introduction.tex 30.2 KB
 Leonard Guetta committed Aug 24, 2020 1 \chapter*{Introduction}  Leonard Guetta committed Oct 20, 2020 2 3 \addcontentsline{toc}{chapter}{Introduction} %% For this chapter to appear in toc  Leonard Guetta committed Oct 21, 2020 4 5 6 7 8 The general framework in which this dissertation takes place is the \emph{homotopy theory of strict $\oo$\nbd{}categories}, and, as the title suggests, its focus is on homological aspects of this theory. The goal is to study and compare two different homological invariants for strict $\oo$\nbd{}categories; that is to say, two different  Leonard Guetta committed Oct 20, 2020 9 10 11 functors $\mathbf{Str}\oo\Cat \to \ho(\Ch)$ from the category of strict $\oo$\nbd{}categories to the homotopy category of non-negatively graded chain complexes (i.e.\ the localization of the category of non-negatively graded chain  Leonard Guetta committed Oct 21, 2020 12 13 14 15 16 complexes with respect to the quasi\nbd{}isomorphisms). Before we enter into the heart of the subject, let us emphasize that, with the sole exception of the end of this introduction, all the $\oo$\nbd{}categories that we consider are strict $\oo$\nbd{}categories. Hence, we drop the adjective  Leonard Guetta committed Oct 20, 2020 17 strict'' and simply say \emph{$\oo$\nbd{}category} instead of \emph{strict  Leonard Guetta committed Oct 21, 2020 18 19  $\oo$\nbd{}category} and we write $\oo\Cat$ instead of $\mathbf{Str}\oo\Cat$ for the category of (strict) $\oo$\nbd{}categories.  Leonard Guetta committed Aug 24, 2020 20 21   Leonard Guetta committed Oct 20, 2020 22 \begin{named}[Background: $\oo$-categories as spaces] The homotopy theory of  Leonard Guetta committed Oct 21, 2020 23 24  $\oo$\nbd{}categories begins with the nerve functor introduced by Street in \cite{street1987algebra}  Leonard Guetta committed Oct 20, 2020 25 26 27  $N_{\omega} : \oo\Cat \to \Psh{\Delta}$  Leonard Guetta committed Oct 25, 2020 28  that associates to every $\oo$\nbd{}category $C$ a simplicial set $N_{\oo}(C)$  Leonard Guetta committed Oct 20, 2020 29  called the \emph{nerve of $C$}, generalizing the usual nerve of (small)  Leonard Guetta committed Oct 21, 2020 30  categories. Using this functor, we can transfer the homotopy theory of  Leonard Guetta committed Oct 20, 2020 31  simplicial sets to $\oo$\nbd{}categories, as it is done for example in the  Leonard Guetta committed Oct 21, 2020 32 33 34  articles \cite{ara2014vers,ara2018theoreme,gagna2018strict,ara2019quillen,ara2020theoreme,ara2020comparaison}. Following the terminology of these articles, a morphism $f : C \to D$ of  Leonard Guetta committed Dec 18, 2020 35  $\oo\Cat$ is a \emph{Thomason equivalence} if $N_{\omega}(f)$ is a Kan--Quillen  Leonard Guetta committed Oct 20, 2020 36 37 38 39 40 41  weak equivalence of simplicial sets. By definition, the nerve functor induces a functor at the level of homotopy categories $\overline{N_{\omega}} : \ho(\oo\Cat^{\Th}) \to \ho(\Psh{\Delta}),$ where $\ho(\oo\Cat^{\Th})$ is the localization of $\oo\Cat$ with respect to  Leonard Guetta committed Oct 21, 2020 42  the Thomason equivalences and $\ho(\Psh{\Delta})$ is the localization of  Leonard Guetta committed Dec 18, 2020 43  $\Psh{\Delta}$ with respect to the Kan--Quillen weak equivalences of simplicial  Leonard Guetta committed Oct 21, 2020 44 45 46 47 48 49 50 51 52 53  sets. As it so happens, the functor $\overline{N_{\omega}}$ is an equivalence of categories, as proved by Gagna in \cite{gagna2018strict}. In other words, the homotopy theory of $\oo$\nbd{}categories induced by Thomason equivalences is the same as the homotopy theory of spaces. Gagna's result is in fact a generalization of the analogous result for the usual nerve of small categories, which is attributed to Quillen in \cite{illusie1972complexe}. In the case of small categories, Thomason even showed the existence of a model structure whose weak equivalences are the ones induced by the nerve functor \cite{thomason1980cat}. The analogous result for $\oo\Cat$ is conjectured but not yet established \cite{ara2014vers}.  Leonard Guetta committed Aug 25, 2020 54 55 \end{named} \begin{named}[Two homologies for $\oo$-categories]  Leonard Guetta committed Oct 21, 2020 56  Keeping in mind the nerve functor of Street, a natural thing to do is to  Leonard Guetta committed Oct 20, 2020 57  define the \emph{$k$-th homology group of an $\oo$\nbd{}category $C$} as the  Leonard Guetta committed Dec 15, 2020 58  $k$\nbd{}th homology group of the nerve of $C$. In light of Gagna's result, these  Leonard Guetta committed Oct 20, 2020 59 60 61 62  homology groups are just another way of looking at the homology groups of spaces. In order to explicitly avoid future confusion, we shall now use the name \emph{singular homology groups} of $C$ for these homology groups and the notation $H^{\sing}_k(C)$.  Leonard Guetta committed Aug 25, 2020 63   Leonard Guetta committed Oct 20, 2020 64 65  On the other hand, Métayer gives a definition in \cite{metayer2003resolutions} of other homology groups for $\oo$\nbd{}categories. This definition is based  Leonard Guetta committed Oct 21, 2020 66 67  on the notion of \emph{$\oo$\nbd{}categories free on a polygraph} (also known as \emph{$\oo$\nbd{}categories free on a computad}), which are  Leonard Guetta committed Oct 20, 2020 68 69  $\oo$\nbd{}categories that are obtained from the empty category by recursively freely adjoining cells. From now on, we simply say \emph{free  Leonard Guetta committed Dec 15, 2020 70  $\oo$\nbd{}category}. Métayer observed that every $\oo$\nbd{}category $C$ admits  Leonard Guetta committed Oct 20, 2020 71 72  what we call a \emph{polygraphic resolution}, which means that there exists a free $\oo$\nbd{}category $P$ and a morphism of $\oo\Cat$  Leonard Guetta committed Aug 25, 2020 73  $ Leonard Guetta committed Dec 15, 2020 74  f : P \to C  Leonard Guetta committed Aug 25, 2020 75 $  Leonard Guetta committed Oct 20, 2020 76  that satisfies properties formally resembling those of trivial fibrations of  Leonard Guetta committed Oct 21, 2020 77  topological spaces (or of simplicial sets). Furthermore, every free  Leonard Guetta committed Dec 15, 2020 78  $\oo$\nbd{}category $P$ can be abelianized'' to a chain complex $\lambda(P)$ and  Leonard Guetta committed Oct 20, 2020 79 80 81 82 83 84  Métayer proved that for two different polygraphic resolutions of the same $\oo$\nbd{}category, $P \to C$ and $P' \to C$, the chain complexes $\lambda(P)$ and $\lambda(P')$ are quasi-isomorphic. Hence, we can define the \emph{$k$-th polygraphic homology group} of $C$, denoted by $H_k^{\pol}(C)$, as the $k$-th homology group of $\lambda(P)$ for any polygraphic resolution $P \to C$.  Leonard Guetta committed Aug 26, 2020 85   Leonard Guetta committed Oct 21, 2020 86  One is then led to the following question:  Leonard Guetta committed Aug 26, 2020 87  \begin{center}  Leonard Guetta committed Oct 21, 2020 88  Do we have $H_{\bullet}^{\pol}(C) \simeq H_{\bullet}^{\sing}(C)$ for every  Leonard Guetta committed Oct 20, 2020 89 90  $\oo$\nbd{}category $C$? \end{center}  Leonard Guetta committed Aug 26, 2020 91 92  \iffalse \label{naivequestion}\tag{\textbf{Q}}  Leonard Guetta committed Oct 25, 2020 93  \text{Do we have }H_k^{\pol}(C) \simeq H_k^{\sing}(C)\text{ for every }\oo\text{-category }C\text{ ? }  Leonard Guetta committed Aug 26, 2020 94   Leonard Guetta committed Dec 15, 2020 95  \fi A first partial answer to this question is given by Lafont and Métayer in  Leonard Guetta committed Oct 21, 2020 96 97  \cite{lafont2009polygraphic}: for a monoid $M$ (seen as category with one object and hence as  Leonard Guetta committed Oct 20, 2020 98 99  an $\oo$\nbd{}category), we have $H_{\bullet}^{\pol}(M) \simeq H_{\bullet}^{\sing}(M)$. In fact, the original motivation for polygraphic  Leonard Guetta committed Oct 21, 2020 100 101  homology was the homology of monoids and is part of a program that generalizes to higher dimension the results of Squier on the rewriting theory of monoids  Leonard Guetta committed Dec 15, 2020 102  \cite{guiraud2006termination,lafont2007algebra,guiraud2009higher,guiraud2018polygraphs}. However, interestingly  Leonard Guetta committed Oct 20, 2020 103 104 105  enough, the general answer to the above question is \emph{no}. A counterexample was found by Maltsiniotis and Ara. Let $B$ be the commutative monoid $(\mathbb{N},+)$, seen as a $2$-category with only one $0$-cell and no  Leonard Guetta committed Oct 26, 2020 106  non-trivial $1$-cells. This $2$-category is free (as an $\oo$\nbd{}category)  Leonard Guetta committed Oct 20, 2020 107  and a quick computation shows that:  Leonard Guetta committed Aug 26, 2020 108  $ Leonard Guetta committed Oct 20, 2020 109 110  H_k^{\pol}(B)=\begin{cases} \mathbb{Z} &\text{ if } k=0,2 \\ 0 &\text{ otherwise. }\end{cases}  Leonard Guetta committed Aug 26, 2020 111 $  Leonard Guetta committed Oct 20, 2020 112  On the other hand, it is shown in \cite[Theorem 4.9 and Example  Leonard Guetta committed Oct 21, 2020 113 114  4.10]{ara2019quillen} that the nerve of $B$ is a $K(\mathbb{Z},2)$; hence, it has non-trivial homology groups in all even dimension.  115   Leonard Guetta committed Oct 14, 2020 116  A question that still remains is:  117  \begin{center}  Leonard Guetta committed Oct 20, 2020 118 119  \textbf{(Q)} Which are the $\oo$\nbd{}categories $C$ such that $H_{\bullet}^{\pol}(C) \simeq H_{\bullet}^{\sing}(C)$ ?  120  \end{center}  Leonard Guetta committed Oct 21, 2020 121 122 123 124 125  This is precisely the question around which this dissertation revolves. Nevertheless, the reader will also find several new notions and results within this document that, although primarily motivated by the above question, are of interest in the theory of $\oo$\nbd{}categories and whose \emph{raisons d'être} go beyond the above considerations.  Leonard Guetta committed Aug 25, 2020 126 \end{named}  Leonard Guetta committed Oct 21, 2020 127 \begin{named}[Another formulation of the problem] One of the achievements of the  Leonard Guetta committed Oct 20, 2020 128 129 130  present work is a more abstract reformulation of the question of comparison of singular and polygraphic homology of $\oo$\nbd{}categories. %As often, the reward for abstraction is a much clearer understanding of the problem.  131   Leonard Guetta committed Dec 15, 2020 132  In order to do so, recall first that by a variation of the Dold--Kan  Leonard Guetta committed Oct 20, 2020 133 134 135  equivalence (see for example \cite{bourn1990another}), the category of abelian group objects in $\oo\Cat$ is equivalent to the category of non-negatively graded chain complexes  136  $ Leonard Guetta committed Oct 20, 2020 137  \Ab(\oo\Cat) \simeq \Ch.  138 $  Leonard Guetta committed Oct 20, 2020 139 140  Hence, we have a forgetful functor $\Ch \simeq \Ab(\oo\Cat) \to \oo\Cat$, which has a left adjoint  141  $ Leonard Guetta committed Oct 20, 2020 142  \lambda : \oo\Cat \to \Ch.  143 $  Leonard Guetta committed Oct 20, 2020 144  Moreover, for a \emph{free} $\oo$\nbd{}category $C$, the chain complex  Leonard Guetta committed Oct 21, 2020 145  $\lambda(C)$ is exactly the one obtained by the abelianization'' process  Leonard Guetta committed Oct 20, 2020 146  considered in Métayer's definition of polygraphic homology.  147   Leonard Guetta committed Oct 20, 2020 148 149 150 151 152 153 154 155 156  Now, the category $\oo\Cat$ admits a model structure, known as the \emph{folk model structure} \cite{lafont2010folk}, whose weak equivalences are the \emph{equivalences of $\oo$\nbd{}categories} (a generalization of the usual notion of equivalence of categories) and whose cofibrant objects are exactly the free $\oo$\nbd{}categories \cite{metayer2008cofibrant}. Polygraphic resolutions are then nothing but cofibrant replacements in this model category. As the definition of polygraphic homology groups strongly suggests, the functor $\lambda$ is left Quillen with respect to this model structure. In particular, it admits a left derived functor  157  $ Leonard Guetta committed Oct 20, 2020 158  \LL \lambda^{\folk} : \ho(\oo\Cat^{\folk}) \to \ho(\Ch)  159 $  Leonard Guetta committed Oct 20, 2020 160 161  and we tautologically have that $H_k^{\pol}(C) = H_k(\LL \lambda^{\folk}(C))$ for every $\oo$\nbd{}category $C$ and every $k \geq 0$. From now on, we set  162  $ Leonard Guetta committed Oct 20, 2020 163  \sH^{\pol}(C):=\LL \lambda^{\folk}(C).  164 $  Leonard Guetta committed Oct 20, 2020 165 166 167 168  This way of understanding polygraphic homology as a left derived functor has been around in the folklore for some time and I claim absolutely no originality for it. %Notice by the way, that the polygraphic homology of an $\oo$\nbd{}category is now an object of $\ho(\Ch)$ and not only a mere sequence of abelian groups.  169   Leonard Guetta committed Oct 20, 2020 170 171  On the other hand, $\lambda$ is also left derivable when $\oo\Cat$ is equipped with Thomason equivalences, yielding a left derived functor  172  $ Leonard Guetta committed Oct 20, 2020 173  \LL \lambda^{\Th} : \ho(\oo\Cat^{\Th}) \to \ho(\Ch).  174 $  Leonard Guetta committed Oct 16, 2020 175 176 177  This left derived functor being such that $H_k^{\sing}(C) = H_k(\LL \lambda^{\Th}(C))$ for every $\oo$\nbd{}category $C$ and every $k \geq 0$. Contrary to the folk'' case, this result is new and first appears within  Leonard Guetta committed Oct 21, 2020 178 179 180 181 182  this document (at least to my knowledge). Note that since, as mentioned earlier, the existence of a Thomason-like model structure on $\oo\Cat$ is still conjectural, usual tools from Quillen's theory of model categories were unavailable to prove the left derivability of $\lambda$ and the difficulty was to find a workaround solution.  Leonard Guetta committed Aug 28, 2020 183   Leonard Guetta committed Oct 26, 2020 184  From now on, we set  185  $ Leonard Guetta committed Oct 20, 2020 186  \sH^{\sing}(C):=\LL \lambda^{\Th}(C).  Leonard Guetta committed Aug 28, 2020 187 $  188   Leonard Guetta committed Oct 20, 2020 189 190 191 192 193 194 195 196  Finally, it can be shown that every equivalence of $\oo$\nbd{}categories is a Thomason equivalence. Hence, the identity functor of $\oo\Cat$ induces a functor $\J$ at the level of homotopy categories $\J : \ho(\oo\Cat^{\folk}) \to \ho(\oo\Cat^{\Th}),$ and altogether we have a triangle $ 197 198  \begin{tikzcd} \ho(\oo\Cat^{\folk}) \ar[rd,"\LL \lambda^{\folk}"'] \ar[r,"\J"] & \ho(\oo\Cat^{\Th}) \ar[d,"\LL \lambda^{\Th}"] \\  Leonard Guetta committed Oct 14, 2020 199 200  & \ho(\Ch). \end{tikzcd}  Leonard Guetta committed Oct 20, 2020 201 $  Leonard Guetta committed Oct 21, 2020 202  This triangle is \emph{not} commutative (even up to isomorphism), since this  Leonard Guetta committed Dec 17, 2020 203  would imply that the singular and polygraphic homology groups coincide for every  Leonard Guetta committed Oct 21, 2020 204 205 206 207 208 209  $\oo$\nbd{}category. However, since both functors $\LL \lambda^{\folk}$ and $\LL \lambda^{\Th}$ are left derived functors of the same functor $\lambda$, the existence of a natural transformation $\pi : \LL \lambda^{\Th} \circ \J \Rightarrow \LL \lambda^{\folk}$ follows by universal property. Since $\J$ is the identity on objects, for every $\oo$\nbd{}category $C$, this natural transformation yields a map  Leonard Guetta committed Oct 20, 2020 210  $ Leonard Guetta committed Oct 14, 2020 211  \pi_C : \sH^{\sing}(C) \to \sH^{\pol}(C),  Leonard Guetta committed Oct 20, 2020 212 213 214 215 216 217 218 219 $ which we refer to as the \emph{canonical comparison map}. Let us say that $C$ is \emph{homologically coherent} if $\pi_C$ is an isomorphism (which means exactly that for every $k\geq 0$, the induced map $H^{\sing}_k(C) \to H_k^{\pol}(C)$ is an isomorphism). The question of study then becomes: \begin{center} \textbf{(Q')} Which $\oo$\nbd{}categories are homologically coherent ? \end{center}  Leonard Guetta committed Oct 21, 2020 220  Note that, in theory, question \textbf{(Q')} is more precise than question  Leonard Guetta committed Oct 20, 2020 221  \textbf{(Q)} since we impose which morphism has to be an isomorphism in the  Leonard Guetta committed Jan 19, 2021 222 223 224 225 226 227 228  comparison of homology groups. However, for all the concrete examples that we shall meet in practice, it is always question \textbf{(Q')} that will be answered. % in practice, when we show that the % polygraphic and singular homology groups of an $\oo$\nbd{}category are % isomorphic, it is always via the above canonical comparison map. Conversely, % when we show that an $\oo$\nbd{}category is not \good{}, it always rules out % any isomorphism possible (not only the canonical comparison map).  Leonard Guetta committed Aug 28, 2020 229   Leonard Guetta committed Oct 21, 2020 230 231 232 233  As will be explained in this thesis, a formal consequence of the above is that polygraphic homology is \emph{not} invariant under Thomason equivalence. This means that there exists at least one Thomason equivalence $f : C \to D$ such that the induced map  Leonard Guetta committed Oct 20, 2020 234  $ Leonard Guetta committed Aug 28, 2020 235  \sH^{\pol}(C) \to \sH^{\pol}(D)  Leonard Guetta committed Oct 20, 2020 236 $  Leonard Guetta committed Oct 21, 2020 237 238 239 240 241 242 243 244 245 246 247 248 249 250  is \emph{not} an isomorphism. % Indeed, if this was not the case, then $\LL % \lambda^{\folk}$ would factor through $\J$, yielding a functor % ${\ho(\oo\Cat^{\Th}) \to \ho(\Ch)}$, which can easily be proved by universal % property to be (canonically isomorphic to) $\LL \lambda^{\Th}$. % In particular, this would imply that every $\oo$\nbd{}category is % homologically coherent, which, as we have already seen, is not true. In other words, if we think of $\oo\Cat$ as a model of homotopy types (via the localization by the Thomason equivalences), then polygraphic homology is \emph{not} a well-defined invariant. Another point of view would be to consider that polygraphic homology is an intrinsic invariant of $\oo$\nbd{}categories (and not up to Thomason equivalence) and in that way is finer than singular homology. This is not the point of view adopted here, and the reason will be motivated at the end of this introduction. The slogan to retain is:  Leonard Guetta committed Oct 20, 2020 251 252 253 254 255 256 257 258 259 260  \begin{center} Polygraphic homology is a way of computing singular homology groups of a homologically coherent $\oo$\nbd{}category. \end{center} The point is that given a \emph{free} $\oo$\nbd{}category $P$ (which is thus its own polygraphic resolution), the chain complex $\lambda(P)$ is much smaller'' than the chain complex associated to the nerve of $P$ and hence the polygraphic homology groups of $P$ are much easier to compute than its singular homology groups. The situation is comparable to using cellular homology for computing singular homology of a CW-complex. The difference is  Leonard Guetta committed Oct 21, 2020 261  that in this last case, such a thing is always possible while in the case of  Leonard Guetta committed Oct 20, 2020 262 263 264  $\oo$\nbd{}categories, one must ensure that the (free) $\oo$\nbd{}category is homologically coherent. %Intuitively speaking, this means that some free $\oo$\nbd{}categories are not cofibrant enough'' for homology.  Leonard Guetta committed Aug 28, 2020 265 266 \end{named} \begin{named}[Finding homologically coherent $\oo$-categories]  Leonard Guetta committed Oct 19, 2020 267  One of the main results presented in this dissertation is:  Leonard Guetta committed Aug 28, 2020 268  \begin{center}  Leonard Guetta committed Oct 20, 2020 269  Every (small) category $C$ is homologically coherent.  Leonard Guetta committed Aug 28, 2020 270  \end{center}  Leonard Guetta committed Oct 20, 2020 271  In order for this result to make sense, one has to consider categories as  Leonard Guetta committed Oct 26, 2020 272  $\oo$\nbd{}categories with only unit cells above dimension $1$. Beware that  Leonard Guetta committed Oct 21, 2020 273 274  this does not make the result trivial because given a polygraphic resolution $P \to C$ of a small category $C$, the $\oo$\nbd{}category $P$ need \emph{not}  Leonard Guetta committed Oct 20, 2020 275  have only unit cells above dimension $1$.  Leonard Guetta committed Aug 28, 2020 276   Leonard Guetta committed Oct 16, 2020 277 278 279  As such, this result is only a small generalization of Lafont and Métayer's result concerning monoids (although this new result, even restricted to monoids, is more precise because it means that the \emph{canonical comparison  Leonard Guetta committed Oct 21, 2020 280 281 282 283 284 285 286 287 288 289 290  map} is an isomorphism). But the true novelty lies in the proof which is more conceptual that the one of Lafont and Métayer. It requires the development of several new concepts and results which in the end combine together smoothly to yield the desired result. This dissertation has been written so that all the elements needed to prove this result are spread over several chapters; a more condensed version of it is the object of the article \cite{guetta2020homology}. Among the new notions developed along the way, that of discrete Conduché $\oo$\nbd{}functor is probably the most significant. An $\oo$\nbd{}functor $f : C \to D$ is a \emph{discrete Conduché $\oo$\nbd{}functor} when for every cell $x$ of $C$, if $f(x)$ can be written as  Leonard Guetta committed Oct 16, 2020 291  $ Leonard Guetta committed Oct 26, 2020 292  f(x)=y'\comp_k y'',  Leonard Guetta committed Oct 16, 2020 293 $  Leonard Guetta committed Oct 26, 2020 294  then there exists a unique pair $(x',x'')$ of cells of $C$ that are  Leonard Guetta committed Oct 16, 2020 295 296  $k$\nbd{}composable and such that $ Leonard Guetta committed Oct 26, 2020 297  f(x')=y',\, f(x'')=y'' \text{ and } x=x'\comp_k x''.  Leonard Guetta committed Oct 20, 2020 298 $  Leonard Guetta committed Oct 21, 2020 299  The main result that we prove concerning discrete Conduché $\oo$\nbd{}functors  Leonard Guetta committed Dec 17, 2020 300  is that for a discrete Conduché $\oo$\nbd{}functor $f : C \to D$, if the  Leonard Guetta committed Oct 21, 2020 301 302 303  $\oo$\nbd{}category $D$ is free, then $C$ is also free. The proof of this result is long and tedious, though conceptually not extremely hard, and first appears in the paper \cite{guetta2020polygraphs}, which is dedicated to it.  Leonard Guetta committed Oct 16, 2020 304   Leonard Guetta committed Oct 21, 2020 305  After having settled the case of ($1$\nbd{})categories, it is natural to move  Leonard Guetta committed Oct 20, 2020 306 307  on to $2$\nbd{}categories. Contrary to the case of ($1$\nbd{})categories, not all $2$\nbd{}categories are \good{} and the situation seems to be much harder  Leonard Guetta committed Oct 21, 2020 308  to understand. As a simplification, one can focus on $2$\nbd{}categories which  Leonard Guetta committed Oct 20, 2020 309 310 311 312 313 314 315 316 317 318 319  are free (as $\oo$\nbd{}categories). This is what is done in this dissertation. With this extra hypothesis, the problem of characterization of \good{} free $2$\nbd{}categories may be reduced to the following question: given a cocartesian square of the form $\begin{tikzcd} \sS_1 \ar[r] \ar[d] & P \ar[d]\\ \sD_2 \ar[r] & P', \ar[from=1-1,to=2-2,"\ulcorner",phantom,very near end] \end{tikzcd}$ where $P$ is a free $2$\nbd{}category, when is it \emph{homotopy cocartesian}  Leonard Guetta committed Oct 26, 2020 320  with respect to the Thomason equivalences? As a consequence, a substantial part of  Leonard Guetta committed Oct 21, 2020 321  the work presented here consists in developing tools to detect homotopy  Leonard Guetta committed Oct 26, 2020 322  cocartesian squares of $2$\nbd{}categories with respect to the Thomason  Leonard Guetta committed Oct 20, 2020 323 324 325 326  equivalences. While it appears that these tools do not allow to completely answer the above question, they still make it possible to detect such homotopy cocartesian squares in many concrete situations. In fact, a whole section of the thesis is dedicated to giving examples of (free) $2$\nbd{}categories and  Leonard Guetta committed Oct 21, 2020 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341  computing the homotopy type of their nerve using these tools. Among all these examples, a particular class of well-behaved $2$\nbd{}categories, which I have coined bubble-free $2$\nbd{}categories'', seems to stand out. This class is easily characterized as follows. Given a $2$\nbd{}category, let us call \emph{bubble} a non-trivial $2$\nbd{}cell whose source and target are units on a $0$\nbd{}cell (necessarily the same). A \emph{bubble-free $2$\nbd{}category} is then nothing but a $2$\nbd{}category that has no bubbles. The archetypal example of a $2$\nbd{}category that is \emph{not} bubble-free is the $2$\nbd{}category $B$ introduced earlier (which is the commutative monoid $(\mathbb{N},+)$ seen as a $2$\nbd{}category). As already said, this $2$\nbd{}category is not \good{} and this does not seem to be a coincidence. It is indeed remarkable that of all the many examples of $2$\nbd{}categories studied in this work, the only ones that are not \good{} are exactly the ones that are \emph{not} bubble-free. This leads to the conjecture below, which stands as a conclusion of the thesis.  Leonard Guetta committed Oct 20, 2020 342 343 344 345  \begin{center} \textbf{(Conjecture)} A free $2$\nbd{}category is \good{} if and only if it is bubble-free. \end{center}  Leonard Guetta committed Aug 28, 2020 346 \end{named}  Leonard Guetta committed Oct 17, 2020 347 \begin{named}[The big picture]  Leonard Guetta committed Oct 20, 2020 348 349  Let us end this introduction with another point of view on the comparison of singular and polygraphic homologies. This point of view is highly conjectural  Leonard Guetta committed Oct 21, 2020 350  and is not addressed at all in the rest of the dissertation. It should be  Leonard Guetta committed Oct 20, 2020 351  thought of as a guideline for future work.  Leonard Guetta committed Aug 29, 2020 352   Leonard Guetta committed Oct 20, 2020 353 354  In the same way that (strict) $2$\nbd{}categories are particular cases of bicategories, strict $\oo$\nbd{}categories are in fact particular cases of  Leonard Guetta committed Oct 21, 2020 355  what are usually called \emph{weak $\oo$\nbd{}categories}. Such mathematical  Leonard Guetta committed Oct 20, 2020 356  objects have been defined, for example, by Batanin using globular operads  Leonard Guetta committed Oct 17, 2020 357  \cite{batanin1998monoidal} or by Maltsiniotis following ideas of Grothendieck  Leonard Guetta committed Oct 20, 2020 358  \cite{maltsiniotis2010grothendieck}. Similarly to the fact that the theory of  Leonard Guetta committed Oct 21, 2020 359  quasi-categories (which is a homotopical model for the theory of weak  Leonard Guetta committed Oct 20, 2020 360 361 362 363 364 365 366  $\oo$\nbd{}categories whose cells are invertible above dimension $1$) may be expressed using the same language as the theory of usual categories, it is generally believed that all intrinsic'' notions (in a precise sense to be defined) of the theory of strict $\oo$\nbd{}categories have weak counterparts. For example, it is believed that there should be a folk model structure on the category of weak $\oo$\nbd{}categories and that there should be a good notion of free weak $\oo$\nbd{}category. In fact, this last notion should be defined  Leonard Guetta committed Dec 17, 2020 367  as weak $\oo$\nbd{}categories that are recursively obtained from the empty  Leonard Guetta committed Jan 27, 2021 368  $\oo$\nbd{}category by freely  Leonard Guetta committed Oct 20, 2020 369 370 371 372  adjoining cells, which is the formal analogue of the strict version but in the weak context. The important point here is that a free strict $\oo$\nbd{}category is \emph{never} free as a weak $\oo$\nbd{}category (except for the empty $\oo$\nbd{}category).  Leonard Guetta committed Oct 19, 2020 373  % For  Leonard Guetta committed Oct 20, 2020 374 375  % example, the $2$\nbd{}category $B$ we have introduced earlier, which is free % as a strict  Leonard Guetta committed Oct 19, 2020 376 377  % $\oo$\nbd{}category, seems to be \emph{not} free as % a weak $\oo$\nbd{}category.  Leonard Guetta committed Oct 20, 2020 378 379  Moreover, there are good candidates for the polygraphic homology of weak $\oo$\nbd{}categories obtained by mimicking the definition in the strict case.  Leonard Guetta committed Oct 21, 2020 380 381 382 383 384 385 386 387 388  But in general the polygraphic homology of a strict $\oo$\nbd{}category need not be the same as its weak polygraphic homology''. Indeed, since free strict $\oo$\nbd{}categories are not free as weak $\oo$\nbd{}categories, taking a weak polygraphic resolution'' of a strict $\oo$\nbd{}category is not the same as taking a polygraphic resolution. In fact, when trying to compute the weak polygraphic homology of $B$, it would seem that it gives the homology groups of a $K(\mathbb{Z},2)$, which is what we would have expected of its polygraphic homology in the first place. From this observation, it is tempting to make the following conjecture:  Leonard Guetta committed Oct 19, 2020 389  \begin{center}  Leonard Guetta committed Dec 17, 2020 390  The weak polygraphic homology of a strict $\oo$\nbd{}category coincides  Leonard Guetta committed Oct 20, 2020 391  with its singular homology.  Leonard Guetta committed Oct 19, 2020 392  \end{center}  Leonard Guetta committed Oct 20, 2020 393 394 395 396  In other words, we conjecture that the fact that polygraphic and singular homologies of strict $\oo$\nbd{}categories do not coincide is a defect due to working in too narrow a setting. The good'' definition of polygraphic homology ought to be the weak one.  Leonard Guetta committed Oct 20, 2020 397   Leonard Guetta committed Oct 19, 2020 398   Leonard Guetta committed Oct 20, 2020 399  We can go even further and conjecture the same thing for weak  Leonard Guetta committed Oct 22, 2020 400 401 402 403 404 405 406  $\oo$\nbd{}categories. In order to do so, we need a definition of singular homology for weak $\oo$\nbd{}categories. This is conjecturally done as follows. To every weak $\oo$\nbd{}category $C$, one can associate a weak $\oo$\nbd{}groupoid $L(C)$ by formally inverting all the cells of $C$. Then, if we believe in Grothendieck's conjecture (see \cite{grothendieck1983pursuing} and \cite[Section 2]{maltsiniotis2010grothendieck}), the category of  Leonard Guetta committed Oct 25, 2020 407  weak $\oo$\nbd{}groupoids equipped with the weak equivalences of weak  Leonard Guetta committed Dec 17, 2020 408 409  $\oo$\nbd{}groupoids (see \cite[Paragraph 2.2]{maltsiniotis2010grothendieck}) is a model for the homotopy  Leonard Guetta committed Oct 22, 2020 410 411 412  theory of spaces. In particular, every weak $\oo$\nbd{}groupoid has homology groups and we can define the singular homology groups of a weak $\oo$\nbd{}category $C$ as the homology groups of $L(C)$.  Leonard Guetta committed Oct 20, 2020 413 414 415  %% This defines a functor %% $ Leonard Guetta committed Oct 20, 2020 416  %% L : \mathbf{W}\oo\Cat \to \mathbf{W}\oo\Grpd  Leonard Guetta committed Oct 20, 2020 417  %%$  Leonard Guetta committed Oct 17, 2020 418 419 \end{named} \begin{named}[Organization of the thesis]  Leonard Guetta committed Oct 20, 2020 420 421 422 423 424 425 426  In the first chapter, we review some aspects of the theory of $\oo$\nbd{}categories. In particular, we study with great care free $\oo$\nbd{}categories, which are at the heart of the present work. It is the only chapter of the thesis that does not contain any reference to homotopy theory whatsoever. It is also there that we introduce the notion of discrete Conduché $\oo$\nbd{}functor and study their relation with free $\oo$\nbd{}categories. The culminating point of the chapter is Theorem  Leonard Guetta committed Oct 21, 2020 427  \ref{thm:conduche}, which states that given a discrete Conduché  Leonard Guetta committed Oct 20, 2020 428 429 430  $\oo$\nbd{}functor $F : C \to D$, if $D$ is free, then so is $C$. The proof of this theorem is long and technical and is broke down into several distinct parts.  Leonard Guetta committed Oct 18, 2020 431   Leonard Guetta committed Oct 20, 2020 432  The second chapter is devoted to recalling some tools of homotopical algebra.  Leonard Guetta committed Oct 20, 2020 433 434 435  More precisely, basic aspects of the theory of homotopy colimits using the formalism of Grothendieck's derivators are quickly presented. Note that this chapter does \emph{not} contain any original result and can be skipped at  Leonard Guetta committed Oct 21, 2020 436  first reading. It is only intended to give the reader a summary of useful  Leonard Guetta committed Oct 20, 2020 437 438  results on homotopy colimits that are used in the rest of the dissertation.  Leonard Guetta committed Oct 21, 2020 439  In the third chapter, we delve into the homotopy theory of  Leonard Guetta committed Oct 20, 2020 440 441 442 443  $\oo$\nbd{}categories. It is there that we define the different notions of weak equivalences for $\oo$\nbd{}categories and compare them. The two most significant new results to be found in this chapter are probably Proposition \ref{prop:folkisthom}, which states that every equivalence of  Leonard Guetta committed Oct 22, 2020 444 445  $\oo$\nbd{}categories is a Thomason equivalence, and Theorem \ref{thm:folkthmA}, which states that equivalences of $\oo$\nbd{}categories  Leonard Guetta committed Oct 21, 2020 446  satisfy a property reminiscent of Quillen's Theorem $A$ \cite[Theorem  Leonard Guetta committed Oct 20, 2020 447 448 449  A]{quillen1973higher} and its $\oo$\nbd{}categorical generalization by Ara and Maltsiniotis \cite{ara2018theoreme,ara2020theoreme}.  Leonard Guetta committed Oct 26, 2020 450  In the fourth chapter, we define the polygraphic and singular homologies of  Leonard Guetta committed Oct 20, 2020 451 452 453 454  $\oo$\nbd{}categories and properly formulate the problem of their comparison. Up to Section \ref{section:polygraphichmlgy} included, all the results were known prior to this thesis (at least in the folklore), but starting from Section \ref{section:singhmlgyderived} all the results are original. Three  Leonard Guetta committed Oct 22, 2020 455  fundamental results of this chapter are: Theorem \ref{thm:hmlgyderived},  Leonard Guetta committed Oct 20, 2020 456 457 458  which states that singular homology is obtained as a derived functor of an abelianization function, Proposition \ref{prop:criteriongoodcat}, which gives an abstract criterion to detect \good{} $\oo$\nbd{}categories, and Proposition  Leonard Guetta committed Oct 21, 2020 459 460  \ref{prop:comphmlgylowdimension}, which states that low-dimensional singular and polygraphic homology groups always coincide.  Leonard Guetta committed Oct 20, 2020 461   Leonard Guetta committed Oct 26, 2020 462  The fifth chapter is mainly geared towards the fundamental Theorem  Leonard Guetta committed Oct 21, 2020 463 464  \ref{thm:categoriesaregood}, which states that every category is \good{}. To prove this theorem, we first focus on a particular class of  Leonard Guetta committed Oct 20, 2020 465  $\oo$\nbd{}categories, which we call \emph{contractible  Leonard Guetta committed Oct 21, 2020 466 467  $\oo$\nbd{}categories}, and show that every contractible $\oo$\nbd{}category is \good{} (Proposition \ref{prop:contractibleisgood}).  Leonard Guetta committed Oct 20, 2020 468   Leonard Guetta committed Oct 20, 2020 469  Finally, the sixth and last chapter of the thesis revolves around the homology  Leonard Guetta committed Oct 21, 2020 470  of free $2$\nbd{}categories. The goal pursued is to try to understand which  Leonard Guetta committed Oct 21, 2020 471  free $2$\nbd{}categories are \good{}. In order to do so, we give a criterion  Leonard Guetta committed Oct 20, 2020 472 473  to detect homotopy cocartesian square with respect to Thomason equivalences (Proposition \ref{prop:critverthorThomhmtpysquare}) based on the homotopy  Leonard Guetta committed Dec 17, 2020 474 475  theory of bisimplicial sets. Then, we apply this criterion and some other \emph{ad hoc} techniques to compute many examples of homotopy type of free  Leonard Guetta committed Oct 20, 2020 476 477  $2$\nbd{}categories. The conclusion of the chapter is Conjecture \ref{conjecture:bubblefree}, which states that a free $2$\nbd{}category is  Leonard Guetta committed Oct 21, 2020 478  \good{} if and only if it is bubble-free.  Leonard Guetta committed Oct 17, 2020 479 \end{named}  Leonard Guetta committed Oct 21, 2020 480 481 %% Let us come back to the canonical $2$-triangle %% $ Leonard Guetta committed Sep 20, 2020 482 %% \begin{tikzcd}  Leonard Guetta committed Oct 21, 2020 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 %% \ho(\oo\Cat^{\folk}) \ar[rd,"\sH^{\folk}=\LL \lambda^{\folk}"',""{name=B,above}] \ar[r,"\J"] & \ho(\oo\Cat^{\Th}) \ar[d,"\LL \lambda^{\Th}=\sH^{\sing}"] \\ %% & \ho(\Ch) \ar[from=1-2,to=B,Rightarrow,"\pi"] %% \end{tikzcd} %%$ %% and ask the question: %% \begin{center} %% What \emph{would} it mean that the natural transformation $\pi$ be an %% isomorphism (i.e.\ that all $\oo$\nbd{}categories be homologically %% coherent) ? %% \end{center} %% For simplification, let us assume that the conjectured Thomason-like model %% structure on $\oo\Cat$ was established and that $\lambda$ was left Quillen %% with respect to this model structure (which is also conjectured). Now, the %% conjectured cofibrations of the Thomason-like model structure (see %% \cite{ara2014vers}) are particular cases of folk cofibrations and thus, all %% Thomason cofibrant objects are folk cofibrant objects. The converse, on the %% other hand, is not true. Consequently, Quillen's theory of derived functors %% tells us that for a \emph{Thomason} cofibrant object $P$, we have %% \tag{$\ast$}\label{equationintro}  Leonard Guetta committed Sep 20, 2020 502 503 %% \LL \lambda^{\Th}(P) \simeq \lambda(P) \simeq \LL \lambda^{\folk}(P), %%  Leonard Guetta committed Oct 21, 2020 504 505 506 507 508 509 510 511 512 %% (and the resulting isomorphism is obviously the canonical comparison map). %% Now, \emph{if} the natural transformation $\pi$ were an isomorphism, then a %% quick 2-out-of-3 reasoning would show that \eqref{equationintro} would also %% be true when $P$ is only \emph{folk} cofibrant. Hence, intuitively %% speaking, if $\pi$ were an isomorphism, then folk cofibrant objects would %% be \emph{sufficiently cofibrant} for the homology, even though there are %% not Thomason cofibrant. (And in fact, using cofibrant replacements, it can %% be shown that this condition is sufficient to ensure that $\pi$ be an %% isomorphism).  Leonard Guetta committed Aug 29, 2020 513   Leonard Guetta committed Oct 21, 2020 514 515 516 517 518 519 520 %% Yet, as we have already seen, such property is not true: there are folk %% cofibrant objects that are \emph{not} enough cofibrant to compute (Street) %% homology. The archetypal example being the bubble'' of Ara and %% Maltsiniotis. However, even if false, the idea that folk cofibrant objects %% are sufficiently cofibrants for homology is seducing and I conjecturally %% believe that this defect is a mere consequence of working in a too narrow %% setting, as I shall now explain.  Leonard Guetta committed Aug 29, 2020 521   Leonard Guetta committed Oct 21, 2020 522 523 524 525 526 527 528 529 530 %% In the same way that bicategories and tricategories are weak'' variations %% of the notions of (strict) $2$-categories and $3$-categories, there exists %% a general notion of \emph{weak $\oo$\nbd{}categories}. These objects can be %% defined, for example, using the formalism of Grothendieck's coherators %% \cite{maltsiniotis2010grothendieck}, or of Batanin's globular operads %% \cite{batanin1998monoidal}. (In fact, each of these formalism give rise to %% many different possible notions of weak $\oo$\nbd{}categories, which are %% conjectured to be all equivalent, at least in some higher categorical %% sense.)  Leonard Guetta committed Sep 20, 2020 531 %% \end{named}  Leonard Guetta committed Sep 25, 2020 532 533 534 535 536  %%% Local Variables: %%% mode: latex %%% TeX-master: "main" %%% End: