oct 27, 2020 322  \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}.  Leonard Guetta committed Sep 09, 2020 323 324 325 $ The first part of the proposition follows then from Lemma \ref{lemma:abelpol}.  Leonard Guetta committed Oct 01, 2020 326  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  Leonard Guetta committed Sep 09, 2020 327 328 329 330  $\mathbb{Z}\Sigma_n \overset{\phi_n}{\longrightarrow} \lambda_n(C) \overset{\lambda_n(F)}{\longrightarrow} \lambda_n(C') \overset{\psi_n}{\longrightarrow} \mathbb{Z}\Sigma'_n.\qedhere$ \end{proof}  Leonard Guetta committed Dec 27, 2020 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345  \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  Leonard Guetta committed Jan 06, 2021 346  $g$, which means that $\gamma^{\Ch}(f)=\gamma^{\Ch}(g).$  Leonard Guetta committed Dec 27, 2020 347 348  \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}).  Leonard Guetta committed May 04, 2020 349  \begin{lemma}\label{lemma:abeloplax}  Leonard Guetta committed Dec 27, 2020 350 351  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 $\lambda(v)$.  Leonard Guetta committed May 04, 2020 352 353  \end{lemma} \begin{proof}  Leonard Guetta committed Oct 25, 2020 354  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.\ $\lambda_n(D)$).  Leonard Guetta committed May 04, 2020 355 356 357 358  Let $h_n$ be the map \begin{aligned}  Leonard Guetta committed Sep 09, 2020 359  h_n : \lambda_n(C) &\to \lambda_{n+1}(D)\\  Leonard Guetta committed May 04, 2020 360 361 362  [x] & \mapsto [\alpha_x]. \end{aligned}  Leonard Guetta committed Dec 27, 2020 363 364 365 366 367 368 369  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  Leonard Guetta committed May 04, 2020 370 371 372 373 374 375  $\partial (h_n(x)) + h_{n-1}(\partial(x)) = [v(x)] - [u(x)].$ Details are left to the reader. \end{proof} \begin{proposition}  Leonard Guetta committed Sep 03, 2020 376  The abelianization functor $\lambda : \oo\Cat \to \Ch$ is left Quillen with respect to the folk model structure on $\oo\Cat$.  Leonard Guetta committed May 04, 2020 377 378 379 380  \end{proposition} \begin{proof} The fact that $\lambda$ is a left adjoint is Lemma \ref{lemma:adjlambda}.  Leonard Guetta committed Sep 09, 2020 381  A simple computation using Lemma \ref{prop:abelianizationfreeoocat} shows that for every $n\in \mathbb{N}$,  Leonard Guetta committed May 04, 2020 382 383 384  $\lambda(i_n) : \lambda(\sS_{n-1}) \to \lambda(\sD_{n})$  Leonard Guetta committed Sep 09, 2020 385  is a monomorphism with projective cokernel. Hence $\lambda$ sends folk cofibrations to cofibrations of chain complexes.  Leonard Guetta committed May 04, 2020 386   Leonard Guetta committed Oct 25, 2020 387  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}).  Leonard Guetta committed Sep 09, 2020 388  From Lemma \ref{lemma:abeloplax}, we conclude that $\lambda$ sends folk trivial cofibrations to trivial cofibrations of chain complexes.  Leonard Guetta committed May 04, 2020 389  \end{proof}  Leonard Guetta committed Sep 09, 2020 390  In particular, $\lambda$ is totally left derivable (when $\oo\Cat$ is equipped with folk weak equivalences). This motivates the following definition.  Leonard Guetta committed May 04, 2020 391 392 393  \begin{definition}\label{de:polhom} The \emph{polygraphic homology functor} $ Leonard Guetta committed Sep 09, 2020 394  \sH^{\pol} : \ho(\oo\Cat^{\folk}) \to \ho(\Ch)  Leonard Guetta committed May 04, 2020 395 $  Leonard Guetta committed Oct 01, 2020 396  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 $C$}.  Leonard Guetta committed May 04, 2020 397 398  \end{definition} \begin{paragr}  Leonard Guetta committed Oct 27, 2020 399  Similarly to singular homology groups, for $k\geq0$ the $k$\nbd{}th polygraphic homology group of an $\oo$\nbd{}category $C$ is defined as  400 401 402  $H^{\pol}_k(C):=H_k(\sH^{\pol}(C))$  Leonard Guetta committed Oct 27, 2020 403  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  Leonard Guetta committed May 04, 2020 404  $ 405 406 407 408 409 410 411 412 413 414 415 416  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}  Leonard Guetta committed May 04, 2020 417 $  Leonard Guetta committed Feb 05, 2021 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 Moreover, the functor $\lambda$ being left Quillen, it is strongly derivable (Definition \ref{def:strnglyder}) and hence induces a morphism of op-prederivators, which we again denote by $\sH^{\pol}$: $\sH^{\pol} : \Ho(\oo\Cat^{\folk}) \to \Ho(\Ch).$ % We also have a universal $2$\nbd{}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}  Leonard Guetta committed Oct 27, 2020 439  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.  Leonard Guetta committed Sep 09, 2020 440 441 442 443 444 445 446 447 448 449  \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 $\epsilon\in\{0,1\}$.  Leonard Guetta committed Dec 27, 2020 450  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  Leonard Guetta committed Sep 09, 2020 451 452 453 454 455  $v \star \alpha' = \alpha \star u.$ \end{lemma} \begin{proof}  Leonard Guetta committed Feb 05, 2021 456  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  Leonard Guetta committed Sep 09, 2020 457 458 459 460 461 462 463 464  $\begin{tikzcd} (\sD_0\amalg \sD_0) \otimes C \simeq C \amalg C \ar[d,"i_1 \otimes C"'] \ar[dr,"{\langle u, v \rangle}"] &\\ \sD_1 \otimes C \ar[r,"\alpha"'] & D \end{tikzcd}$ (where $i_1 : \sD_0 \amalg \sD_0 \simeq \sS_0 \to \sD_1$ is the morphism introduced in \ref{paragr:inclusionsphereglobe}) is commutative.  Leonard Guetta committed Oct 25, 2020 465  Now, the hypotheses of the lemma yield the following commutative square  Leonard Guetta committed Sep 09, 2020 466 467 468 469 470 471  $\begin{tikzcd} (\sD_0 \amalg \sD_0)\otimes C' \ar[d,"{i_1\otimes C'}"'] \ar[rr,"{\langle f'_0, f_1' \rangle}"] && D' \ar[d,"v"] \\ \sD_1\otimes C'\ar[r,"\sD_1 \otimes u"'] & \sD_1\otimes C \ar[r,"\alpha"] & D \end{tikzcd}$  Leonard Guetta committed Dec 27, 2020 472 473 474 475 476  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  Leonard Guetta committed Sep 09, 2020 477 478 479 480 481  $\alpha' : \sD_1\otimes C' \to D'.$ The commutativity of the two induced triangles shows what we needed to prove. \end{proof}  Leonard Guetta committed Jan 20, 2021 482 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.  Leonard Guetta committed Oct 27, 2020 483 \begin{lemma}\label{lemma:oplaxpolhmlgy}  Leonard Guetta committed Dec 27, 2020 484  Let $u,v : C \to D$ be two $\oo$\nbd{}functors. If there exists an oplax transformation $u\Rightarrow v$, then  Leonard Guetta committed Sep 09, 2020 485 486 487 488 489  $\sH^{\pol}(u)=\sH^{\pol}(v).$ \end{lemma} \begin{proof}  Leonard Guetta committed Dec 27, 2020 490 491 492  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}).  Leonard Guetta committed Sep 09, 2020 493 494 495 496 497 498 499 500 501  In the general case, let $p : C' \to C$ and $q : D' \to D$  Leonard Guetta committed Feb 05, 2021 502  be folk trivial fibrations with $C'$ and $D'$ cofibrant. Using that $q$ is a trivial fibration and $C'$ is cofibrant, we know that there exist $u' : C' \to D'$ and $v' : C' \to D'$ such that the squares  Leonard Guetta committed Sep 09, 2020 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518  $\begin{tikzcd} C' \ar[d,"p"] \ar[r,"u'"] & D' \ar[d,"q"] \\ C \ar[r,"u"] & D \end{tikzcd} \text{ and } \begin{tikzcd} C' \ar[d,"p"] \ar[r,"v'"] & D' \ar[d,"q"] \\ C \ar[r,"v"] & D \end{tikzcd}$ 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.  Leonard Guetta committed Sep 17, 2020 519 \begin{proposition}\label{prop:oplaxhmtpypolhmlgy}  Leonard Guetta committed Oct 01, 2020 520  Let $u : C \to D$ be an $\oo$\nbd{}functor. If $u$ is an oplax homotopy equivalence, then the induced morphism  Leonard Guetta committed Sep 09, 2020 521  $ 522  \sH^{\pol}(u) : \sH^{\pol}(C) \to \sH^{\pol}(D)  Leonard Guetta committed Sep 09, 2020 523 524 525 $ is an isomorphism. \end{proposition}  526 \begin{paragr}\label{paragr:polhmlgythomeq}  Leonard Guetta committed Oct 25, 2020 527  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.  Leonard Guetta committed Sep 09, 2020 528 529 \end{paragr} \begin{remark}  Leonard Guetta committed Oct 27, 2020 530  Lemma \ref{lemma:liftingoplax}, Lemma \ref{lemma:oplaxpolhmlgy} and Proposition \ref{prop:oplaxhmtpypolhmlgy} are also true if we replace oplax'' by lax'' everywhere.  Leonard Guetta committed Sep 09, 2020 531 532 \end{remark}  Leonard Guetta committed Feb 05, 2021 533   Leonard Guetta committed Oct 20, 2020 534  \section{Singular homology as derived abelianization}\label{section:singhmlgyderived}  Leonard Guetta committed Oct 27, 2020 535  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.  Leonard Guetta committed May 08, 2020 536  \begin{lemma}\label{lemma:nuhomotopical}  Leonard Guetta committed Feb 05, 2021 537 538  Let $\nu : \Ch \to \oo\Cat$ be the right adjoint of the abelianization functor (see Lemma \ref{lemma:adjlambda}). This functor sends the quasi-isomorphisms to Thomason equivalences.  Leonard Guetta committed May 04, 2020 539 540  \end{lemma} \begin{proof}  Leonard Guetta committed Oct 25, 2020 541  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}).  Leonard Guetta committed May 04, 2020 542 543  \end{proof} \begin{remark}  Leonard Guetta committed Feb 05, 2021 544 545  The proof of the previous lemma shows the stronger result that $\nu$ sends the quasi-ismorphisms to folk weak equivalences. This will be of no use in the sequel.  Leonard Guetta committed May 04, 2020 546  \end{remark}  Leonard Guetta committed Oct 25, 2020 547  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}).  Leonard Guetta committed May 08, 2020 548 549 550 551 552 553 554 555 556  \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).  Leonard Guetta committed May 04, 2020 557  \end{lemma}  Leonard Guetta committed May 08, 2020 558  \begin{proof}  Leonard Guetta committed May 25, 2020 559  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}.  Leonard Guetta committed May 08, 2020 560  \end{proof}  Leonard Guetta committed Feb 05, 2021 561  Recall now that the notions of adjunction and equivalence are valid in every $2$\nbd{}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 $2$\nbd{}category is the $2$\nbd{}category of categories.  Leonard Guetta committed Sep 09, 2020 562  \begin{lemma}\label{lemma:adjeq}  Leonard Guetta committed May 25, 2020 563  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 $kg$.  Leonard Guetta committed May 08, 2020 564 565  \end{lemma} We can now state and prove the promised result.  Leonard Guetta committed Oct 22, 2020 566  \begin{theorem}\label{thm:hmlgyderived}  Leonard Guetta committed Oct 25, 2020 567  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  Leonard Guetta committed May 08, 2020 568  $ Leonard Guetta committed Sep 09, 2020 569  \LL \lambda^{\Th} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch)  Leonard Guetta committed May 08, 2020 570 $  Leonard Guetta committed Oct 25, 2020 571  is isomorphic to the singular homology  Leonard Guetta committed May 08, 2020 572  $ Leonard Guetta committed Sep 09, 2020 573  \sH^{\sing} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Ch).  Leonard Guetta committed May 08, 2020 574 $  Leonard Guetta committed Oct 22, 2020 575  \end{theorem}  Leonard Guetta committed May 08, 2020 576  \begin{proof}  Leonard Guetta committed May 25, 2020 577  Let $\nu$ be the right adjoint of the abelianization functor (see Lemma \ref{lemma:adjlambda}) and consider the following adjunctions  Leonard Guetta committed May 08, 2020 578  $ Leonard Guetta committed May 25, 2020 579 580 581  \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}  Leonard Guetta committed May 08, 2020 582 $  Leonard Guetta committed May 25, 2020 583 584 585 586 587 588 589  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 $\nu : (\Ch,\W_{\Ch}) \to (\oo\Cat,\W^{\Th}),$  Leonard Guetta committed Sep 09, 2020 590  thanks to Lemma \ref{lemma:nuhomotopical}.  Leonard Guetta committed May 25, 2020 591 592  \item The functor $N_{\omega}$ induces a morphism of localizers $ Leonard Guetta committed May 27, 2020 593  N_{\omega} : (\oo\Cat,\W^{\Th}) \to (\Psh{\Delta},\W_{\Delta}),  Leonard Guetta committed May 25, 2020 594 $  Leonard Guetta committed Sep 09, 2020 595  by definition of Thomason equivalences.  Leonard Guetta committed May 25, 2020 596 597 598 599 600 601 602 603 604 605 606 607  \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: $\begin{tikzcd} \overline{\kappa} \simeq \overline{\lambda c_{\omega}} : \Ho(\Psh{\Delta}) \ar[r,shift left] & \ar[l,shift left] \Ho(\Ch) :\overline{N_{\omega}}\overline{\nu}. \end{tikzcd}$  Leonard Guetta committed Oct 01, 2020 608  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:  Leonard Guetta committed May 25, 2020 609 610 611 612 613  $\begin{tikzcd} \overline{\kappa} \overline{N_{\oo}} : \Ho(\oo\Cat^{\Th}) \ar[r,shift left]& \ar[l,shift left] \Ho(\Ch) : M \overline{N_{\oo}} \overline{\nu} \simeq \overline{\nu}. \end{tikzcd}$  Leonard Guetta committed Dec 27, 2020 614  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.  Leonard Guetta committed May 08, 2020 615  \end{proof}  Leonard Guetta committed May 25, 2020 616  \begin{remark}  Leonard Guetta committed Feb 05, 2021 617 618 619 620  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. However, 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.  Leonard Guetta committed May 25, 2020 621  \end{remark}  Leonard Guetta committed Jun 01, 2020 622  \begin{paragr}\label{paragr:univmor}  Leonard Guetta committed Feb 05, 2021 623 624  Since $\sH^{\sing} : \ho(\oo\Cat^{\Th}) \to \ho(\Ch)$ is the left derived functor of the abelianization functor, it comes with a universal natural transformation  Leonard Guetta committed Jun 01, 2020 625 626  $\begin{tikzcd}  627  \oo\Cat \ar[d,"\gamma^{\Th}"'] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\  Leonard Guetta committed Feb 05, 2021 628  \ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"'] & \ho(\Ch).  629  \ar[from=2-1,to=1-2,"\alpha^{\sing}",shorten <= 1em, shorten >= 1em, Rightarrow]  Leonard Guetta committed Jun 01, 2020 630 631  \end{tikzcd}$  Leonard Guetta committed Feb 05, 2021 632 633 634 635 636  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 a natural transformation  Leonard Guetta committed Jun 01, 2020 637  $ Leonard Guetta committed Sep 09, 2020 638  \lambda c_{\oo} N_{\oo} \Rightarrow \lambda.  Leonard Guetta committed Jun 01, 2020 639 $  Leonard Guetta committed Feb 05, 2021 640  Then $\alpha^{\sing}$ is nothing but the following composition of natural transformations  Leonard Guetta committed Jun 01, 2020 641 642  $\begin{tikzcd}[column sep=huge]  Leonard Guetta committed Oct 27, 2020 643  \oo\Cat \ar[d,"\gamma^{\Th}"]\ar[r,bend left,"\lambda",""{name=A,below}] \ar[r,"\lambda c_{\oo} N_{\oo}"',""{name=B,above}] & \Ch \ar[d,"\gamma^{\Ch}"] \\  Leonard Guetta committed Feb 05, 2021 644  \ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"'] & \ho(\Ch),  Leonard Guetta committed Sep 09, 2020 645  \ar[from=B,to=A,Rightarrow]\ar[from=1-1,to=2-2,phantom,"\simeq" description]  Leonard Guetta committed Jun 01, 2020 646 647  \end{tikzcd}$  Leonard Guetta committed Jan 20, 2021 648  where the square is commutative (up to an isomorphism) because $\sH^{\sing}\simeq\overline{\lambda c_{\oo}} \overline{N_{\oo}}$.  Leonard Guetta committed Jun 01, 2020 649 650  \end{paragr} \section{Comparing homologies}  651  \begin{paragr}\label{paragr:cmparisonmap}  Leonard Guetta committed Sep 09, 2020 652  Recall from Proposition \ref{prop:folkisthom} that the identity functor on $\oo\Cat$ induces a morphism of localizers  Leonard Guetta committed Jun 01, 2020 653  $(\oo\Cat,\W^{\folk}) \to (\oo\Cat,\W^{\Th}),$  Leonard Guetta committed Sep 09, 2020 654  which in turn induces a functor  Leonard Guetta committed Sep 14, 2020 655 656 657 658 659 660  $\J : \ho(\oo\Cat^{\folk}) \to \ho(\oo\Cat^{\Th}),$ such that $\gamma^{\Th} = \J \circ \gamma^{\folk}.$ Now, consider the following triangle  661  \label{cmprisontrngle}  Leonard Guetta committed May 27, 2020 662  \begin{tikzcd}  Leonard Guetta committed Sep 09, 2020 663 664  \ho(\oo\Cat^{\folk}) \ar[r,"\J"] \ar[rd,"\sH^{\pol}"'] & \ho(\oo\Cat^{\Th}) \ar[d,"\sH^{\sing}"] \\ & \ho(\Ch).  Leonard Guetta committed May 27, 2020 665  \end{tikzcd}  666   Leonard Guetta committed Oct 01, 2020 667  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 $C$)  668  $ Leonard Guetta committed Sep 14, 2020 669  \sH^{\pol}(C)\simeq \sH^{\sing}(C).  670 $  Leonard Guetta committed Sep 14, 2020 671 672  As it happens, this is not possible as the following counter-example, due to Ara and Maltsiniotis, shows. \end{paragr}  Leonard Guetta committed Oct 09, 2020 673 \begin{paragr}[Ara and Maltsiniotis' counter-example]\label{paragr:bubble}  Leonard Guetta committed Oct 01, 2020 674  Write $\mathbb{N}=(\mathbb{N},+,0)$ for the commutative monoid of non-negative integers and let $C$ be the $2$\nbd{}category defined as  Leonard Guetta committed Sep 10, 2020 675 676 677  $C:=B^2\mathbb{N}$  Leonard Guetta committed Oct 27, 2020 678  (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 $\ho(\Ch)$)  Leonard Guetta committed Sep 10, 2020 679  $ Leonard Guetta committed May 27, 2020 680 681 682 683 684 685  \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 $ 686  H^{\pol}_k(C)=\begin{cases} \mathbb{Z} \text{ if } k=0,2\\ 0 \text{ in other cases.}\end{cases}  Leonard Guetta committed May 27, 2020 687 $  Leonard Guetta committed Oct 21, 2020 688  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).  Leonard Guetta committed Sep 10, 2020 689 \end{paragr}  Leonard Guetta committed Sep 14, 2020 690 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 $\sH^{\pol}(\gamma^{\folk}(u))$.  Leonard Guetta committed Sep 15, 2020 691 \begin{proposition}\label{prop:polhmlgynotinvariant}  692  There exists at least one Thomason equivalence $u : C \to D$ such that the induced morphism  Leonard Guetta committed Sep 10, 2020 693  $ 694  \sH^{\pol}(u) : \sH^{\pol}(C) \to \sH^{\pol}(D)  Leonard Guetta committed Sep 10, 2020 695 $  696  is not an isomorphism of $\ho(\Ch)$.  Leonard Guetta committed Sep 10, 2020 697 698 \end{proposition} \begin{proof}  Leonard Guetta committed Oct 25, 2020 699 700 701  Suppose the converse, which is that the functor $\sH^{\pol} \circ \gamma^{\folk} : \oo\Cat \to \ho(\Ch)  Leonard Guetta committed Feb 05, 2021 702 $sends the 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  703  $ 704  \J : \ho(\oo\Cat^{\folk}) \to \ho(\oo\Cat^{\Th}).  705 $  706  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  707  $ 708  \overline{\sH^{\pol}} : \ho(\oo\Cat^{\Th}) \to \ho(\Ch)  709 $  710 711 712 713 714 715 716 717 718 719 720 721  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 $\begin{tikzcd} \oo\Cat \ar[r,"\lambda"] \ar[d,"\gamma^{\Th}"] & \Ch \ar[d,"\gamma^{\Ch}"] \\ \ho(\oo\Cat^{\Th}) \ar[r,"\overline{\sH^{\pol}}"'] & \ho(\Ch). \ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >=1em,Rightarrow] \end{tikzcd}$  Leonard Guetta committed Oct 27, 2020 722  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  723 724 725 726 727 728 729  $\begin{tikzcd} \oo\Cat \ar[r,"\lambda"] \ar[d,"\gamma^{\Th}=\J\circ \gamma^{\folk}"'] & \Ch \ar[d,"\gamma^{\Ch}"] \\ \ho(\oo\Cat^{\Th}) \ar[r,"G"'] & \ho(\Ch). \ar[from=2-1,to=1-2,"\beta",shorten <= 1em, shorten >=1em,Rightarrow] \end{tikzcd}$  Leonard Guetta committed Oct 27, 2020 730  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  731 732 733 734  $\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)\\  Leonard Guetta committed Oct 14, 2020 735  \ho(\oo\Cat^{\Th}) \ar[ru,"G"',bend right] &.  736  \ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >=1em,Rightarrow]\\  Leonard Guetta committed Sep 15, 2020 737  \ar[from=3-1,to=2-2,"\delta"',shorten <= 1em, shorten >= 1em,Rightarrow]  Leonard Guetta committed Oct 14, 2020 738  \end{tikzcd}  739 $  Leonard Guetta committed Dec 27, 2020 740  But since $\J$ acts as a localization functor, $\delta$ also factorizes uniquely as  741  $ Leonard Guetta committed Oct 14, 2020 742  \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}  743 744 745 746 747 748 749 750 $ Altogether we have that $\beta$ factorizes as $\begin{tikzcd} \oo\Cat \ar[r,"\lambda"] \ar[d,"\gamma^{\Th}"] & \Ch \ar[d,"\gamma^{\Ch}"] \\ \ho(\oo\Cat^{\Th}) \ar[r,"\overline{\sH^{\pol}}",""{name=B,below}] & \ho(\Ch). \ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >=1em,Rightarrow] \ar[from=2-1,to=2-2,"G"',pos=16/30,bend right,""{name=A,above}]  Leonard Guetta committed Oct 14, 2020 751  \ar[from=A,to=B,Rightarrow,"\delta'"]  752 753  \end{tikzcd}$  Leonard Guetta committed Oct 27, 2020 754  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  755 756 757  $\sH^{\sing}\simeq \overline{\sH^{\pol}}.$  Leonard Guetta committed Oct 01, 2020 758  But since $\J$ is the identity on objects, this implies that for every $\oo$\nbd{}category $C$ we have  759 760 761 762  $\sH^{\sing}(C)\simeq \overline{\sH^{\pol}}(C)=\sH^{\pol}(C),$ which we know is impossible.  Leonard Guetta committed Oct 13, 2020 763 764 765 766  \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  Leonard Guetta committed Oct 25, 2020 767  homology of an $\oo$\nbd{}category is \emph{not} a well defined invariant. This  Leonard Guetta committed Oct 13, 2020 768 769 770  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}  Leonard Guetta committed Sep 16, 2020 771 \begin{paragr}\label{paragr:defcancompmap}  Leonard Guetta committed Feb 05, 2021 772  Even though triangle \eqref{cmprisontrngle} is not commutative (even up to an isomorphism), it can be filled up with a $2$\nbd{}morphism. Indeed, consider the following $2$\nbd{}square  Leonard Guetta committed Sep 14, 2020 773 774  $\begin{tikzcd}  Leonard Guetta committed Sep 15, 2020 775 776 777  \oo\Cat \ar[d,"\gamma^{\Th}"] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\ \ho(\oo\Cat^{\Th}) \ar[r,"\sH^{\sing}"] & \ho(\Ch), \ar[from=2-1,to=1-2,"\alpha^{\sing}",shorten <= 1em, shorten >= 1em, Rightarrow]  Leonard Guetta committed Sep 14, 2020 778 779  \end{tikzcd}$  Leonard Guetta committed Feb 05, 2021 780 781 782  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 the folk weak equivalences, we obtain by universal property (see \ref{paragr:defleftderived}) a unique natural transformation  Leonard Guetta committed Sep 14, 2020 783 784  \label{cmparisonmapdiag} \begin{tikzcd}  Leonard Guetta committed Sep 15, 2020 785 786  \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]  Leonard Guetta committed Sep 14, 2020 787 788  \end{tikzcd}  Leonard Guetta committed Sep 15, 2020 789 790 791 792 793 794 795 796 797 798  such that $\alpha^{\sing}$ factorizes as $\begin{tikzcd} \oo\Cat \ar[d,"\gamma^{\folk}"] \ar[r,"\lambda"] & \Ch \ar[d,"\gamma^{\Ch}"] \\ \ho(\oo\Cat^{\folk})\ar[d,"\J"] \ar[r,"\sH^{\pol}",""{name=B,below}] & \ho(\Ch)\\ \ho(\oo\Cat^{\Th}) \ar[ru,"\sH^{\sing}"',bend right=15] & \ar[from=2-1,to=1-2,"\alpha^{\pol}",shorten <= 1em, shorten >= 1em, Rightarrow] \ar[from=3-1,to=B,Rightarrow,"\pi",shorten <= 1em, shorten >= 1em] \end{tikzcd}$  Leonard Guetta committed Sep 16, 2020 799 800 801 802 803 804 805 806  %% \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} %% %% is commutative.  Leonard Guetta committed Oct 25, 2020 807  Since $\J$ is nothing but the identity on objects, for every $\oo$\nbd{}category $C$, the natural transformation $\pi$ yields a map  Leonard Guetta committed Sep 14, 2020 808  $ Leonard Guetta committed Sep 15, 2020 809  \pi_C : \sH^{\sing}(C) \to \sH^{\pol}(C),  Leonard Guetta committed Sep 14, 2020 810 811 812 $ which we shall refer to as the \emph{canonical comparison map.} \end{paragr}  Leonard Guetta committed Sep 16, 2020 813  \begin{remark}  Leonard Guetta committed Oct 25, 2020 814  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 $\Ch$  Leonard Guetta committed Sep 16, 2020 815 816 817 818 819  $\lambda c_{\oo}N_{\oo}(C) \to \lambda(C)$ induced by the co-unit of $c_{\oo} \dashv N_{\oo}$. \end{remark}  Leonard Guetta committed Sep 14, 2020 820 821  % This motivates the following definition. \begin{definition}  Leonard Guetta committed Oct 01, 2020 822  An $\oo$\nbd{}category $C$ is said to be \emph{\good{}} when the canonical comparison map  Leonard Guetta committed Sep 14, 2020 823  $ Leonard Guetta committed Sep 15, 2020 824  \pi_C : \sH^{\sing}(C) \to \sH^{\pol}(C)  Leonard Guetta committed Sep 14, 2020 825 826 827 $ is an isomorphism of $\ho(\Ch)$. \end{definition}  Leonard Guetta committed Sep 16, 2020 828 829  \begin{paragr}  Leonard Guetta committed Oct 25, 2020 830 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.  Leonard Guetta committed Sep 15, 2020 831  \end{paragr}  Leonard Guetta committed Oct 01, 2020 832 %% \section{A criterion to detect \good{} $\oo$\nbd{}categories}  Leonard Guetta committed Oct 25, 2020 833 %% 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.  834   Leonard Guetta committed Sep 17, 2020 835 836 \begin{paragr}\label{paragr:prelimcriteriongoodcat} Similarly to \ref{paragr:cmparisonmap}, the morphism of localizers  837  $ Leonard Guetta committed Sep 17, 2020 838 839 840 841 842 843 844 845 846 847 848 849  (\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}  850 $  Leonard Guetta committed Sep 17, 2020 851 852  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 $\ho(\oo\Cat^{\Th})$  Leonard Guetta committed Oct 27, 2020 853  \label{equation:Jhocolim