hmtpy.tex 47.8 KB
 Leonard Guetta committed Apr 21, 2020 1 2 3 \chapter{Homotopy theory of $\oo$-categories} \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 12  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 $\sigma^i : [n+1] \to [n]$ the only sujerctive non-decreasing map such that the pre-image of $i \in [n]$ contains exactly two elements.  Leonard Guetta committed Apr 21, 2020 13 14 15 16 17  The category $\Psh{\Delta}$ of simplicial sets is the category of presheaves on $\Delta$. For a simplicial set $X$, we use the notations \begin{aligned} X_n &:= X([n]) \\  Leonard Guetta committed Aug 21, 2020 18 19  \partial_i &:= X(\delta^i): X_n \to X_{n\shortminus 1}\\ s_i &:= X(\sigma^i): X_{n+1} \to X_n.  Leonard Guetta committed Apr 21, 2020 20 21  \end{aligned}  Leonard Guetta committed Sep 01, 2020 22  Elements of $X_n$ are referred to as \emph{$n$-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 23 \end{paragr}  Leonard Guetta committed Sep 26, 2020 24 \begin{paragr}\label{paragr:orientals}  Leonard Guetta committed Sep 01, 2020 25  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{$n$-oriental}. There are various ways to give a precise definition of the orientals, but each of them needs 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 litterature on the subject (such as \cite{street1987algebra}, \cite{street1991parity,street1994parity}, \cite{steiner2004omega}, \cite{buckley2016orientals} or \cite[chapitre 7]{ara2016joint}) for details.  Leonard Guetta committed Apr 21, 2020 26 27  The two main points to retain are:  Leonard Guetta committed Aug 21, 2020 28 29  \begin{description} \item[(OR1)] Each $\Or_n$ is a free $\omega$-category whose set of generating $k$-cells is canonically isomorphic 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 37 38 39 40 41 42 43 44  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=-] \item There is no generating $k$-cells for $k>n$. Hence, $\Or_n$ is an $n$-category. \item There is exactly one generating $n$-cell of $\Or_n$, which is $\langle 0 \,1 \cdots n\rangle$. We refer to this cell as the \emph{principal cell of $\Or_n$}. \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 Sep 01, 2020 46  \item[(OR2)] For $n>0$, the source (resp.\ target) of the principal cell of $\Or_n$ can be expressed as a composite of all the generating $(n-1)$\nbd-cells corresponding to $\delta^i$ with $i$ odd (resp.\ even); each of these generating $(n-1)$\nbd-cell appearing exactly once in the composite.  Leonard Guetta committed Aug 21, 2020 47  \end{description}  Leonard Guetta committed Sep 01, 2020 48  Another way of formulating \textbf{(OR2)} is: for $n>0$ the weight of the $(n-1)$-cell corresponding to $\delta_i$ in the \emph{source} of the principal cell of $\Or_n$ (see \ref{paragr:weight}) 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 62 63 64 65 66  $\Or_0 = \langle 0 \rangle,$ $\Or_1=\begin{tikzcd} \langle 0 \rangle \ar[r,"\langle 01 \rangle"] &\langle 1 \rangle, \end{tikzcd}$ $\Or_2= \begin{tikzcd} &\langle 1 \rangle \ar[rd,"\langle 12 \rangle"]& \\ \langle 0 \rangle \ar[ru,"\langle 01 \rangle"]\ar[rr,"\langle 02 \rangle"',""{name=A,above}]&&\langle 2 \rangle \ar[Rightarrow,from=A,to=1-2,"\langle 012 \rangle"] \end{tikzcd},$ $ 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 78  & \langle 1 \rangle \ar[rd,"\langle 12 \rangle"] \ar[dd,"\langle 13 \rangle"' description,""{name=B,right}] & \\ \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 Sep 01, 2020 89 90  N_{\omega}(C) : \Delta^{op} &\to \Set\\ [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 Sep 01, 2020 100  which we refer to as the \emph{nerve functor for $\oo$-categories}. Furthermore, for every $n \in \mathbb{N}$, we also define a nerve functor for $n$-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 Sep 01, 2020 104  By the usual Kan extension technique, we obtain for each $n \in \nbar$ a functor $c_n : \Psh{\Delta} \to n\Cat,$ left adjoint of $N_n$.  Leonard Guetta committed Sep 02, 2020 105 106 \end{paragr} \iffalse  Leonard Guetta committed Apr 21, 2020 107 108 109  \begin{lemma} Let $X$ be a simplicial set. The $\oo$-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 $X$. \end{lemma}  Leonard Guetta committed Sep 02, 2020 110 111  \fi \todo{Mettre lemme qui dit que la realisation oo-categorique d'un ensemble simplicial est libre ?}  Leonard Guetta committed Apr 21, 2020 112  \begin{paragr}  Leonard Guetta committed Sep 02, 2020 113 114  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 115  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 116 $  Leonard Guetta committed Sep 02, 2020 117  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 118   Leonard Guetta committed Sep 02, 2020 119  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 $C$, a $m$-simplex $X$ of $N_2(C)$ consists of:  Leonard Guetta committed Apr 21, 2020 120  \begin{itemize}[label=-]  Leonard Guetta committed Sep 02, 2020 121  \item for every $0\leq i \leq m$, an object $X_i$ of $C$,  Leonard Guetta committed Sep 02, 2020 122 123  \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 124 125  $\begin{tikzcd}  Leonard Guetta committed Sep 02, 2020 126 127 128  & X_j \ar[rd,"{X_{j,k}}"]& \\ X_i \ar[ru,"X_{i,j}"]\ar[rr,"X_{i,k}"',""{name=A,above}]&&X_k \ar[Rightarrow,from=A,to=1-2,"X_{i,j,k}"]  Leonard Guetta committed Apr 21, 2020 129 130 131 132 133  \end{tikzcd},$ \end{itemize} subject to the following axiom: \begin{itemize}[label=-]  Leonard Guetta committed Sep 02, 2020 134 135 136 137 138 139 140 141 142 143 144 145  \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 146  \end{itemize}  Leonard Guetta committed Sep 02, 2020 147 148 149 150 151 152 153 154 155  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 156  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 157  $ Leonard Guetta committed Sep 02, 2020 158  \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 225  \end{remark}  Leonard Guetta committed Sep 02, 2020 226 227  By definition, for all 1 \leq n \leq m \leq \omega, the canonical inclusion {n\Cat \hookrightarrow m\Cat} sends Thomason equivalences of n\Cat to Thomason equivalences of m\Cat. Hence, it induces a morphism of localizers and then a morphism of op-prederivator \Ho(n\Cat^\Th) \to \Ho(m\Cat^{\Th}). \begin{proposition}  Leonard Guetta committed Apr 21, 2020 228 229 230 231 232  For all 1 \leq n \leq m \leq \omega, the canonical morphism \[ \Ho(n\Cat^\Th) \to \Ho(m\Cat^{\Th})$ is an equivalence of op-prederivators.  Leonard Guetta committed Sep 02, 2020 233  \end{proposition}  Leonard Guetta committed Apr 21, 2020 234 235 236 237 238 239 240 241 242  \begin{proof} This follows from Theorem \ref{thm:gagna} and the commutativity of the triangle $\begin{tikzcd}[column sep=tiny] \Ho(n\Cat^{\Th}) \ar[rr] \ar[rd,"\overline{N_n}"'] & & \Ho(m\Cat) \ar[dl,"\overline{N_m}"] \\ &\Ho(\Psh{\Delta})& \end{tikzcd}.$ \end{proof}  Leonard Guetta committed Sep 02, 2020 243 244  \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 245  \begin{paragr}  Leonard Guetta committed Sep 02, 2020 246  The Gray tensor product makes $\oo\Cat$ into a monoidal category for which the unit is the $\oo$-category $\sD_0$ (which is the terminal $\oo$-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 247  $ Leonard Guetta committed Sep 02, 2020 248  \underline{\hom}_{\mathrm{oplax}}(-,-),\, \underline{\hom}_{\mathrm{lax}}(-,-) : \oo\Cat^{\op}\times\oo\Cat \to \oo\Cat  Leonard Guetta committed Apr 21, 2020 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 $ such that for all $\oo$-categories $X,Y$ and $Z$, we have isomorphisms $\Hom_{\oo\Cat}(X\otimes Y , Z) \simeq \Hom_{\oo\Cat}(X, \underline{\hom}_{\mathrm{oplax}}(Y,Z)) \simeq \Hom_{\oo\Cat}(Y, \underline{\hom}_{\mathrm{lax}}(X,Z))$ natural in $X,Y$ and $Z$. When $X=\sD_0$, using $\sD_0 \otimes Y \simeq Y$, we obtain $\Hom_{\oo\Cat}(Y,Z)\simeq \Hom_{\oo\Cat}(\sD_0,\underline{\hom}_{\mathrm{oplax}}(Y,Z)).$ Hence, the $0$-cells of the $\oo$-category $\underline{\hom}_{\mathrm{oplax}}(Y,Z)$ are the $\oo$-functors $Y \to Z$. \end{paragr} \begin{paragr} Let $u,v : X \to Y$ be two $\oo$-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 $\alpha : u \Rightarrow v$ 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=-] \item As an $\oo$-functor $\alpha : \sD_1\otimes X \to Y$ such that the following diagram $\begin{tikzcd} X\ar[rd,"u"] \ar[d,"i_0^X"']& \\ \sD_1\otimes X \ar[r,"\alpha"] & Y \\ X \ar[ru,"v"'] \ar[u,"i_1^X"] \end{tikzcd},$  Leonard Guetta committed Sep 02, 2020 279  where $i_0^X$ and $i_1^X$ are induced by the two $\oo$-functors $\sD_0 \to \sD_1$ and where we implicitly used the isomorphism $\sD_0 \otimes X \simeq X$, is commutative.  Leonard Guetta committed Apr 21, 2020 280 281 282 283 284 285 286 287  \item As an $\oo$-functor $\alpha : X \to \homlax(\sD_1,Y)$ such that the following diagram $\begin{tikzcd} & Y \\ X \ar[ru,"u"] \ar[r,"\alpha"] \ar[rd,"v"']& \homlax(\sD_1,Y) \ar[u,"\pi_0^Y"'] \ar[d,"\pi_1^Y"] \\ & Y \end{tikzcd}$  Leonard Guetta committed Sep 02, 2020 288  where $\pi^Y_0$ and $\pi^Y_1$ are induced by the two $\oo$-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 289 290 291 292 293  \end{itemize} The $\oo$-category $\homlax(\sD_1,Y)$ is sometimes referred to as the $\oo$-category of cylinders in $Y$. An explicit description of this $\oo$-category can be found, for example, in \cite[Appendix A]{metayer2003resolutions}, \cite[Section 4]{lafont2009polygraphic} or \cite[Appendice B.1]{ara2016joint}. \end{paragr}  Leonard Guetta committed Sep 02, 2020 294  \begin{paragr}\label{paragr:formulasoplax}[Formulas for oplax tranformations] 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 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320  Let $u, v : X \to Y$ two $\oo$-functors. An oplax transformation $\alpha : u \Rightarrow v$ is given by the data of: \begin{itemize}[label=-] \item for every $0$-cell $x$ of $X$, a $1$-cell of $Y$ $\alpha_x : u(x) \to v(x),$ \item for every $n$-cell of $x$ of $X$ with $n>0$, an $(n+1)$-cell of $Y$ $\alpha_x : \alpha_{t_{n-1}(x)}\comp_{n-1}\cdots\comp_1\alpha_{t_0(x)}\comp_0u(x) \to v(x)\comp_0\alpha_{s_0(x)}\comp_1\cdots\comp_{n-1}\alpha_{s_{n-1}(x)}$ subject to the following axioms: \begin{enumerate} \item for every $n$-cell $x$ of $X$, $\alpha_{1_x}=1_{\alpha_x},$ \item for all $0\leq k < n$, for all $n$-cells $x$ and $y$ of $X$ that are $k$-composable, $\begin{multlined} \alpha_{x \comp_k y}={\left(v(t_{k+1}(x))\comp_0\alpha_{s_0(x)}\comp_1\cdots\comp_{n-1}\alpha_{s_{n-1}(x)}\comp_k\alpha_y\right)}\\ {\comp_{k+1}\left(\alpha_{t_{n-1}(x)}\comp_{n-1}\cdots\comp_1\alpha_{t_0(x)}\comp_0u(s_{k+1}(y))\right)}. \end{multlined}$ \end{enumerate} \end{itemize} \end{paragr}  Leonard Guetta committed Sep 02, 2020 321 322 323 324 325 326 327 328 329 330 331 332 333 334  \begin{example}\label{example:natisoplax} Let $u,v : C \to D$ be two functors between (small) categories. Using the formulas of the previous paragraph, it is straightforward to check that the data of an oplax transformation from $u$ to $v$ consists exactly of the data of a natural transformation from $u$ to $v$. \end{example} \begin{paragr} Let $u : C \to D$ be an $\oo$-functor. There is an oplax transformation from $u$ to $u$, denoted by $1_u$, which is defined as $(1_u)_{x}:=1_{u(x)}$ for every $n$-cell $x$ of $C$. More abstractly, this oplax transformation corresponds to the $\oo$-functor $\sD_1 \otimes C \overset{p\otimes i}{\longrightarrow} \sD_0 \otimes D \simeq D,$ where $p$ is the only $\oo$-functor $\sD_1\to \sD_0$. \end{paragr}  Leonard Guetta committed Apr 21, 2020 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350  \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 Sep 02, 2020 351 352 353 354 355 356 357 358 359 360  for each cell $x$ of $B$) defines an oplax transformation from $g \circ u$ to $g \circ v$ (resp. $u \circ f$ to $v\circ f$) that we denote $g\star \alpha$ (resp. $\alpha \star f$). 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 $\sD_1 \otimes C \overset{\alpha}{\longrightarrow} D \overset{f}{\longrightarrow} E$ (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 361  \end{paragr}  Leonard Guetta committed Sep 02, 2020 362 363 364  \begin{remark} All the above descriptions of oplax transformations can be easily dualized for \emph{lax transformations} (that is to say $1$\nbd-cells of the $\oo$\nbd-category $\underline{\hom}_{\mathrm{lax}}(X,Y)$ for some $\oo$-categories $X$ and $Y$). Habit is the only reason why we put emphasis on oplax transformations rather than lax transformations. \end{remark}  Leonard Guetta committed Apr 21, 2020 365  \section{Homotopy equivalences and deformation retracts}  Leonard Guetta committed Sep 02, 2020 366 367 368 369 370 371 372  \begin{paragr}\label{paragr:hmtpyequiv} 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 $\oo$-functors $u, v : C \to D$ are \emph{oplax homotopic} if there are equivalent under this equivalence relation. \end{paragr} \begin{definition}\label{def:oplaxhmtpyequiv} 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 $\mathrm{id}_C$. \end{definition} In the following lemma, we denote by $\gamma : \oo\Cat \to \ho(\oo\Cat^{\Th})$ the localization functor with respect to the Thomason equivalences.  Leonard Guetta committed Apr 21, 2020 373  \begin{lemma}\label{lemma:oplaxloc}  Leonard Guetta committed Sep 02, 2020 374  Let $u, v : C \to D$ be two $\oo$-functors. If there exists an oplax transformation $\alpha : u \Rightarrow v$, then $\gamma(u)=\gamma(v)$.  Leonard Guetta committed Apr 21, 2020 375 376  \end{lemma} \begin{proof}  Leonard Guetta committed Sep 02, 2020 377  This follows immediately from \cite[Théorème B.11]{ara2020theoreme}.  Leonard Guetta committed Apr 21, 2020 378  \end{proof}  Leonard Guetta committed Sep 02, 2020 379  From this lemma and the fact that Thomason equivalences are saturated (Corollary \ref{cor:thomsaturated}) we deduce the following proposition.  Leonard Guetta committed Sep 17, 2020 380 381  \begin{proposition}\label{prop:oplaxhmtpyisthom} Every oplax homotopy equivalence is a Thomason equivalence.  Leonard Guetta committed Sep 02, 2020 382  \end{proposition}  Leonard Guetta committed May 04, 2020 383  \begin{paragr}\label{paragr:defrtract}  Leonard Guetta committed Sep 02, 2020 384  An $\oo$-functor $i : C \to D$ is an \emph{oplax deformation retract} if there exists an $\oo$-functor $r : C \to D$ such that:  Leonard Guetta committed Apr 21, 2020 385  \begin{enumerate}[label=(\alph*)]  Leonard Guetta committed Sep 02, 2020 386  \item $r\circ i=\mathrm{id}_C$,  Leonard Guetta committed Sep 03, 2020 387  \item there exists an oplax transformation $\alpha : \mathrm{id}_B \Rightarrow i\circ r$.  Leonard Guetta committed Apr 21, 2020 388  \end{enumerate}  Leonard Guetta committed Sep 02, 2020 389 390  Furthermore, $i$ is a \emph{strong oplax deformation retract} if $\alpha$ can be chosen such that: \begin{enumerate}[label=(\alph*),resume]  Leonard Guetta committed Apr 21, 2020 391 392  \item $\alpha \ast i = 1_i$. \end{enumerate}  Leonard Guetta committed Sep 02, 2020 393  An oplax deformation retract is a particular case of homotopy equivalence and thus of Thomason equivalence.  Leonard Guetta committed Apr 21, 2020 394  \end{paragr}  395  \begin{lemma}\label{lemma:pushoutstrngdefrtract}  Leonard Guetta committed Sep 02, 2020 396  The pushout of a strong oplax deformation retract is a strong oplax deformation retract.  Leonard Guetta committed Apr 21, 2020 397 398  \end{lemma} \begin{proof}  Leonard Guetta committed Sep 03, 2020 399 400  Let $i : A \to B$ be a strong oplax deformation retract and \label{cocartsquareretract}\tag{i}  Leonard Guetta committed Apr 21, 2020 401 402  \begin{tikzcd} A \ar[d,"i"] \ar[r,"u"] & A' \ar[d,"i'"] \\  Leonard Guetta committed Sep 03, 2020 403 404 405  B \ar[r,"v"] & B'\ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"] \end{tikzcd}  Leonard Guetta committed Sep 05, 2020 406  be a cocartesian square. We have to show that $i'$ is also a strong oplax deformation retract. By hypothesis there exists $r : B \to A$ 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 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463  \label{diagramtransf}\tag{ii} \begin{tikzcd} B\ar[rd,"\mathrm{id}_B"] \ar[d,"i_0^B"']& \\ \sD_1\otimes B \ar[r,"\alpha"] & B \\ B \ar[ru,"i\circ r"'] \ar[u,"i_1^B"] \end{tikzcd}, and \label{diagramstrong}\tag{iii} \begin{tikzcd} \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}, where $p$ is the unique morphism $\sD_1 \to \sD_0$, are commutative. From the commutativity of the following solid arrow diagram $\begin{tikzcd} A \ar[r,"u"] \ar[d,"i"] & A' \ar[d,"i'"] \ar[dd,bend left=75,"\mathrm{id}_{B'}"] \\ B \ar[d,"r"] \ar[r,"v"] & B' \ar[d,"r'",dashed ] \\ A \ar[r,"u"] & A' \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end] \end{tikzcd},$ we deduce the existence of $r' : B' \to A'$ that makes the whole diagram commutes. In particular, we have $r' \circ i' = \mathrm{id}_{B'}$. From the commutativity of (\ref{diagramstrong}), we easily deduce the commutativity of the following solid arrow diagram $\begin{tikzcd} \sD_1\otimes A \ar[r,"\sD_1\otimes u"] \ar[d,"\sD_1\otimes i"] & \sD_1 \otimes A' \ar[d,"\sD_1 \otimes i'"] \ar[dd,bend left=75,"p\otimes i'"] \\ \sD_1\otimes B \ar[d,"\alpha"] \ar[r,"\sD_1 \otimes v"] & \sD_1 \otimes B' \ar[d,"\alpha'",dashed ] \\ \sD_1 \otimes B \ar[r,"v"] & \sD_1 \otimes B' \end{tikzcd}.$ 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 $\alpha' \circ (\sD_1 \otimes i') = p \otimes i'$. Now, notice that for any $\oo$-category $C$, the maps $i^C_0 : C \to \sD_1 \otimes C \text{ and } i^C_1 : C \to \sD_1 \otimes C$ 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 $\begin{tikzcd} B \ar[r,"i_{\epsilon}^{B}"] \ar[d,"v"] & \sD_1 \otimes B \ar[d,"\sD_1 \otimes v"] \ar[r,"\alpha"] & B \ar[d,"v"] \\ B' \ar[r,"i_{\epsilon}^{B'}"] & \sD_1 \otimes B' \ar[r,"\alpha'"] & B'  Leonard Guetta committed Apr 21, 2020 464 465  \end{tikzcd}$  Leonard Guetta committed Sep 03, 2020 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488  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 489  \end{proof}  Leonard Guetta committed Sep 03, 2020 490 491  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 492  Every strong oplax deformation retract is a co-universal Thomason equivalence.  Leonard Guetta committed Sep 03, 2020 493 494 495 496 497 498 499 500  \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} \section{Equivalence of $\omega$-categories and the folk model structure}  Leonard Guetta committed Sep 24, 2020 501 \begin{paragr}\label{paragr:ooequivalence}  Leonard Guetta committed Sep 03, 2020 502 503 504  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 Sep 05, 2020 505  \item[-] there exists $r, s \in C_{n+1}$ such that $r : x \to y$, $s : y \to x$,  Leonard Guetta committed Sep 03, 2020 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522  $r\ast_{n}s \sim_{\omega} 1_y$ 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} Let $x$ and $y$ be two $0$-cells of an $n$-category $C$. \begin{itemize}[label=-] \item When $n=1$, $x \sim_{\omega} y$ means that $x$ and $y$ are isomorphic. \item When $n=2$, $x \sim_{\omega} y$ means that $x$ and $y$ are equivalent, i.e.\ there exists $f : x \to y$ and $g : y \to x$ such that $fg$ is isomorphic to $1_y$ and $gf$ is isomorphic to $1_x$. \end{itemize}  Leonard Guetta committed Sep 04, 2020 523 524 525 526 527 \end{example} For later reference, we put here the following trivial but important lemma, whose proof is ommited. \begin{lemma} Let $F : C \to D$ be an $\oo$\nbd-functor and $x$,$y$ be $n$-cells of $C$ for some $n \geq 0$. If $x \sim_{\oo} y$, then $F(x) \sim_{\oo} F(y)$. \end{lemma}  Leonard Guetta committed Sep 03, 2020 528  \begin{definition}\label{def:eqomegacat}  Leonard Guetta committed Sep 04, 2020 529  An $\omega$-functor $D : C \to D$ is an \emph{equivalence of $\oo$\nbd-categories} when:  Leonard Guetta committed Sep 03, 2020 530 531  \begin{itemize} \item[-] for every $y \in D_0$, there exists a $x \in C_0$ such that  Leonard Guetta committed Sep 04, 2020 532 533  $F(x)\sim_{\omega}y,$ \item[-] for every $x,y \in C_n$ that are \emph{parallel} and 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 534 535 536  $\alpha : x \to y$ and  Leonard Guetta committed Sep 04, 2020 537  $F(\alpha)\sim_{\omega}\beta.$  Leonard Guetta committed Sep 03, 2020 538 539 540  \end{itemize} \end{definition} \begin{example}\label{example:equivalencecategories}  Leonard Guetta committed Sep 04, 2020 541  If $C$ and $D$ are (small) categories seen as $\oo$-categories, then a functor $F : C \to D$ is an equivalence of $\oo$\nbd-categories if and only if it is fully faithful, hence, an equivalence of categories.  Leonard Guetta committed Sep 03, 2020 542  \end{example}  Leonard Guetta committed Sep 10, 2020 543  For the next theorem, recall that the canonical maps $i_n : \sS_{n-1} \to \sD_n$ for $n \geq 0$ have been defined in \ref{paragr:defglobe}.  Leonard Guetta committed Sep 03, 2020 544 545 546 547 548 549 550 551 552  \begin{theorem}\label{thm:folkms} There exists a cofibrantely generated model structure on $\omega\Cat$ such that the weak equivalences are the equivalences of $\omega$-categories, and the set $\{i_n : \sS_{n-1} \to \sD_n \vert n \in \mathbb{N}\}$ is a set of generating cofibrations. \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 $\omega\Cat$.  553 554 555 556 557 558 559 560 561  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. Similarly to the Thomason case (see \ref{paragr:notationthom}), we will usually make reference to the word folk'' in homotopic construction induced by the folk weak equivalences. For example, we write $\W^{\folk}$ the class of folk weak equivalences, $\Ho(\oo\Cat^{\folk})$ for the homotopy op-prederivator of $(\oo\Cat,\W_{\oo}^{\folk})$ and $\gamma^{\folk} : \oo\Cat \to \Ho(\oo\Cat^{\folk})$ for the localization morphism. It follows from the previous theorem and Theorem \ref{thm:cisinskiI} that the localizer $(\oo\Cat,\W^{\folk})$ is homotopy cocomplete.  Leonard Guetta committed Sep 03, 2020 562  \end{paragr}  Leonard Guetta committed Sep 10, 2020 563  \begin{proposition}\label{prop:freeiscofibrant}  Leonard Guetta committed Sep 03, 2020 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615  An $\omega$-category is cofibrant for the folk model structure if and only if it is free. \end{proposition} \begin{proof} 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 $\omega$-category $C$ is the colimit of the canonical diagram (Lemma \ref{lemma:filtration}) $\sk_{n}(C) \to \sk_{1}(C) \to \cdots \to \sk_n(C) \to \sk_{n+1}(C) \to \cdots$ For the converse, see \cite{metayer2008cofibrant}. \end{proof} \todo{Dire que le structure folk est monoidale ?} \iffalse \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 \section{Equivalences of $\omega$-categories vs Thomason equivalences} \begin{lemma}\label{lemma:nervehomotopical} The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends equivalences of $\omega$-categories to weak equivalences of simplicial sets. \end{lemma} \begin{proof} 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 folk trivial fibrations to weak equivalences of simplicial sets. In particular, it suffices to show the stronger condition that the nerve sends folk trivial fibrations to trivial fibrations of simplicial sets. By adjunction, this is equivalent to showing that the functor $c_{\omega} : \Psh{\Delta} \to \omega\Cat$ sends cofibrations of simplicial sets to folk cofibrations. Since $c_{\omega}$ is cocontinuous and the cofibrations of simplicial sets are generated by the inclusions $\partial \Delta_n \to \Delta_n$ 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 $(\Or_n)_{\leq n-1} \to \Or_n.$ Since $\Or_n$ is free, this last morphism is by definition a push-out of a coproduct of folk cofibrations, hence a folk cofibration. \end{proof} As an immediate consequence of the previous lemma, we have the following proposition. \begin{proposition}\label{prop:folkisthom} Every equivalence of $\oo$-categories is a Thomason equivalence. \end{proposition} \begin{remark} The converse of the above proposition is false. For example, the unique $\oo$\nbd-functor $\sD_1 \to \sD_0$ 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}). \end{remark} \begin{paragr}\label{paragr:compweakeq}  Leonard Guetta committed Sep 17, 2020 616  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 617 618  % \label{cantoTh} $ Leonard Guetta committed Sep 17, 2020 619  \mathcal{J} : \ho(\oo\Cat^{\folk}) \to \ho(\oo\Cat^{\Th}).  Leonard Guetta committed Sep 03, 2020 620 621 $ %  Leonard Guetta committed Sep 17, 2020 622 623 624 625 626 627  %% 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. This functor cannot be an equivalence since this would imply that every Thomason equivalence is an equivalence of $\oo$-categories.  Leonard Guetta committed Sep 03, 2020 628 629  \end{paragr} \section{Slices of $\oo$-category and a folk Theorem $A$}  Leonard Guetta committed Jun 03, 2020 630  \begin{paragr}\label{paragr:slices}  Leonard Guetta committed Apr 21, 2020 631 632 633 634 635 636 637 638  Let $A$ be an $\oo$-category and $a_0$ an object of $A$. We define the slice $\oo$-category $A/a_0$ as the following fibred product: $\begin{tikzcd} A/a_0 \ar[d] \ar[r] & \homlax(\sD_1,A) \ar[d,"\pi_1^A"] \\ \sD_0 \ar[r,"\langle a_0 \rangle"'] & A. \ar[from=1-1,to=2-2,phantom,very near start,"\lrcorner"] \end{tikzcd}$  Leonard Guetta committed Sep 03, 2020 639  We also define an $\oo$\nbd-functor $\pi : A/a_0 \to A$ as the following composition  Leonard Guetta committed Apr 21, 2020 640 641 642  $\pi : A/a_0 \to \homlax(\sD_1,A) \overset{\pi^A_0}{\longrightarrow} A.$  Leonard Guetta committed Sep 03, 2020 643  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 644  \begin{itemize}[label=-]  Leonard Guetta committed Sep 04, 2020 645  \item An $n$-cell of $A/a_0$ is a matrix \todo{le mot matrix'' est-il maladroit ?}  Leonard Guetta committed Apr 21, 2020 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663  $(x,a)=\begin{pmatrix} \begin{matrix} (x_0,a_1) & (x_1,a_2) & \cdots & (x_{n-1},a_n) \\[0.5em] (x_0',a_1') & (x_1',a_2') & \cdots & (x_{n-1}',a_n') \end{matrix} & (x_n,a_{n+1}) \end{pmatrix}$ where $x_0$ and $x_0'$ are $0$-cells of $A$, and: \begin{tabular}{ll} $x_i : x_{i-1} \longrightarrow x'_{i-1}$ &for every $1 \leq i \leq n$,\$0.75em] x_i': x_{i-1} \longrightarrow x'_{i-1} &for every 1 \leq i < n,\\[0.75em] a_i : a'_{i-1}\comp_{i-2} a'_{i-2} \comp_{i-3} \cdots \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_0 x'_{i-1} \longrightarrow a_{i-1}, &for every 1 \leq i \leq n\\ \end{tabular}  Leonard Guetta committed Sep 03, 2020 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684  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 Apr 21, 2020 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733  \item The source, target of the n-cell (a,x) are given by the matrices: \[ 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}$ $t(x,a)=\begin{pmatrix} \begin{matrix} (x_0,a_1) & (x_1,a_2) & \cdots & (x_{n-2},a_{n-1}) \\[0.5em] (x_0',a_1') & (x_1',a_2') & \cdots & (x_{n-2}',a_{n-1}') \end{matrix} & (x'_{n-1},a'_{n}). \end{pmatrix}$ % It is understood that when $n=1$, the source is simply $(x_0,a_1)$ and the target $(x_0,a_1')$ \item The unit of the $n$-cell $(a,x)$ is given by the matrix: $1_{(x,a)}=\begin{pmatrix} \begin{matrix} (x_0,a_1) & (x_1,a_2) & \cdots & (x_{n-1},a_n) & (x_n,a_{n+1}) \\[0.5em] (x_0',a_1') & (x_1',a_2') & \cdots & (x_{n-1}',a_n') & (x_n,a_{n+1}) \end{matrix} & (1_{x_n},1_{a_{n+1}}) \end{pmatrix}$ \item The composition of $n$-cells $(x,a)$ and $(y,b)$ such that $s_k(y,b)=t_k(a,x)$, is given by the matrix: $(y,b)\comp_k (x,a)=\begin{pmatrix} \begin{matrix} (x_0,a_1) & \cdots & (x_k,a_k) & (z_{k+1},c_{k+2}) & \cdots &(z_{n-1},c_n) \\[0.5em] (y_0',b_1') & \cdots & (y'_k,b'_k) &(z'_{k+1},c'_{k+2}) & \cdots & (z'_{n-1},c'_n) \\ \end{matrix} & (z_n,c_{n+1}) \end{pmatrix},$ where: \begin{tabular}{ll} $z_{i}=y_i\comp_k x_i$ & 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] c_i=a_i\comp_k b_i \comp_{k-1} a'_{k-1} \comp_{k-2} a'_{k-2} \comp_{k-3} \cdots \comp_{1} a'_1\comp_0 x_k&for every k+1 \leq i \leq n+1, \\[0.75em] c'_i=a'_i\comp_k b'_i \comp_{k-1} a'_{k-1} \comp_{k-2} a'_{k-2} \comp_{k-3} \cdots \comp_{1} a'_1\comp_0 x'_k&for every k+1 \leq i \leq n.\\ \end{tabular} \end{itemize}  Leonard Guetta committed Sep 03, 2020 734  We leave it to the reader to check that the formulas are well defined and that the axioms of \oo-category are satisfied. The canonical forgetful \oo\nbd-functor \pi : A/a_0 \to A is simply expressed as:  Leonard Guetta committed Apr 21, 2020 735 736 737 738  \begin{align*} A/a_0 &\to A \\ (x,a) &\mapsto x_n. \end{align*}  Leonard Guetta committed Apr 24, 2020 739  Notice that if A is an n-category then so is A/a_0. In this case, for an n-cell (x,a), a_{n+1} is an identity, hence  Leonard Guetta committed Apr 21, 2020 740 741 742 743  \[ 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 744 745 746  \begin{example}\label{example:slicecategories} 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 $a_0$. \end{example}  Leonard Guetta committed Apr 21, 2020 747  \begin{paragr}  Leonard Guetta committed Sep 03, 2020 748  Let $u : A \to B$ be a morphism of $\oo\Cat$ and $b_0$ an object of $B$. We define the $\oo$-category $A/b_0$ (also denoted $u\downarrow b_0$) as the following fibred product:  Leonard Guetta committed Apr 21, 2020 749 750  $\begin{tikzcd}  Leonard Guetta committed Apr 24, 2020 751 752  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 753  \ar[from=1-1,to=2-2,phantom,description,very near start,"\lrcorner"]  Leonard Guetta committed Apr 24, 2020 754  \end{tikzcd}  Leonard Guetta committed Apr 21, 2020 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 $ More explicitly, an $n$-cell $(x,b)$ of $A/b_0$ is a matrix $(x,b)=\begin{pmatrix} \begin{matrix} (x_0,b_1) & (x_1,b_2) & \cdots & (x_{n-1},b_n) \\[0.5em] (x_0',b_1') & (x_1',b_2') & \cdots & (x_{n-1}',b_n') \end{matrix} & (x_n,b_{n+1}) \end{pmatrix}$ where the $x_i$ and $x'_i$ are $i$-cells of $A$, the $b_i$ and $b'_i$ are $i$-cells of $B$, such that $\begin{pmatrix} \begin{matrix} (u(x_0),b_1) & (u(x_1),b_2) & \cdots & (u(x_{n-1}),b_n) \\[0.5em] (u(x_0'),b_1') & (u(x'_1),b_2') & \cdots & (u(x'_{n-1}),b_n') \end{matrix} & (u(x_n),b_{n+1}) \end{pmatrix}$ is an $n$-cell of $B/b_0$.  Leonard Guetta committed Sep 04, 2020 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799  The canonical $\oo$\nbd-functor $A/b_0 \to A$ is simply expressed as \begin{align*} A/b_0 &\to A\\ (x,b) &\mapsto x_n, \end{align*} and the $\oo$\nbd-functor $u/b_0$ as \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 $\oo\Cat$ $\begin{tikzcd}[column sep=small] A \ar[rr,"u"] \ar[dr,"v"'] & &B \ar[dl,"w"] \\ &C& \end{tikzcd},$ then for any object $c_0$ of $C$, we have a functor $u/c_0 : A/c_0 \to B/c_0$ defined as \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 800 801  \end{paragr}  Leonard Guetta committed Apr 24, 2020 802 803 804  \begin{proposition}(Folk Theorem $A$) Let $\begin{tikzcd}[column sep=small]  Leonard Guetta committed Sep 04, 2020 805  A \ar[rr,"u"] \ar[dr,"v"'] & &B \ar[dl,"w"] \\  Leonard Guetta committed Apr 24, 2020 806 807 808  &C& \end{tikzcd}$  Leonard Guetta committed Sep 05, 2020 809  be a commutative triangle in $\oo\Cat$. If for every object $c_0$ of $C$ the induced morphism  Leonard Guetta committed Apr 24, 2020 810 811 812 813 814 815  $u/c_0 : A/c_0 \to B/c_0$ is an equivalence of $\oo$-categories, then so is $u$. \end{proposition} \begin{proof}  Leonard Guetta committed Sep 04, 2020 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869  Before anything else, let us note the following trivial but important fact: for any $\oo$\nbd-functor $F : X \to Y$ and any $n$-cells $x$ and $y$ of $X$, if $x \sim_{\oo} y$, then $F(x) \sim_{\oo} F(y)$. \begin{enumerate}[label=(\roman*)] \item Let $b_0$ be $0$\nbd-cell of $B$ and set $c_0:=v(b_0)$. By definition, the pair $(b_0,1_{c_0})$ is a $0$-cell of $B/c_0$. By hypothesis, we know that there exists a $0$\nbd-cell $(a_0,c_1)$ of $A/c_0$ such that $(u(a_0),c_1)\sim_{\oo} (b_0,1_{c_0})$. Hence, by applying the canonical functor $B/c_0 \to B$, we obtain that $u(a_0) \sim_{\oo} b_0$. \item Let $f$ and $f'$ be parallel $n$\nbd-cells of $A$ and $\beta : u(f) \to u(f')$ an $(n+1)$\nbd-cell of $B$. We need to show that there exists an $(n+1)$\nbd-cell $\alpha : f \to f'$ of $A$ such that $u(\alpha) \sim_{\oo} \beta$. Let us use the notations: \begin{itemize}[label=-] \item $a_i := \src_i(f)=\src_i(f')$ for \$0 \leq i