hmtpy.tex 51.6 KB
 Leonard Guetta committed Oct 22, 2020 1 \chapter{Homotopy theory of \texorpdfstring{$\oo$}{ω}-categories}  Leonard Guetta committed Apr 21, 2020 2 3 \section{Nerve} \begin{paragr}\label{paragr:simpset}  Leonard Guetta committed Aug 21, 2020 4  We denote by $\Delta$ the category whose objects are the finite non-empty totally ordered sets $[n]=\{0<\cdots 0$ and $0\leq i\leq n$, we denote by  Leonard Guetta committed Apr 21, 2020 5 6 7  $\delta^i : [n-1] \to [n]$  Leonard Guetta committed Aug 21, 2020 8 9 10 11  the only injective increasing map whose image does not contain $i$, and for $n\geq 0$ and $0 \leq i \leq n$, we denote by $\sigma^i : [n+1] \to [n]$  Leonard Guetta committed Oct 09, 2020 12  the only surjective non-decreasing map such that the pre-image of $i \in [n]$ contains exactly two elements.  Leonard Guetta committed Oct 25, 2020 13  The category $\Psh{\Delta}$ of \emph{simplicial sets} is the category of presheaves on $\Delta$. For a simplicial set $X$, we use the notations  Leonard Guetta committed Apr 21, 2020 14 15 16  \begin{aligned} X_n &:= X([n]) \\  Leonard Guetta committed Aug 21, 2020 17  \partial_i &:= X(\delta^i): X_n \to X_{n\shortminus 1}\\  Leonard Guetta committed Oct 26, 2020 18  s_i &:= X(\sigma^i): X_{n} \to X_{n+1}.  Leonard Guetta committed Apr 21, 2020 19 20  \end{aligned}  Leonard Guetta committed Oct 26, 2020 21  Elements of $X_n$ are referred to as \emph{$n$\nbd{}simplices of $X$}, the maps $\partial_i$ are the \emph{face maps} and the maps $s_i$ are the \emph{degeneracy maps}.  Leonard Guetta committed Apr 21, 2020 22 \end{paragr}  Leonard Guetta committed Sep 26, 2020 23 \begin{paragr}\label{paragr:orientals}  Leonard Guetta committed Oct 26, 2020 24  We denote by $\Or : \Delta \to \omega\Cat$ the cosimplicial object introduced by Street in \cite{street1987algebra}. The $\omega$-category $\Or_n$ is called the \emph{$n$\nbd{}oriental}. There are various ways to give a precise definition of the orientals, but each of them requires some machinery that we don't want to introduce here. Instead, we only recall some important facts on orientals that we shall need in the sequel and refer to the literature on the subject (such as \cite{street1987algebra,street1991parity,street1994parity,steiner2004omega,buckley2016orientals} or \cite[chapitre 7]{ara2016joint}) for details.  Leonard Guetta committed Apr 21, 2020 25 26  The two main points to retain are:  Leonard Guetta committed Aug 21, 2020 27  \begin{description}  Leonard Guetta committed Dec 27, 2020 28 29  \item[(OR1)] Each $\Or_n$ is a free $\oo$\nbd{}category whose set of generating $k$\nbd{}cells is canonically isomorphic to the sets of increasing sequences  Leonard Guetta committed Apr 21, 2020 30 31 32 33  $0 \leq i_1 < i_2 < \cdots < i_k \leq n,$ or, which is equivalent, to injective increasing maps $[k] \to [n]$.  Leonard Guetta committed Aug 21, 2020 34  \end{description}  Leonard Guetta committed Apr 21, 2020 35 36  We use the notation $\langle i_1\, i_2\cdots i_k\rangle$ for such a cell. In particular, we have that: \begin{itemize}[label=-]  Leonard Guetta committed Dec 27, 2020 37  \item There are no generating $k$-cells for $k>n$. Hence, $\Or_n$ is an $n$\nbd{}category.  Leonard Guetta committed Oct 26, 2020 38  \item There is exactly one generating $n$\nbd{}cell of $\Or_n$, which is $\langle 0 \,1 \cdots n\rangle$. We refer to this cell as the \emph{principal cell of $\Or_n$}.  Leonard Guetta committed Apr 21, 2020 39 40 41 42 43 44  \item There are exactly $n+1$ generating $(n-1)$-cells of $\Or_n$. They correspond to the maps $\delta^i : [n-1] \to [n]$ for $i \in \{0,\cdots,n\}$. \end{itemize}  Leonard Guetta committed Aug 21, 2020 45  \begin{description}  Leonard Guetta committed Dec 27, 2020 46  \item[(OR2)] For $n>0$, the source (resp.\ target) of the principal cell of $\Or_n$ can be expressed as a composition of all the generating $(n-1)$\nbd{}cells corresponding to $\delta^i$ with $i$ odd (resp.\ even); each of these generating $(n-1)$\nbd{}cells appearing exactly once in the composite.  Leonard Guetta committed Aug 21, 2020 47  \end{description}  Leonard Guetta committed Feb 04, 2021 48  Another way of formulating \textbf{(OR2)} is: for $n>0$ the weight (\ref{paragr:weight}) of the $(n-1)$\nbd{}cell corresponding to $\delta_i$ in the \emph{source} of the principal cell of $\Or_n$ is $1$ if $i$ is odd and $0$ if $i$ is even and the other way around for the \emph{target} of the principal cell of $\Or_n$.  Leonard Guetta committed Aug 21, 2020 49  Here are some pictures in low dimension:  Leonard Guetta committed Apr 21, 2020 50 51 52 53 54 55 56 57 58 59 60 61  $\Or_0 = \langle 0 \rangle,$ $\Or_1=\begin{tikzcd} \langle 0 \rangle \ar[r,"\langle 01 \rangle"] &\langle 1 \rangle, \end{tikzcd}$ $\Or_2= \begin{tikzcd} &\langle 1 \rangle \ar[rd,"\langle 12 \rangle"]& \\  Leonard Guetta committed Oct 14, 2020 62  \langle 0 \rangle \ar[ru,"\langle 01 \rangle"]\ar[rr,"\langle 02 \rangle"',""{name=A,above}]&&\langle 2 \rangle,  Leonard Guetta committed Apr 21, 2020 63  \ar[Rightarrow,from=A,to=1-2,"\langle 012 \rangle"]  Leonard Guetta committed Oct 14, 2020 64  \end{tikzcd}  Leonard Guetta committed Apr 21, 2020 65 66 $ $ Leonard Guetta committed Sep 01, 2020 67  \Or_3=  Leonard Guetta committed Apr 21, 2020 68 69 70 71  \begin{tikzcd} & \langle 1 \rangle \ar[rd,"\langle 12 \rangle"]& \\ \langle 0 \rangle \ar[ru,"\langle 01 \rangle"] \ar[rd,"\langle 03 \rangle"',""{name=B,above}] \ar[rr,"\langle 02 \rangle" description,""{name=A,above}]& & \langle 2 \rangle \ar[ld,"\langle 23 \rangle"]\\ & \langle 3 \rangle &  Leonard Guetta committed Sep 01, 2020 72 73  \ar[from=A,to=1-2,Rightarrow,"\langle 012 \rangle", shorten <= 0.25em, shorten >= 0.25em] \ar[from=B,to=2-3,Rightarrow,"\langle 023 \rangle"', near start, shorten <= 1.1em, shorten >= 1.5em]  Leonard Guetta committed Apr 21, 2020 74 75 76  \end{tikzcd} \overset{\langle 0123 \rangle}{\Rrightarrow} \begin{tikzcd}  Leonard Guetta committed Sep 01, 2020 77  & \langle 1 \rangle \ar[rd,"\langle 12 \rangle"] \ar[dd,"\langle 13 \rangle"' description,""{name=B,right}] & \\  Leonard Guetta committed Oct 25, 2020 78  \langle 0 \rangle \ar[ru,"\langle 01 \rangle"] \ar[rd,"\langle 03 \rangle"',""{name=A,above}] & & \langle 2 \rangle. \ar[ld,"\langle 23 \rangle"]\\  Leonard Guetta committed Apr 21, 2020 79  & \langle 3 \rangle &  Leonard Guetta committed Sep 01, 2020 80 81  \ar[from=A,to=1-2,Rightarrow,"\langle 013 \rangle", near start, shorten <= 1em, shorten >= 1.5em] \ar[from=B,to=2-3,Rightarrow,"\langle 123 \rangle", shorten <= 0.75em, shorten >=0.75em]  Leonard Guetta committed Sep 01, 2020 82  \end{tikzcd}  Leonard Guetta committed Apr 21, 2020 83 84 85 $ \end{paragr} \begin{paragr}\label{paragr:nerve}  Leonard Guetta committed Sep 01, 2020 86  For every $\omega$-category $C$, the \emph{nerve of $C$} is the simplicial set $N_{\omega}(C)$ defined as  Leonard Guetta committed Apr 21, 2020 87 88  \begin{aligned}  Leonard Guetta committed Jan 18, 2021 89  N_{\omega}(C) : \Delta^{\op} &\to \Set\\  Leonard Guetta committed Sep 01, 2020 90  [n] &\mapsto \Hom_{\omega\Cat}(\Or_n,C).  Leonard Guetta committed Apr 21, 2020 91 92 93 94 95 96  \end{aligned} By post-composition, this yields a functor \begin{aligned} N_{\omega} : \omega\Cat &\to \Psh{\Delta} \\  Leonard Guetta committed Sep 01, 2020 97  C &\mapsto N_{\omega}(C),  Leonard Guetta committed Apr 21, 2020 98 99  \end{aligned}  Leonard Guetta committed Oct 26, 2020 100  which we refer to as the \emph{nerve functor for $\oo$\nbd{}categories}. Furthermore, for every $n \in \mathbb{N}$, we also define a nerve functor for $n$\nbd{}categories as the restriction of $N_{\oo}$ to $n\Cat$ (seen as a full subcategory of $\oo\Cat$)  Leonard Guetta committed Apr 21, 2020 101  $ Leonard Guetta committed Sep 02, 2020 102  N_n := N_{\oo}{\big |}_{n\Cat} : n\Cat \to \Psh{\Delta}.  Leonard Guetta committed Apr 21, 2020 103 $  Leonard Guetta committed Oct 25, 2020 104  By the usual Kan extension technique, we obtain for every $n \in \nbar$ a functor $c_n : \Psh{\Delta} \to n\Cat,$ left adjoint to $N_n$.  Leonard Guetta committed Sep 02, 2020 105 106 \end{paragr} \iffalse  Leonard Guetta committed Apr 21, 2020 107  \begin{lemma}  Leonard Guetta committed Oct 25, 2020 108  Let $X$ be a simplicial set. The $\oo$\nbd{}category $c_{\oo}(X)$ is free and the set of generating $k$-cells of $c_{\oo}(X)$ is canonically isomorphic the to set of non-degenerate $k$-simplices of $X$.  Leonard Guetta committed Apr 21, 2020 109  \end{lemma}  Leonard Guetta committed Sep 02, 2020 110  \fi  Leonard Guetta committed Apr 21, 2020 111  \begin{paragr}  Leonard Guetta committed Sep 02, 2020 112 113  For $n=1$, the functor $N_1$ is the usual nerve of categories. Recall that for a (small) category $C$, an $m$-simplex $X$ of $N_1(C)$ is a sequence of composable arrows of $C$ $ Leonard Guetta committed Sep 02, 2020 114  X_0 \overset{X_{0,1}}{\longrightarrow} X_1 \overset{X_{1,2}}{\longrightarrow} \cdots \longrightarrow X_{m-1}\overset{X_{m-1,m}}{\longrightarrow} X_m.  Leonard Guetta committed Sep 02, 2020 115 $  Leonard Guetta committed Sep 02, 2020 116  For $m > 0$ and $0 \leq i \leq m$, the $(m-1)$-simplex $\partial_i(X)$ is obtained by composing arrows at $X_i$ (or simply deleting it for $i=0$ or $m$). For $m \geq 0$ and $0 \leq i \leq m$, the $(m+1)$-simplex $s_i(X)$ is obtained by inserting a unit map at $X_i$.  Leonard Guetta committed Sep 02, 2020 117   Leonard Guetta committed Jan 20, 2021 118  For $n=2$, the functor $N_2$ is what is sometimes known as the \emph{Duskin nerve} \cite{duskin2002simplicial} (restricted from bicategories to $2$-categories). For a $2$-category $C$, an $m$\nbd{}simplex $X$ of $N_2(C)$ consists of:  Leonard Guetta committed Apr 21, 2020 119  \begin{itemize}[label=-]  Leonard Guetta committed Sep 02, 2020 120  \item for every $0\leq i \leq m$, an object $X_i$ of $C$,  Leonard Guetta committed Sep 02, 2020 121 122  \item for all $0\leq i \leq j \leq m$, an arrow $X_{i,j} : X_i \to X_j$ of $C$, \item for all $0 \leq i \leq j \leq k \leq m$, a $2$-triangle  Leonard Guetta committed Apr 21, 2020 123 124  $\begin{tikzcd}  Leonard Guetta committed Sep 02, 2020 125  & X_j \ar[rd,"{X_{j,k}}"]& \\  Leonard Guetta committed Oct 14, 2020 126  X_i \ar[ru,"X_{i,j}"]\ar[rr,"X_{i,k}"',""{name=A,above}]&&X_k,  Leonard Guetta committed Sep 02, 2020 127  \ar[Rightarrow,from=A,to=1-2,"X_{i,j,k}"]  Leonard Guetta committed Oct 14, 2020 128  \end{tikzcd}  Leonard Guetta committed Apr 21, 2020 129 130 $ \end{itemize}  Leonard Guetta committed Oct 25, 2020 131  subject to the following axioms:  Leonard Guetta committed Apr 21, 2020 132  \begin{itemize}[label=-]  Leonard Guetta committed Sep 02, 2020 133 134 135 136 137 138 139 140 141 142 143 144  \item for all $0 \leq i \leq m$, we have $X_{i,i}=1_{X_i},$ \item for all $0 \leq i \leq j \leq m$, we have $X_{i,i,j}=X_{i,j,j}=1_{X_{i,j}},$ \item for all $0 \leq i < j < k < l \leq m$, we have the equality (known as the \emph{cocycle condition}) $(X_{k,l} \comp_0 X_{i,j,k})\comp_1 X_{i,k,l} = (X_{j,k,l} \comp_0 X_{i,j})\comp_1 X_{i,j,l}.$  Leonard Guetta committed Apr 21, 2020 145  \end{itemize}  Leonard Guetta committed Sep 02, 2020 146 147 148 149 150 151 152 153 154  For $m> 0$ and $0\leq l \leq m$, the $(m-1)$-simplex $\partial_l(X)$ is defined as $\partial_l(X)_{i}=X_{\delta_l(j)}, \quad \partial_l(X)_{i,j}=X_{\delta_l(i),\delta_l(j)} \text{ and } \partial_l(X)_{i,j,k}=X_{\delta_l(i),\delta_l(j),\delta_l(k)}.$ And similarly, for $m \geq 0$ and $0\leq l \leq m$, the $(m+1)$-simplex $s_l(X)$ is defined as $s_l(X)_{i}=X_{\sigma_l(j)}, \quad s_l(X)_{i,j}=X_{\sigma_l(i),\sigma_l(j)} \text{ and } s_l(X)_{i,j,k}=X_{\sigma_l(i),\sigma_l(j),\sigma_l(k)}.$ \iffalse  Leonard Guetta committed Sep 02, 2020 155  Let $X$ be a $m$-simplex with $m>0$ and $0\leq l \leq m$. The $(m-1)$-simplex $\partial_l(X)$ is described as follows:  Leonard Guetta committed Apr 21, 2020 156  $ Leonard Guetta committed Sep 02, 2020 157  \partial_l(X)_i = \begin{cases} X_i &\text{ if } 0 \leq i3, the existence of such a model structure is conjectured but not yet established.  Leonard Guetta committed Apr 21, 2020 233  \end{remark}  Leonard Guetta committed Feb 04, 2021 234  By definition, for all 1 \leq n \leq m \leq \omega, the canonical inclusion \[n\Cat \hookrightarrow m\Cat$ sends the Thomason equivalences of $n\Cat$ to Thomason equivalences of $m\Cat$. Hence, it induces a morphism of localizers and then a morphism of op\nbd{}prederivators $\Ho(n\Cat^\Th) \to \Ho(m\Cat^{\Th}).$  Leonard Guetta committed Jan 05, 2021 235  \begin{proposition}\label{prop:nthomeqder}  Leonard Guetta committed Apr 21, 2020 236 237 238 239  For all $1 \leq n \leq m \leq \omega$, the canonical morphism $\Ho(n\Cat^\Th) \to \Ho(m\Cat^{\Th})$  Leonard Guetta committed Oct 25, 2020 240  is an equivalence of op\nbd{}prederivators.  Leonard Guetta committed Sep 02, 2020 241  \end{proposition}  Leonard Guetta committed Apr 21, 2020 242 243 244 245  \begin{proof} This follows from Theorem \ref{thm:gagna} and the commutativity of the triangle $\begin{tikzcd}[column sep=tiny]  Leonard Guetta committed Jan 06, 2021 246  \Ho(n\Cat^{\Th}) \ar[rr] \ar[rd,"\overline{N_n}"'] & & \Ho(m\Cat^{\Th}) \ar[dl,"\overline{N_m}"] \\  Leonard Guetta committed Oct 14, 2020 247 248  &\Ho(\Psh{\Delta})&. \end{tikzcd}  Leonard Guetta committed Apr 21, 2020 249 250 $ \end{proof}  Leonard Guetta committed Jan 05, 2021 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265  \begin{paragr}\label{paragr:thomhmtpycol} It follows from the previous proposition that for all $1 \leq n \leq m \leq \omega$, the morphism $\Ho(n\Cat^\Th) \to \Ho(m\Cat^{\Th})$ of op\nbd{}prederivators is homotopy cocontinuous and reflects homotopy colimits (in an obvious sense). Hence, given a diagram $d : I \to n\Cat$ with $n>0$, we can harmlessly use the notation $\hocolim^{\Th}_{i \in I}(d)$ for both the Thomason homotopy colimits in $n\Cat$ and in $\oo\Cat$ (or any $m\Cat$ with $n\leq m$). Similarly, a commutative square of $n\Cat$ is Thomason homotopy cocartesian in $n\Cat$ if and only if it is so in $\oo\Cat$. Hence, there is really no ambiguity when simply calling such a square \emph{Thomason homotopy cocartesian}. \end{paragr}  Leonard Guetta committed Sep 02, 2020 266 267  \section{Tensor product and oplax transformations} Recall that $\oo\Cat$ can be equipped with a monoidal product $\otimes$, introduced by Al-Agl and Steiner in \cite{al1993nerves} and by Crans in \cite{crans1995combinatorial}, commonly referred to as the \emph{Gray tensor product}. The implicit reference for this section is \cite[Appendices A and B]{ara2016joint}.  Leonard Guetta committed Apr 21, 2020 268  \begin{paragr}  Leonard Guetta committed Oct 25, 2020 269  The Gray tensor product makes $\oo\Cat$ into a monoidal category for which the unit is the $\oo$\nbd{}category $\sD_0$ (which is the terminal $\oo$\nbd{}category). This monoidal category is \emph{not} symmetric but it is biclosed \cite[Theorem A.15]{ara2016joint}, meaning that there exist two functors  Leonard Guetta committed Apr 21, 2020 270  $ Leonard Guetta committed Sep 02, 2020 271  \underline{\hom}_{\mathrm{oplax}}(-,-),\, \underline{\hom}_{\mathrm{lax}}(-,-) : \oo\Cat^{\op}\times\oo\Cat \to \oo\Cat  Leonard Guetta committed Apr 21, 2020 272 $  Leonard Guetta committed Oct 25, 2020 273 274 275 276 277 278  such that for all $\oo$\nbd{}categories $X,Y$ and $Z$, we have isomorphisms \begin{align*} \Hom_{\oo\Cat}(X\otimes Y , Z) &\simeq \Hom_{\oo\Cat}(X, \underline{\hom}_{\mathrm{oplax}}(Y,Z))\\ &\simeq \Hom_{\oo\Cat}(Y, \underline{\hom}_{\mathrm{lax}}(X,Z)) \end{align*} natural in $X,Y$ and $Z$. When $X=\sD_0$, we have $\sD_0 \otimes Y \simeq Y$, and thus  Leonard Guetta committed Apr 21, 2020 279 280 281  $\Hom_{\oo\Cat}(Y,Z)\simeq \Hom_{\oo\Cat}(\sD_0,\underline{\hom}_{\mathrm{oplax}}(Y,Z)).$  Leonard Guetta committed Oct 25, 2020 282  Hence, the $0$-cells of the $\oo$\nbd{}category $\underline{\hom}_{\mathrm{oplax}}(Y,Z)$ are the $\oo$\nbd{}functors $Y \to Z$.  Leonard Guetta committed Apr 21, 2020 283 284  \end{paragr} \begin{paragr}  Leonard Guetta committed Oct 25, 2020 285  Let $u,v : X \to Y$ be two $\oo$\nbd{}functors. An \emph{oplax transformation} from $u$ to $v$ is a $1$-cell $\alpha$ of $\homoplax(X,Y)$ with source $u$ and target $v$. We usually use the double arrow notation $ Leonard Guetta committed Apr 21, 2020 286 287 288 289 290 291 292 293 294  \alpha : u \Rightarrow v$ for oplax transformations. By adjunction, we have \begin{align*} \Hom_{\oo\Cat}(\sD_1,\homoplax(X,Y)) &\simeq \Hom_{\oo\Cat}(\sD_1\otimes X , Y)\\ &\simeq \Hom_{\oo\Cat}(X,\homlax(\sD_1,Y)). \end{align*} Hence, $\alpha : u \Rightarrow v$ can be encoded in the following two ways: \begin{itemize}[label=-]  Leonard Guetta committed Oct 25, 2020 295  \item As an $\oo$\nbd{}functor $\alpha : \sD_1\otimes X \to Y$ such that the following diagram  Leonard Guetta committed Apr 21, 2020 296 297 298  $\begin{tikzcd} X\ar[rd,"u"] \ar[d,"i_0^X"']& \\  Leonard Guetta committed Oct 14, 2020 299 300 301  \sD_1\otimes X \ar[r,"\alpha"] & Y, \\ X \ar[ru,"v"'] \ar[u,"i_1^X"]& \end{tikzcd}  Leonard Guetta committed Apr 21, 2020 302 $  Leonard Guetta committed Oct 25, 2020 303 304  where $i_0^X$ and $i_1^X$ are induced by the two $\oo$\nbd{}functors $\sD_0 \to \sD_1$ and where we implicitly used the isomorphism $\sD_0 \otimes X \simeq X$, is commutative. \item As an $\oo$\nbd{}functor $\alpha : X \to \homlax(\sD_1,Y)$ such that the following diagram  Leonard Guetta committed Apr 21, 2020 305 306 307 308  $\begin{tikzcd} & Y \\ X \ar[ru,"u"] \ar[r,"\alpha"] \ar[rd,"v"']& \homlax(\sD_1,Y) \ar[u,"\pi_0^Y"'] \ar[d,"\pi_1^Y"] \\  Leonard Guetta committed Jan 05, 2021 309  & Y,  Leonard Guetta committed Apr 21, 2020 310 311  \end{tikzcd}$  Leonard Guetta committed Oct 25, 2020 312  where $\pi^Y_0$ and $\pi^Y_1$ are induced by the two $\oo$\nbd{}functors $\sD_0 \to \sD_1$ and where we implicitly used the isomorphism $\homlax(\sD_0,Y)\simeq Y$, is commutative.  Leonard Guetta committed Apr 21, 2020 313  \end{itemize}  Leonard Guetta committed Oct 25, 2020 314  The $\oo$\nbd{}category $\homlax(\sD_1,Y)$ is sometimes referred to as the $\oo$\nbd{}category of cylinders in $Y$. An explicit description of this $\oo$\nbd{}category can be found, for example, in \cite[Appendix A]{metayer2003resolutions}, \cite[Section 4]{lafont2009polygraphic} or \cite[Appendice B.1]{ara2016joint}.  Leonard Guetta committed Apr 21, 2020 315 316 317  \end{paragr}  Leonard Guetta committed Oct 09, 2020 318  \begin{paragr}\label{paragr:formulasoplax}[Formulas for oplax transformations] We now give a third way of describing oplax transformations based on explicit formulas. The proof that this description is equivalent to those given in the previous paragraph can be found in \cite[Appendice B.2]{ara2016joint}.  Leonard Guetta committed Apr 21, 2020 319   Leonard Guetta committed Oct 25, 2020 320  Let $u, v : X \to Y$ two $\oo$\nbd{}functors. An oplax transformation $\alpha : u \Rightarrow v$ is given by the data of:  Leonard Guetta committed Apr 21, 2020 321 322 323 324 325  \begin{itemize}[label=-] \item for every $0$-cell $x$ of $X$, a $1$-cell of $Y$ $\alpha_x : u(x) \to v(x),$  Leonard Guetta committed Oct 26, 2020 326  \item for every $n$\nbd{}cell of $x$ of $X$ with $n>0$, an $(n+1)$-cell of $Y$  Leonard Guetta committed Apr 21, 2020 327  $ Leonard Guetta committed Oct 25, 2020 328  \alpha_x : \alpha_{\trgt_{n-1}(x)}\comp_{n-1}\cdots\comp_1\alpha_{\trgt_0(x)}\comp_0u(x) \to v(x)\comp_0\alpha_{\src_0(x)}\comp_1\cdots\comp_{n-1}\alpha_{\src_{n-1}(x)}  Leonard Guetta committed Apr 21, 2020 329 330 331 $ subject to the following axioms: \begin{enumerate}  Leonard Guetta committed Oct 26, 2020 332  \item for every $n$\nbd{}cell $x$ of $X$,  Leonard Guetta committed Apr 21, 2020 333  $\alpha_{1_x}=1_{\alpha_x},$  Leonard Guetta committed Oct 26, 2020 334  \item for all $0\leq k < n$, for all $n$\nbd{}cells $x$ and $y$ of $X$ that are $k$-composable,  Leonard Guetta committed Apr 21, 2020 335 336  $\begin{multlined}  Leonard Guetta committed Feb 04, 2021 337 338  \alpha_{x \comp_k y}={\left(v(\trgt_{k+1}(x))\comp_0\alpha_{\src_0(y)}\comp_1\cdots\comp_{k-1}\alpha_{\src_{k-1}(y)}\comp_k\alpha_y\right)}\\ {\comp_{k+1}\left(\alpha_x \comp_k\alpha_{\trgt_{k-1}(x)}\comp_{k-1}\cdots\comp_1\alpha_{\trgt_0(x)}\comp_0u(\src_{k+1}(y))\right)}.  Leonard Guetta committed Apr 21, 2020 339 340 341 342 343 344  \end{multlined}$ \end{enumerate} \end{itemize} \end{paragr}  Leonard Guetta committed Sep 02, 2020 345  \begin{example}\label{example:natisoplax}  Leonard Guetta committed Oct 04, 2020 346 347 348 349 350  When $C$ and $D$ are $n$\nbd{}categories with $n$ finite and $u,v :C \to D$ are two $n$\nbd{}functors, an oplax transformation $\alpha : u \Rightarrow v$ amounts to the data of a $(k+1)$\nbd{}cell $\alpha_x$ of $D$ for each $k$\nbd{}cell $x$ of $C$ with $0 \leq k \leq n$, with source and target as in the previous paragraph. These data being subject to the axioms of the previous paragraph. Note that when $x$ is an $n$\nbd{}cell of $C$, $\alpha_x$ is necessarily a unit, which can be expressed as the equality $\alpha_{t_{n-1}(x)}\comp_{n-1}\cdots\comp_1\alpha_{t_0(x)}\comp_0u(x) = v(x)\comp_0\alpha_{s_0(x)}\comp_1\cdots\comp_{n-1}\alpha_{s_{n-1}(x)}$ In particular, when $n=1$ and $C$ and $D$ are thus (small) categories, an oplax transformation $u \Rightarrow v$ is nothing but a natural transformation from $u$ to $v$.  Leonard Guetta committed Sep 02, 2020 351 352  \end{example} \begin{paragr}  Leonard Guetta committed Oct 25, 2020 353  Let $u : C \to D$ be an $\oo$\nbd{}functor. There is an oplax transformation from $u$ to $u$, denoted by $1_u$, which is defined as  Leonard Guetta committed Sep 02, 2020 354 355 356  $(1_u)_{x}:=1_{u(x)}$  Leonard Guetta committed Oct 26, 2020 357  for every cell $x$ of $C$. More abstractly, this oplax transformation corresponds to the $\oo$\nbd{}functor  Leonard Guetta committed Sep 02, 2020 358  $ Leonard Guetta committed Dec 27, 2020 359  \sD_1 \otimes C \overset{p\otimes u}{\longrightarrow} \sD_0 \otimes D \simeq D,  Leonard Guetta committed Sep 02, 2020 360 $  Leonard Guetta committed Oct 25, 2020 361  where $p$ is the only $\oo$\nbd{}functor $\sD_1\to \sD_0$.  Leonard Guetta committed Sep 02, 2020 362  \end{paragr}  Leonard Guetta committed Apr 21, 2020 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378  \begin{paragr} Let $\begin{tikzcd} B \ar[r,"f"] & C \ar[r,shift left,"u"] \ar[r,shift right,"v"']&D \ar[r,"g"] &E \end{tikzcd}$ be a diagram in $\omega\Cat$ and $\alpha : u \Rightarrow v$ an oplax transformation. The data of $(g\star \alpha)_x := g(\alpha_x)$ for each cell $x$ of $C$ (resp. $(\alpha \star f)_x :=\alpha_{f(x)}$  Leonard Guetta committed Oct 25, 2020 379  for each cell $x$ of $B$) defines an oplax transformation from $g \circ u$ to $g \circ v$ (resp. $u \circ f$ to $v\circ f$) which we denote by $g\star \alpha$ (resp. $\alpha \star f$).  Leonard Guetta committed Sep 02, 2020 380   Leonard Guetta committed Oct 01, 2020 381  More abstractly, if $\alpha$ is seen as an $\oo$\nbd{}functor $\sD_1 \otimes C \to D$, then $g \star \alpha$ (resp.\ $\alpha \star f)$ corresponds to the $\oo$\nbd{}functor obtained as the following composition  Leonard Guetta committed Sep 02, 2020 382 383 384 385 386 387 388  $\sD_1 \otimes C \overset{\alpha}{\longrightarrow} D \overset{f}{\longrightarrow} E$ (resp.\ $\sD_1 \otimes B \overset{\sD_1 \otimes f}{\longrightarrow} \sD_1 \otimes C \overset{\alpha}{\longrightarrow} D).$  Leonard Guetta committed Apr 21, 2020 389  \end{paragr}  Leonard Guetta committed Sep 02, 2020 390  \begin{remark}  Leonard Guetta committed Jan 20, 2021 391 392  All the above descriptions of oplax transformations can be easily dualized for \emph{lax transformations} (i.e.\ the $1$\nbd{}cells of the $\oo$\nbd{}category $\underline{\hom}_{\mathrm{lax}}(X,Y)$ for some $\oo$\nbd{}categories $X$ and $Y$). Habit is the only reason why we put emphasis on oplax transformations rather than lax transformations.  Leonard Guetta committed Sep 02, 2020 393  \end{remark}  Leonard Guetta committed Apr 21, 2020 394  \section{Homotopy equivalences and deformation retracts}  Leonard Guetta committed Sep 02, 2020 395  \begin{paragr}\label{paragr:hmtpyequiv}  Leonard Guetta committed Oct 25, 2020 396  Let $C$ and $D$ be two $\oo$\nbd{}categories and consider the smallest equivalence relation on the set $\Hom_{\oo\Cat}(C,D)$ such that two $\oo$\nbd{}functors from $C$ to $D$ are equivalent if there is an oplax direction between them (in any direction). Let us say that two $\oo$\nbd{}functors $u, v : C \to D$ are \emph{oplax homotopic} if they are equivalent for this equivalence relation.  Leonard Guetta committed Sep 02, 2020 397 398  \end{paragr} \begin{definition}\label{def:oplaxhmtpyequiv}  Leonard Guetta committed Oct 01, 2020 399  An $\oo$\nbd{}functor $u : C \to D$ is an \emph{oplax homotopy equivalence} if there exists an $\oo$\nbd{}functor $v : D \to C$ such that $u\circ v$ is oplax homotopic to $\mathrm{id}_D$ and $v\circ u$ is oplax homotopic to $\mathrm{id}_C$.  Leonard Guetta committed Sep 02, 2020 400  \end{definition}  Leonard Guetta committed Feb 04, 2021 401  Recall that we write $\gamma^{\Th} : \oo\Cat \to \ho(\oo\Cat^{\Th})$ for the localization functor with respect to the Thomason equivalences.  Leonard Guetta committed Apr 21, 2020 402  \begin{lemma}\label{lemma:oplaxloc}  Leonard Guetta committed Oct 25, 2020 403  Let $u, v : C \to D$ be two $\oo$\nbd{}functors. If there exists an oplax transformation $\alpha : u \Rightarrow v$, then $\gamma^{\Th}(u)=\gamma^{\Th}(v)$.  Leonard Guetta committed Apr 21, 2020 404 405  \end{lemma} \begin{proof}  Leonard Guetta committed Sep 02, 2020 406  This follows immediately from \cite[Théorème B.11]{ara2020theoreme}.  Leonard Guetta committed Apr 21, 2020 407  \end{proof}  Leonard Guetta committed Oct 25, 2020 408  From this lemma and the fact that the Thomason equivalences are saturated (Corollary \ref{cor:thomsaturated}), we deduce the following proposition.  Leonard Guetta committed Sep 17, 2020 409 410  \begin{proposition}\label{prop:oplaxhmtpyisthom} Every oplax homotopy equivalence is a Thomason equivalence.  Leonard Guetta committed Sep 02, 2020 411  \end{proposition}  Leonard Guetta committed May 04, 2020 412  \begin{paragr}\label{paragr:defrtract}  Leonard Guetta committed Oct 26, 2020 413  An $\oo$\nbd{}functor $i : C \to D$ is an \emph{oplax deformation retract} if there exists an $\oo$\nbd{}functor $r : D \to C$ such that:  Leonard Guetta committed Apr 21, 2020 414  \begin{enumerate}[label=(\alph*)]  Leonard Guetta committed Sep 02, 2020 415  \item $r\circ i=\mathrm{id}_C$,  Leonard Guetta committed Oct 26, 2020 416  \item there exists an oplax transformation $\alpha : \mathrm{id}_D \Rightarrow i\circ r$.  Leonard Guetta committed Apr 21, 2020 417  \end{enumerate}  Leonard Guetta committed Sep 02, 2020 418 419  Furthermore, $i$ is a \emph{strong oplax deformation retract} if $\alpha$ can be chosen such that: \begin{enumerate}[label=(\alph*),resume]  Leonard Guetta committed Dec 27, 2020 420  \item $\alpha \star i = 1_i$.  Leonard Guetta committed Apr 21, 2020 421  \end{enumerate}  Leonard Guetta committed Sep 02, 2020 422  An oplax deformation retract is a particular case of homotopy equivalence and thus of Thomason equivalence.  Leonard Guetta committed Apr 21, 2020 423  \end{paragr}  424  \begin{lemma}\label{lemma:pushoutstrngdefrtract}  Leonard Guetta committed Sep 02, 2020 425  The pushout of a strong oplax deformation retract is a strong oplax deformation retract.  Leonard Guetta committed Apr 21, 2020 426 427  \end{lemma} \begin{proof}  Leonard Guetta committed Sep 03, 2020 428 429  Let $i : A \to B$ be a strong oplax deformation retract and \label{cocartsquareretract}\tag{i}  Leonard Guetta committed Apr 21, 2020 430 431  \begin{tikzcd} A \ar[d,"i"] \ar[r,"u"] & A' \ar[d,"i'"] \\  Leonard Guetta committed Sep 03, 2020 432 433 434  B \ar[r,"v"] & B'\ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"] \end{tikzcd}  Leonard Guetta committed Jan 20, 2021 435  be a cocartesian square. We have to show that $i'$ is also a strong oplax deformation retract. By hypothesis there exist $r : B \to A$ such that $r \circ i = \mathrm{id}_A$ and $\alpha : \sD_1 \otimes B \to B$ such that the diagrams  Leonard Guetta committed Sep 03, 2020 436 437 438  \label{diagramtransf}\tag{ii} \begin{tikzcd} B\ar[rd,"\mathrm{id}_B"] \ar[d,"i_0^B"']& \\  Leonard Guetta committed Oct 14, 2020 439 440 441  \sD_1\otimes B \ar[r,"\alpha"] & B, \\ B \ar[ru,"i\circ r"'] \ar[u,"i_1^B"]& \end{tikzcd}  Leonard Guetta committed Sep 03, 2020 442 443 444 445  and \label{diagramstrong}\tag{iii} \begin{tikzcd}  Leonard Guetta committed Oct 14, 2020 446 447  \sD_1 \otimes A \ar[rr, bend right,"p\otimes i"']\ar[r,"\sD_1 \otimes i"] & \sD_1 \otimes B \ar[r,"\alpha"] & B, \end{tikzcd}  Leonard Guetta committed Sep 03, 2020 448 449 450 451 452 453  where $p$ is the unique morphism $\sD_1 \to \sD_0$, are commutative. From the commutativity of the following solid arrow diagram $\begin{tikzcd}  Leonard Guetta committed Oct 26, 2020 454  A \ar[r,"u"] \ar[d,"i"] & A' \ar[d,"i'"] \ar[dd,bend left=75,"\mathrm{id}_{A'}"] \\  Leonard Guetta committed Sep 03, 2020 455  B \ar[d,"r"] \ar[r,"v"] & B' \ar[d,"r'",dashed ] \\  Leonard Guetta committed Oct 14, 2020 456  A \ar[r,"u"] & A',  Leonard Guetta committed Sep 03, 2020 457  \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]  Leonard Guetta committed Oct 14, 2020 458  \end{tikzcd}  Leonard Guetta committed Sep 03, 2020 459 $  Leonard Guetta committed Dec 27, 2020 460  we deduce the existence of $r' : B' \to A'$ that makes the whole diagram commute. In particular, we have $r' \circ i' = \mathrm{id}_{A'}$.  Leonard Guetta committed Sep 03, 2020 461 462 463 464 465 466  From the commutativity of (\ref{diagramstrong}), we easily deduce the commutativity of the following solid arrow diagram $\begin{tikzcd} \sD_1\otimes A \ar[r,"\sD_1\otimes u"] \ar[d,"\sD_1\otimes i"] & \sD_1 \otimes A' \ar[d,"\sD_1 \otimes i'"] \ar[dd,bend left=75,"p\otimes i'"] \\ \sD_1\otimes B \ar[d,"\alpha"] \ar[r,"\sD_1 \otimes v"] & \sD_1 \otimes B' \ar[d,"\alpha'",dashed ] \\  Leonard Guetta committed Feb 04, 2021 467  B \ar[r,"v"] & B'.  Leonard Guetta committed Oct 14, 2020 468  \end{tikzcd}  Leonard Guetta committed Sep 03, 2020 469 $  Leonard Guetta committed Feb 04, 2021 470  The existence of $\alpha' : \sD_1 \otimes B' \to B'$ that makes the whole diagram commutes follows from the fact that the functor $\sD_1 \otimes \shortminus$ preserves colimits. In particular, we have $\alpha' \circ (\sD_1 \otimes i') = p \otimes i'.$  Leonard Guetta committed Sep 03, 2020 471   Leonard Guetta committed Oct 25, 2020 472  Now, notice that for every $\oo$\nbd{}category $C$, the maps  Leonard Guetta committed Sep 03, 2020 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492  $i^C_0 : C \to \sD_1 \otimes C \text{ and } i^C_1 : C \to \sD_1 \otimes C$ are natural in $C$. Using this naturality and simple diagram chasing (left to the reader), we obtain the equalities $\alpha ' \circ i_0^{B'} \circ v= v,$ $\alpha' \circ i^{B'}_0 \circ i'=i',$ and the equalities $\alpha ' \circ i_1^{B'} \circ v= i' \circ r' \circ v$ $\alpha' \circ i^{B'}_1 \circ i'=i' \circ r' \circ i'.$ Using the fact that square (\ref{cocartsquareretract}) is cocartesian, we deduce that $\alpha ' \circ i_0^{B'} = \mathrm{id}_{B'}$ and $\alpha' \circ i^{B'}_1 = i' \circ r'$. This proves that $i'$ is an oplax deformation retract, which is furthermore strong because of the equality $\alpha' \circ (\sD_1 \otimes i') = p \otimes i'$. \iffalse Now, we have commutative diagrams $\begin{tikzcd} B \ar[r,"i_{\epsilon}^{B}"] \ar[d,"v"] & \sD_1 \otimes B \ar[d,"\sD_1 \otimes v"] \ar[r,"\alpha"] & B \ar[d,"v"] \\ B' \ar[r,"i_{\epsilon}^{B'}"] & \sD_1 \otimes B' \ar[r,"\alpha'"] & B'  Leonard Guetta committed Apr 21, 2020 493 494  \end{tikzcd}$  Leonard Guetta committed Sep 03, 2020 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517  with $\epsilon=0$ or $1$, which proves that $\alpha ' \circ i_0^{B'} \circ v = v \circ \alpha \circ i_0^B = v$ and $\alpha ' \circ i_1^{B'} \circ v = v \circ \alpha \circ i_1^B = v \circ i \circ r = i' \circ u \circ r = i' \circ r' \circ v.$ Similarly, we have commutative diagrams $\begin{tikzcd} A' \ar[d,"i'"] \ar[r,"i_{\epsilon}^{A'}"] &\sD_1 \otimes A' \ar[d,"\sD_1 \otimes A'"] \ar[rd,"p\otimes i'"] &\\ B' \ar[r,"i_{\epsilon}^{B'}"] & \sD_1 \otimes B' \ar[r,"\alpha'"] & B' \end{tikzcd}$ for $\epsilon = 0$ or $1$, which proves that $\alpha' \circ i^{B'}_0 \circ i' = p\otimes i' \circ i^{A'}_0 = i'$ $\alpha' \circ i^{B'}_1 \circ i' = p\otimes i' \circ i^{A'}_1 = i' = i' \circ r' \circ i'.$ \fi  Leonard Guetta committed Apr 21, 2020 518  \end{proof}  Leonard Guetta committed Sep 03, 2020 519 520  In the following proposition, a \emph{co-universal Thomason equivalence} means a co-universal weak equivalence for the localizer $(\oo\Cat, \W^{\Th}_{\oo})$ (Definition \ref{def:couniversalwe}). \begin{proposition}  Leonard Guetta committed Sep 17, 2020 521  Every strong oplax deformation retract is a co-universal Thomason equivalence.  Leonard Guetta committed Sep 03, 2020 522 523 524 525 526 527 528  \end{proposition} \begin{proof} Immediate consequence of Lemma \ref{lemma:pushoutstrngdefrtract} and the fact that oplax transformation retracts are Thomason equivalences. \end{proof} \begin{remark} All the results we have seen in this section are still true if we replace oplax'' by lax'' everywhere. \end{remark}  Leonard Guetta committed Jan 21, 2021 529 530 531 532 \section[Equivalences of \texorpdfstring{$\oo$}{ω}-categories and the folk model structure]{Equivalences of \texorpdfstring{$\oo$}{ω}-categories and the folk model structure% \sectionmark{The folk model structure}}  Leonard Guetta committed Jan 20, 2021 533 \sectionmark{The folk model structure}  Leonard Guetta committed Sep 24, 2020 534 \begin{paragr}\label{paragr:ooequivalence}  Leonard Guetta committed Sep 03, 2020 535 536 537  Let $C$ be an $\omega$-category. We define the equivalence relation $\sim_{\omega}$ on the set $C_n$ by co-induction on $n \in \mathbb{N}$. For $x, y \in C_n$, we have $x \sim_{\omega} y$ when: \begin{itemize} \item[-] $x$ and $y$ are parallel,  Leonard Guetta committed Oct 26, 2020 538  \item[-] there exist $r, s \in C_{n+1}$ such that $r : x \to y$, $s : y \to x$,  Leonard Guetta committed Sep 03, 2020 539 540 541 542 543 544 545 546 547 548 549 550  $r\ast_{n}s \sim_{\omega} 1_y$ and $s\ast_nr \sim_{\omega} 1_x.$ \end{itemize} For details on this definition and the proof that it is an equivalence relation, see \cite[section 4.2]{lafont2010folk}. \end{paragr} \begin{example}  Leonard Guetta committed Oct 26, 2020 551  Let $x$ and $y$ be two $0$-cells of an $n$\nbd{}category $C$.  Leonard Guetta committed Sep 03, 2020 552 553  \begin{itemize}[label=-] \item When $n=1$, $x \sim_{\omega} y$ means that $x$ and $y$ are isomorphic.  Leonard Guetta committed Oct 26, 2020 554  \item When $n=2$, $x \sim_{\omega} y$ means that $x$ and $y$ are equivalent, i.e.\ there exist $f : x \to y$ and $g : y \to x$ such that $fg$ is isomorphic to $1_y$ and $gf$ is isomorphic to $1_x$.  Leonard Guetta committed Sep 03, 2020 555  \end{itemize}  Leonard Guetta committed Sep 04, 2020 556 \end{example}  Leonard Guetta committed Oct 09, 2020 557 For later reference, we put here the following trivial but important lemma, whose proof is omitted.  Leonard Guetta committed Oct 25, 2020 558 \begin{lemma}\label{lemma:ooequivalenceisfunctorial}  Leonard Guetta committed Oct 26, 2020 559  Let $F : C \to D$ be an $\oo$\nbd{}functor, $n \geq 0$ and $x,y$ be $n$\nbd{}cells of $C$. If $x \sim_{\oo} y$, then $F(x) \sim_{\oo} F(y)$.  Leonard Guetta committed Sep 04, 2020 560  \end{lemma}  Leonard Guetta committed Sep 03, 2020 561  \begin{definition}\label{def:eqomegacat}  Leonard Guetta committed Oct 25, 2020 562  An $\omega$-functor $F : C \to D$ is an \emph{equivalence of $\oo$\nbd{}categories} when:  Leonard Guetta committed Sep 03, 2020 563  \begin{itemize}  Leonard Guetta committed Sep 30, 2020 564  \item[-] for every $y \in D_0$, there exists $x \in C_0$ such that  Leonard Guetta committed Sep 04, 2020 565  $F(x)\sim_{\omega}y,$  Leonard Guetta committed Sep 30, 2020 566  \item[-] for every $n \geq 0$, for all $x,y \in C_n$ that are parallel and for every $\beta~\in~D_{n+1}$ such that $\beta : F(x) \to F(y),$ there exists $\alpha \in C_{n+1}$ such that  Leonard Guetta committed Sep 03, 2020 567 568 569  $\alpha : x \to y$ and  Leonard Guetta committed Sep 04, 2020 570  $F(\alpha)\sim_{\omega}\beta.$  Leonard Guetta committed Sep 03, 2020 571 572 573  \end{itemize} \end{definition} \begin{example}\label{example:equivalencecategories}  Leonard Guetta committed Oct 25, 2020 574  If $C$ and $D$ are (small) categories seen as $\oo$\nbd{}categories, then a functor $F : C \to D$ is an equivalence of $\oo$\nbd{}categories if and only if it is fully faithful and essentially surjective, i.e.\ an equivalence of categories.  Leonard Guetta committed Sep 03, 2020 575  \end{example}  Leonard Guetta committed Oct 25, 2020 576   Leonard Guetta committed Sep 03, 2020 577  \begin{theorem}\label{thm:folkms}  Leonard Guetta committed Feb 04, 2021 578 579 580  There exists a cofibrantly generated model structure on $\omega\Cat$ whose weak equivalences are the equivalences of $\oo$\nbd{}categories, and whose cofibrations are generated by the set $\{i_n : \sS_{n-1} \to \sD_n \vert n \in \mathbb{N}\}$ (see \ref{paragr:defglobe}).  Leonard Guetta committed Sep 03, 2020 581 582 583 584 585 586 587  \end{theorem} \begin{proof} This is the main result of \cite{lafont2010folk}. \end{proof} \begin{paragr}\label{paragr:folkms} The model structure of the previous theorem is commonly referred to as \emph{folk model structure} on $\omega\Cat$.  Leonard Guetta committed Oct 01, 2020 588  Data of this model structure will often be referred to by using the adjective folk, e.g.\ \emph{folk cofibration}. Consequently \emph{folk weak equivalence} and \emph{equivalence of $\oo$\nbd{}categories} mean the same thing.  589   Leonard Guetta committed Dec 27, 2020 590 591 592 593  Furthermore, as in the Thomason case (see \ref{paragr:notationthom}), we usually make reference to the word folk'' in the notations of homotopic constructions induced by the folk weak equivalences. For example, we write $\W^{\folk}$ for the class of folk weak equivalences, $\Ho(\oo\Cat^{\folk})$ for the homotopy op\nbd{}prederivator of $(\oo\Cat,\W_{\oo}^{\folk})$ and  594 595 596  $\gamma^{\folk} : \oo\Cat \to \Ho(\oo\Cat^{\folk})$  Leonard Guetta committed Nov 04, 2020 597 598 599 600 601  for the localization morphism. It follows from the previous theorem and Theorem \ref{thm:cisinskiI} that the localizer $(\oo\Cat,\W_{\oo}^{\folk})$ is homotopy cocomplete. We will speak of folk homotopy colimits'' and folk homotopy cocartesian squares'' for homotopy colimits and homotopy cocartesian squares in this localizer.  Leonard Guetta committed Sep 30, 2020 602 603  \end{paragr} \begin{paragr}\label{paragr:folktrivialfib}  Leonard Guetta committed Oct 01, 2020 604  Using the set $\{i_n : \sS_{n-1} \to \sD_n \vert n \in \mathbb{N} \}$ of generating folk cofibrations, we obtain that an $\oo$\nbd{}functor $F : C \to D$ is a \emph{folk trivial fibration} when:  Leonard Guetta committed Sep 30, 2020 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621  \begin{itemize}[label=-] \item for every $y \in D_0$, there exists $x \in C_0$ such that $F(x)=y$ \item for every $n\geq 0$, for all $x,y \in C_n$ that are parallel and for every $\beta~\in~D_{n+1}$ such that $\beta : F(x) \to F(y)$ there exists $\alpha \in C_{n+1}$ such that $\alpha : x \to y$ and $F(\alpha)=\beta.$  Leonard Guetta committed Oct 01, 2020 622  This characterization of folk trivial fibrations is to be compared with Definition \ref{def:eqomegacat} of equivalences of $\oo$\nbd{}categories.  Leonard Guetta committed Sep 30, 2020 623 624  \end{itemize} \end{paragr}  Leonard Guetta committed Sep 10, 2020 625  \begin{proposition}\label{prop:freeiscofibrant}  Leonard Guetta committed Sep 03, 2020 626 627 628  An $\omega$-category is cofibrant for the folk model structure if and only if it is free. \end{proposition} \begin{proof}  Leonard Guetta committed Oct 26, 2020 629  The fact that every free $\omega$-category is cofibrant follows immediately from the fact that the $i_n : \sS_{n-1} \to \sD_n$ are cofibrations and that every $\oo$\nbd{}category $C$ is the colimit of the canonical diagram (Lemma \ref{lemma:filtration})  Leonard Guetta committed Sep 03, 2020 630  $ Leonard Guetta committed Oct 25, 2020 631  \sk_{0}(C) \to \sk_{1}(C) \to \cdots \to \sk_n(C) \to \sk_{n+1}(C) \to \cdots  Leonard Guetta committed Sep 03, 2020 632 633 634 $ For the converse, see \cite{metayer2008cofibrant}. \end{proof}  Leonard Guetta committed Oct 09, 2020 635  \iffalse  Leonard Guetta committed Sep 03, 2020 636 637 638 639 640 641 642 643 644 645  \begin{proposition} Let $f : A \to B$ and $g : C \to D$ be morphisms of $\oo\Cat$. If $f$ and $g$ are cofibrations for the folk model structure, then so is $f\otimes g : A \otimes B \to C \otimes D.$ \end{proposition} \begin{proof} See \cite[Proposition 5.1.2.7]{lucas2017cubical} or \cite{ara2019folk}. \end{proof} \fi  Leonard Guetta committed Jan 21, 2021 646 647 648 649 650  \section[Equivalences of \texorpdfstring{$\oo$}{ω}-categories vs Thomason equivalences ]{Equivalences of \texorpdfstring{$\oo$}{ω}-categories vs Thomason equivalences% \sectionmark{Folk vs Thomason}} \sectionmark{Folk vs Thomason}  Leonard Guetta committed Sep 03, 2020 651  \begin{lemma}\label{lemma:nervehomotopical}  Leonard Guetta committed Dec 27, 2020 652 The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalences of $\oo$\nbd{}categories to weak equivalences of simplicial sets.  Leonard Guetta committed Sep 03, 2020 653 654  \end{lemma} \begin{proof}  Leonard Guetta committed Dec 27, 2020 655 656 657 658  Since every $\omega$-category is fibrant for the folk model structure \cite[Proposition 9]{lafont2010folk}, it follows from Ken Brown's Lemma \cite[Lemma 1.1.12]{hovey2007model} that it suffices to show that the nerve sends the folk trivial fibrations to weak equivalences of simplicial sets. In particular, it suffices to show the stronger condition that the nerve sends the folk trivial fibrations to trivial fibrations of simplicial sets.  Leonard Guetta committed Sep 03, 2020 659   Leonard Guetta committed Oct 27, 2020 660  By adjunction, this is equivalent to showing that the functor $c_{\omega} : \Psh{\Delta} \to \omega\Cat$ sends the cofibrations of simplicial sets to folk cofibrations. Since $c_{\omega}$ is cocontinuous and the cofibrations of simplicial sets are generated by the inclusions  Leonard Guetta committed Sep 03, 2020 661 662 663 664 665 666 667  $\partial \Delta_n \to \Delta_n$ for $n \in \mathbb{N}$, it suffices to show that $c_{\omega}$ sends these inclusions to folk cofibrations. Now, it follows from \cite[Lemma 5.1]{street1987algebra} that the image of the inclusion $\partial \Delta_n \to \Delta_n$ by $c_{\omega}$ can be identified with the canonical inclusion $ Leonard Guetta committed Oct 12, 2020 668  \sk_{n-1}(\Or_n) \to \Or_n.  Leonard Guetta committed Sep 03, 2020 669 $  Leonard Guetta committed Oct 12, 2020 670  Since $\Or_n$ is free, this last morphism is by definition a push-out of a coproduct of folk cofibrations (see Definition \ref{def:nbasis}), hence a folk cofibration.  Leonard Guetta committed Sep 03, 2020 671 672 673  \end{proof} As an immediate consequence of the previous lemma, we have the following proposition. \begin{proposition}\label{prop:folkisthom}  Leonard Guetta committed Oct 25, 2020 674  Every equivalence of $\oo$\nbd{}categories is a Thomason equivalence.  Leonard Guetta committed Sep 03, 2020 675 676  \end{proposition} \begin{remark}  Leonard Guetta committed Oct 01, 2020 677  The converse of the above proposition is false. For example, the unique $\oo$\nbd{}functor  Leonard Guetta committed Sep 03, 2020 678 679 680  $\sD_1 \to \sD_0$  Leonard Guetta committed Oct 01, 2020 681  is a Thomason equivalence because its image by the nerve is the unique morphism of simplicial sets $\Delta_1 \to \Delta_0$ (which obviously is a weak equivalence), but it is \emph{not} an equivalence of $\oo$\nbd{}categories because $\sD_1$ and $\sD_0$ are not equivalent as categories (see Example \ref{example:equivalencecategories}).  Leonard Guetta committed Sep 03, 2020 682 683  \end{remark} \begin{paragr}\label{paragr:compweakeq}  Leonard Guetta committed Sep 17, 2020 684  Proposition \ref{prop:folkisthom} implies that the identity functor on $\oo\Cat$ induces a morphism of localizers $(\oo\Cat,\W^{\folk}) \to (\oo\Cat,\W^{\Th})$, which in turn induces a functor between localized categories  Leonard Guetta committed Sep 03, 2020 685 686  % \label{cantoTh} $ Leonard Guetta committed Sep 17, 2020 687  \mathcal{J} : \ho(\oo\Cat^{\folk}) \to \ho(\oo\Cat^{\Th}).  Leonard Guetta committed Sep 03, 2020 688 689 $ %  Leonard Guetta committed Sep 17, 2020 690 691 692 693 694  %% Note that for every small category $A$, the functor %% $%% \ho(\oo\Cat(A)^{\folk}) \to \ho(\oo\Cat(A)^{\Th}) %%$ %% is the identity on objects.  Leonard Guetta committed Oct 25, 2020 695  This functor cannot be an equivalence since this would imply that every Thomason equivalence is an equivalence of $\oo$\nbd{}categories.  Leonard Guetta committed Sep 03, 2020 696  \end{paragr}  Leonard Guetta committed Jan 21, 2021 697  \section{{Slice \texorpdfstring{$\oo$}{ω}-categories and folk Theorem~A}}  Leonard Guetta committed Jun 03, 2020 698  \begin{paragr}\label{paragr:slices}  Leonard Guetta committed Oct 25, 2020 699  Let $A$ be an $\oo$\nbd{}category and $a_0$ an object of $A$. We define the slice $\oo$\nbd{}category $A/a_0$ as the following fibred product:  Leonard Guetta committed Apr 21, 2020 700 701 702 703 704 705 706  $\begin{tikzcd} A/a_0 \ar[d] \ar[r] & \homlax(\sD_1,A) \ar[d,"\pi_1^A"] \\ \sD_0 \ar[r,"\langle a_0 \rangle"'] & A. \ar[from=1-1,to=2-2,phantom,very near start,"\lrcorner"] \end{tikzcd}$  Leonard Guetta committed Oct 01, 2020 707  We also define an $\oo$\nbd{}functor $\pi : A/a_0 \to A$ as the following composition  Leonard Guetta committed Apr 21, 2020 708 709 710  $\pi : A/a_0 \to \homlax(\sD_1,A) \overset{\pi^A_0}{\longrightarrow} A.$  Leonard Guetta committed Oct 01, 2020 711  Let us now give an alternative definition of the $\oo$\nbd{}category $A/a_0$ using explicit formulas. The equivalence with the previous definition follows from the dual of \cite[Proposition B.5.2]{ara2016joint}  Leonard Guetta committed Apr 21, 2020 712  \begin{itemize}[label=-]  Leonard Guetta committed Oct 26, 2020 713  \item An $n$\nbd{}cell of $A/a_0$ is a table  Leonard Guetta committed Apr 21, 2020 714 715 716 717 718 719 720 721 722 723 724 725  $(x,a)=\begin{pmatrix} \begin{matrix} (x_0,a_1) & (x_1,a_2) & \cdots & (x_{n-1},a_n) \\[0.5em] (x_0',a_1') & (x_1',a_2') & \cdots & (x_{n-1}',a_n') \end{matrix} & (x_n,a_{n+1}) \end{pmatrix}$ where $x_0$ and $x_0'$ are $0$-cells of $A$, and: \begin{tabular}{ll}  Leonard Guetta committed Oct 26, 2020 726 727 728 729  $x_i : x_{i-1} \longrightarrow x'_{i-1}$, &for every $1 \leq i \leq n$,\$0.75em] x_i': x_{i-1} \longrightarrow x'_{i-1}, &for every 1 \leq i \leq n-1,\\[0.75em] a_i : a'_{i-1}\comp_{i-2} a'_{i-2} \comp_{i-3} \cdots \comp_1 a'_1\comp_0 x_{i-1} \longrightarrow a_{i-1}, &for every 1 \leq i \leq n+1,\\[0.75em] a'_i : a'_{i-1}\comp_{i-2} a'_{i-2} \comp_{i-3} \cdots \comp_1 a'_1 \comp_0 x'_{i-1} \longrightarrow a_{i-1}, &for every 1 \leq i \leq n\\  Leonard Guetta committed Apr 21, 2020 730 731  \end{tabular}  Leonard Guetta committed Sep 03, 2020 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752  are i-cells of A. In low dimension, this gives: \begin{tabular}{ll} (x_0,a_1) :& {\begin{tikzcd} x_0 \ar[d,"a_1"] \\ a_0 \end{tikzcd}} \\[2.75em] {\begin{pmatrix} \begin{matrix} (x_0,a_1) \\[0.5em] (x_0',a_1') \end{matrix} & (x_1,a_{2}) \end{pmatrix}} :& {\begin{tikzcd}[column sep=small] x_0 \ar[rr,"x_1"] \ar[rd,"a_1"',""{name=A,left}] && x_0' \ar[ld,"a_1'"] \\ &a_0 & \ar[from=1-3,to=A,Rightarrow,"a_2", shorten <=1em, shorten >=1em]\end{tikzcd}} \\[2.75em] {\begin{pmatrix} \begin{matrix} (x_0,a_1) & (x_1,a_2) \\[0.5em] (x_0',a_1') & (x_1',a_2') \end{matrix} & (x_2,a_{3}) \end{pmatrix}}:&{\begin{tikzcd}[column sep=small] x_0 \ar[rr,"x_1"] \ar[rd,"a_1"',""{name=A,left}] && x_0' \ar[ld,"a_1'"] \\ &a_0 & \ar[from=1-3,to=A,Rightarrow,"a_2", shorten <=1em, shorten >=1em]\end{tikzcd}\; \overset{a_3}{\Lleftarrow} \; \begin{tikzcd}[column sep=small] x_0\ar[rr,bend left=50,"x_1",pos=11/20,""{name=toto,below}] \ar[rr,"x_1'"description,""{name=titi,above}] \ar[rd,"a_1"',""{name=A,left}] && x_0' \ar[ld,"a_1'"] \\ &a_0 & \ar[from=1-3,to=A,Rightarrow,"a_2'", shorten <=1em, shorten >=1em] \ar[from=toto,to=titi,Rightarrow,"x_2",pos=1/5]\end{tikzcd}} \end{tabular}  Leonard Guetta committed Feb 04, 2021 753  \item The source and target of the n\nbd{}cell (a,x) are given by the matrices:  Leonard Guetta committed Apr 21, 2020 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768  \[ s(x,a)=\begin{pmatrix} \begin{matrix} (x_0,a_1) & (x_1,a_2) & \cdots & (x_{n-2},a_{n-1}) \\[0.5em] (x_0',a_1') & (x_1',a_2') & \cdots & (x_{n-2}',a_{n-1}') \end{matrix} & (x_{n-1},a_{n}) \end{pmatrix}$ $t(x,a)=\begin{pmatrix} \begin{matrix} (x_0,a_1) & (x_1,a_2) & \cdots & (x_{n-2},a_{n-1}) \\[0.5em] (x_0',a_1') & (x_1',a_2') & \cdots & (x_{n-2}',a_{n-1}') \end{matrix}  Leonard Guetta committed Feb 04, 2021 769 770  & (x'_{n-1},a'_{n}) \end{pmatrix}.  Leonard Guetta committed Apr 21, 2020 771 772 $ % It is understood that when $n=1$, the source is simply $(x_0,a_1)$ and the target $(x_0,a_1')$  Leonard Guetta committed Oct 26, 2020 773  \item The unit of the $n$\nbd{}cell $(a,x)$ is given by the table:  Leonard Guetta committed Apr 21, 2020 774 775 776 777 778 779 780  $1_{(x,a)}=\begin{pmatrix} \begin{matrix} (x_0,a_1) & (x_1,a_2) & \cdots & (x_{n-1},a_n) & (x_n,a_{n+1}) \\[0.5em] (x_0',a_1') & (x_1',a_2') & \cdots & (x_{n-1}',a_n') & (x_n,a_{n+1}) \end{matrix} & (1_{x_n},1_{a_{n+1}})  Leonard Guetta committed Feb 04, 2021 781  \end{pmatrix}.  Leonard Guetta committed Apr 21, 2020 782 $  Leonard Guetta committed Oct 26, 2020 783  \item The composition of $n$\nbd{}cells $(x,a)$ and $(y,b)$ such that $\src_k(y,b)=\trgt_k(a,x)$, is given by the table:  Leonard Guetta committed Apr 21, 2020 784 785 786 787 788 789 790 791 792 793 794 795  $(y,b)\comp_k (x,a)=\begin{pmatrix} \begin{matrix} (x_0,a_1) & \cdots & (x_k,a_k) & (z_{k+1},c_{k+2}) & \cdots &(z_{n-1},c_n) \\[0.5em] (y_0',b_1') & \cdots & (y'_k,b'_k) &(z'_{k+1},c'_{k+2}) & \cdots & (z'_{n-1},c'_n) \\ \end{matrix} & (z_n,c_{n+1}) \end{pmatrix},$ where: \begin{tabular}{ll}  Leonard Guetta committed Oct 26, 2020 796 797  $z_{i}=y_i\comp_k x_i$, & for every $k+1 \leq i \leq n$, \0.75em] z'_i=y'_i \comp_k x'_i, & for every k+1 \leq i \leq n-1, \\[0.75em]  Leonard Guetta committed Feb 11, 2021 798 799  c_i=a_i\comp_{k+1} b_i \comp_{k} a'_{k} \comp_{k-1} a'_{k-1} \comp_{k-2} \cdots \comp_{1} a'_1\comp_0 x_{k+1},&for every k+2 \leq i \leq n+1, \\[0.75em] c'_i=a'_i\comp_{k+1} b'_i \comp_{k} a'_{k} \comp_{k-1} a'_{k-1} \comp_{k-2} \cdots \comp_{1} a'_1\comp_0 x'_{k+1},&for every k+2 \leq i \leq n.\\  Leonard Guetta committed Apr 21, 2020 800 801  \end{tabular} \end{itemize}  Leonard Guetta committed Feb 11, 2021 802  We leave it to the reader to check that the formulas are well defined and that the axioms for \oo\nbd{}categories are satisfied. The canonical forgetful \oo\nbd{}functor \pi : A/a_0 \to A is simply expressed as:  Leonard Guetta committed Apr 21, 2020 803 804 805 806  \begin{align*} A/a_0 &\to A \\ (x,a) &\mapsto x_n. \end{align*}  Leonard Guetta committed Oct 26, 2020 807  Notice that if A is an n\nbd{}category, then so is A/a_0. In this case, for an n\nbd{}cell (x,a), a_{n+1} is a unit, hence  Leonard Guetta committed Apr 21, 2020 808 809 810 811  \[ a'_n \comp_{n-1} a'_{n-1} \comp_{n-2} \cdots \comp_1 a'_1 \comp_0 x_n = a_n. \end{paragr}  Leonard Guetta committed Sep 03, 2020 812  \begin{example}\label{example:slicecategories}  Leonard Guetta committed Oct 01, 2020 813  For a small category $A$ (considered as an $\oo$\nbd{}category) and an object $a_0$ of $A$, the category $A/a_0$ in the sense of the previous paragraph is nothing but the usual slice category of $A$ over $a_0$.  Leonard Guetta committed Sep 03, 2020 814  \end{example}  Leonard Guetta committed Oct 11, 2020 815  \begin{paragr}\label{paragr:comma}  Leonard Guetta committed Oct 25, 2020 816  Let $u : A \to B$ be a morphism of $\oo\Cat$ and $b_0$ an object of $B$. We define the $\oo$\nbd{}category $A/b_0$ (also denoted by $u\downarrow b_0$) as the following fibred product:  Leonard Guetta committed Apr 21, 2020 817 818  $\begin{tikzcd}  Leonard Guetta committed Apr 24, 2020 819 820  A/b_0 \ar[d,"u/b_0"'] \ar[r] & A \ar[d,"u"] \\ B/b_0 \ar[r,"\pi"'] & B.  Leonard Guetta committed Apr 21, 2020 821  \ar[from=1-1,to=2-2,phantom,description,very near start,"\lrcorner"]  Leonard Guetta committed Apr 24, 2020 822  \end{tikzcd}  Leonard Guetta committed Apr 21, 2020 823 $  Leonard Guetta committed Oct 26, 2020 824  More explicitly, an $n$\nbd{}cell $(x,b)$ of $A/b_0$ is a table  Leonard Guetta committed Apr 21, 2020 825 826 827 828 829 830 831 832 833  $(x,b)=\begin{pmatrix} \begin{matrix} (x_0,b_1) & (x_1,b_2) & \cdots & (x_{n-1},b_n) \\[0.5em] (x_0',b_1') & (x_1',b_2') & \cdots & (x_{n-1}',b_n') \end{matrix} & (x_n,b_{n+1}) \end{pmatrix}$  Leonard Guetta committed Feb 11, 2021 834 835 836 837 838 839 840 841  where the $x_i$ and $x'_i$ are $i$-cells of $A$ such that \begin{tabular}{ll} $x_i : x_{i-1} \longrightarrow x'_{i-1}$, &for every $1 \leq i \leq n$,\$0.75em] x_i': x_{i-1} \longrightarrow x'_{i-1}, &for every 1 \leq i \leq n-1,\\[0.75em] \end{tabular} and the b_i and b'_i are i-cells of B such that  Leonard Guetta committed Apr 21, 2020 842 843 844 845 846 847 848 849 850  \[ \begin{pmatrix} \begin{matrix} (u(x_0),b_1) & (u(x_1),b_2) & \cdots & (u(x_{n-1}),b_n) \\[0.5em] (u(x_0'),b_1') & (u(x'_1),b_2') & \cdots & (u(x'_{n-1}),b_n') \end{matrix} & (u(x_n),b_{n+1}) \end{pmatrix}$  Leonard Guetta committed Oct 26, 2020 851  is an $n$\nbd{}cell of $B/b_0$.  Leonard Guetta committed Sep 04, 2020 852   Leonard Guetta committed Oct 01, 2020 853  The canonical $\oo$\nbd{}functor $A/b_0 \to A$ is simply expressed as  Leonard Guetta committed Sep 04, 2020 854 855 856 857  \begin{align*} A/b_0 &\to A\\ (x,b) &\mapsto x_n, \end{align*}  Leonard Guetta committed Oct 01, 2020 858  and the $\oo$\nbd{}functor $u/b_0$ as  Leonard Guetta committed Sep 04, 2020 859 860 861 862 863 864 865 866  \begin{align*} u/b_0 : A/b_0 &\to B/b_0 \\ (x,b) &\mapsto (u(x),b). \end{align*} More generally, if we have a commutative triangle in $\oo\Cat$ $\begin{tikzcd}[column sep=small] A \ar[rr,"u"] \ar[dr,"v"'] & &B \ar[dl,"w"] \\  Leonard Guetta committed Oct 14, 2020 867 868  &C&, \end{tikzcd}  Leonard Guetta committed Sep 04, 2020 869 $  Leonard Guetta committed Oct 25, 2020 870  then for every object $c_0$ of $C$, we have a functor $u/c_0 : A/c_0 \to B/c_0$ defined as  Leonard Guetta committed Sep 04, 2020 871 872 873 874  \begin{align*} u/c_0 : A/c_0 &\to B/c_0 \\ (x,c) &\mapsto (u(x),c). \end{align*}  Leonard Guetta committed Apr 21, 2020 875 876  \end{paragr}