hmtpy.tex 51.6 KB
Newer Older
Leonard Guetta's avatar
Leonard Guetta committed
1
\chapter{Homotopy theory of \texorpdfstring{$\oo$}{ω}-categories}
Leonard Guetta's avatar
Leonard Guetta committed
2
3
\section{Nerve}
\begin{paragr}\label{paragr:simpset}
Leonard Guetta's avatar
Leonard Guetta committed
4
  We denote by $\Delta$ the category whose objects are the finite non-empty totally ordered sets $[n]=\{0<\cdots<n\}$ and whose morphisms are the non-decreasing maps. For $n > 0$ and $0\leq i\leq n$, we denote by
Leonard Guetta's avatar
Leonard Guetta committed
5
6
7
  \[
  \delta^i : [n-1] \to [n] 
  \]
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
12
  the only surjective non-decreasing map such that the pre-image of $i \in [n]$ contains exactly two elements.
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
14
15
16
  \[
  \begin{aligned}
    X_n &:= X([n]) \\
Leonard Guetta's avatar
Leonard Guetta committed
17
    \partial_i &:= X(\delta^i): X_n \to X_{n\shortminus 1}\\
Leonard Guetta's avatar
Leonard Guetta committed
18
    s_i &:= X(\sigma^i): X_{n} \to X_{n+1}.
Leonard Guetta's avatar
Leonard Guetta committed
19
20
  \end{aligned}
  \]
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
22
\end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
23
\begin{paragr}\label{paragr:orientals}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
25
26

  The two main points to retain are:
Leonard Guetta's avatar
Leonard Guetta committed
27
  \begin{description}
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's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
34
    \end{description}
Leonard Guetta's avatar
Leonard Guetta committed
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=-]
37
  \item There are no generating $k$-cells for $k>n$. Hence, $\Or_n$ is an $n$\nbd{}category.
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
45
  \begin{description}
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's avatar
Leonard Guetta committed
47
  \end{description}
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's avatar
Leonard Guetta committed
49
  Here are some pictures in low dimension:
Leonard Guetta's avatar
Leonard Guetta committed
50
51
52
53
54
55
56
57
58
59
60
61
  \[
  \Or_0 = \langle 0 \rangle,
  \]
  \[
  \Or_1=\begin{tikzcd}
    \langle 0 \rangle \ar[r,"\langle 01 \rangle"] &\langle 1 \rangle,
    \end{tikzcd}
  \]
  \[
  \Or_2=
  \begin{tikzcd}
    &\langle 1 \rangle \ar[rd,"\langle 12 \rangle"]& \\
Leonard Guetta's avatar
Leonard Guetta committed
62
    \langle 0 \rangle \ar[ru,"\langle 01 \rangle"]\ar[rr,"\langle 02 \rangle"',""{name=A,above}]&&\langle 2 \rangle,
Leonard Guetta's avatar
Leonard Guetta committed
63
    \ar[Rightarrow,from=A,to=1-2,"\langle 012 \rangle"]
Leonard Guetta's avatar
Leonard Guetta committed
64
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
65
66
  \]
  \[
Leonard Guetta's avatar
Leonard Guetta committed
67
  \Or_3=
Leonard Guetta's avatar
Leonard Guetta committed
68
69
70
71
  \begin{tikzcd}
    & \langle 1 \rangle \ar[rd,"\langle 12 \rangle"]& \\
    \langle 0 \rangle \ar[ru,"\langle 01 \rangle"] \ar[rd,"\langle 03 \rangle"',""{name=B,above}] \ar[rr,"\langle 02 \rangle" description,""{name=A,above}]& & \langle 2 \rangle  \ar[ld,"\langle 23 \rangle"]\\
    & \langle 3 \rangle &
Leonard Guetta's avatar
Leonard Guetta committed
72
73
    \ar[from=A,to=1-2,Rightarrow,"\langle 012 \rangle", shorten <= 0.25em, shorten >= 0.25em]
    \ar[from=B,to=2-3,Rightarrow,"\langle 023 \rangle"', near start, shorten <= 1.1em, shorten >= 1.5em]
Leonard Guetta's avatar
Leonard Guetta committed
74
75
76
  \end{tikzcd}
  \overset{\langle 0123 \rangle}{\Rrightarrow}
    \begin{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
77
    & \langle 1 \rangle \ar[rd,"\langle 12 \rangle"] \ar[dd,"\langle 13 \rangle"' description,""{name=B,right}] & \\
Leonard Guetta's avatar
Leonard Guetta committed
78
    \langle 0 \rangle \ar[ru,"\langle 01 \rangle"] \ar[rd,"\langle 03 \rangle"',""{name=A,above}] & & \langle 2 \rangle.  \ar[ld,"\langle 23 \rangle"]\\
Leonard Guetta's avatar
Leonard Guetta committed
79
    & \langle 3 \rangle &
Leonard Guetta's avatar
Leonard Guetta committed
80
81
    \ar[from=A,to=1-2,Rightarrow,"\langle 013 \rangle", near start, shorten <= 1em, shorten >= 1.5em]
    \ar[from=B,to=2-3,Rightarrow,"\langle 123 \rangle", shorten <= 0.75em, shorten >=0.75em]
Leonard Guetta's avatar
Leonard Guetta committed
82
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
83
84
85
    \]
\end{paragr}
\begin{paragr}\label{paragr:nerve}
Leonard Guetta's avatar
Leonard Guetta committed
86
    For every $\omega$-category $C$, the \emph{nerve of $C$} is the simplicial set $N_{\omega}(C)$ defined as
Leonard Guetta's avatar
Leonard Guetta committed
87
88
    \[
    \begin{aligned}
Leonard Guetta's avatar
Leonard Guetta committed
89
     N_{\omega}(C) : \Delta^{\op} &\to \Set\\
Leonard Guetta's avatar
Leonard Guetta committed
90
      [n] &\mapsto \Hom_{\omega\Cat}(\Or_n,C).
Leonard Guetta's avatar
Leonard Guetta committed
91
92
93
94
95
96
      \end{aligned}
    \]
   By post-composition, this yields a functor
  \[
  \begin{aligned}
  N_{\omega} : \omega\Cat &\to \Psh{\Delta} \\
Leonard Guetta's avatar
Leonard Guetta committed
97
  C &\mapsto N_{\omega}(C),
Leonard Guetta's avatar
Leonard Guetta committed
98
99
  \end{aligned}
  \]
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
101
  \[
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
102
  N_n := N_{\oo}{\big |}_{n\Cat} : n\Cat \to \Psh{\Delta}.
Leonard Guetta's avatar
Leonard Guetta committed
103
  \]
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
dodo    
Leonard Guetta committed
105
106
\end{paragr}
\iffalse
Leonard Guetta's avatar
Leonard Guetta committed
107
  \begin{lemma}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
109
  \end{lemma}
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
110
  \fi
Leonard Guetta's avatar
Leonard Guetta committed
111
  \begin{paragr}
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
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 $C$
    \[
Leonard Guetta's avatar
Leonard Guetta committed
114
    X_0 \overset{X_{0,1}}{\longrightarrow} X_1 \overset{X_{1,2}}{\longrightarrow} \cdots \longrightarrow X_{m-1}\overset{X_{m-1,m}}{\longrightarrow} X_m.
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
115
        \]
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
dodo    
Leonard Guetta committed
117
        
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
119
  \begin{itemize}[label=-]
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
120
  \item for every $0\leq i \leq m$, an object $X_i$ of $C$,
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
123
124
    \[
      \begin{tikzcd}
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
125
    & X_j \ar[rd,"{X_{j,k}}"]& \\
Leonard Guetta's avatar
Leonard Guetta committed
126
    X_i \ar[ru,"X_{i,j}"]\ar[rr,"X_{i,k}"',""{name=A,above}]&&X_k,
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
127
    \ar[Rightarrow,from=A,to=1-2,"X_{i,j,k}"]
Leonard Guetta's avatar
Leonard Guetta committed
128
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
129
130
  \]
  \end{itemize}
Leonard Guetta's avatar
Leonard Guetta committed
131
  subject to the following axioms:
Leonard Guetta's avatar
Leonard Guetta committed
132
  \begin{itemize}[label=-]
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
145
  \end{itemize}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
dodo    
Leonard Guetta committed
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's avatar
Leonard Guetta committed
156
  \[
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
157
  \partial_l(X)_i = \begin{cases} X_i &\text{ if } 0 \leq i<l \leq m-1 \\ X_{i+1} &\text{ if } 0 \leq l\leq i \leq m-1\end{cases}    
Leonard Guetta's avatar
Leonard Guetta committed
158
159
    \]
    \[
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
160
    \partial_l(X)_{i,j}=\begin{cases} X_{i,j} &\text{ if } 0 \leq i < j < l \leq m-1  \\ X_{i,j+1} & \text{ if } 0 \leq i < l \leq j \leq m-1 \\ X_{i+1,j+1} & \text{ if } 0 \leq l \leq i < j \leq m-1 \end{cases}
Leonard Guetta's avatar
Leonard Guetta committed
161
162
    \]
    \[
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
163
    \partial_l(X)_{i,j,k}=\begin{cases} X_{i,j,k} &\text{ if } 0 \leq i < j < k < l \leq m-1 \\ X_{i,j,k+1} &\text{ if } 0 \leq i < j < l \leq k \leq m-1 \\ X_{i,j+1,k+1} &\text{ if } 0 \leq i < l \leq j < k \leq m-1 \\ X_{i+1,j+1,k+1} &\text{ if } 0 \leq l \leq i < j < k \leq m-1.\end{cases}
Leonard Guetta's avatar
Leonard Guetta committed
164
    \]
Leonard Guetta's avatar
Leonard Guetta committed
165
166
    \remtt{ Ai-je besoin de mettre les formules ci-dessus ? Rajouter les formules des dégénerescences ?}
    \fi
Leonard Guetta's avatar
Leonard Guetta committed
167
\end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
168
  \section{Thomason equivalences}
Leonard Guetta's avatar
Leonard Guetta committed
169
170
171
172
  \begin{paragr}
From now on, we will consider that the category $\Psh{\Delta}$ is equipped with the model structure defined by Quillen in \cite{quillen1967homotopical}. A \emph{weak equivalence of simplicial sets} is a weak equivalence for this model structure. The cofibrations for this model structure are the monomorphisms.
  \end{paragr}
  \begin{definition}
Leonard Guetta's avatar
Leonard Guetta committed
173
   Let $n \in \nbar$. A morphism $f : X \to Y$ of $n\Cat$ is a \emph{Thomason equivalence} when ${N_n(f) : N_n(X) \to N_n(Y)}$ is a weak equivalence of simplicial sets. We denote by $\W_n^{\mathrm{Th}}$ the class of Thomason equivalences.
Leonard Guetta's avatar
Leonard Guetta committed
174
  \end{definition}
175
  \begin{paragr}\label{paragr:notationthom}
Leonard Guetta's avatar
Leonard Guetta committed
176
    We usually make reference to the name ``Thomason'' in the notations of homotopic constructions induced by Thomason equivalences. For example, we write $\Ho(n\Cat^{\Th})$ for the homotopy op\nbd{}prederivator of $(n\Cat,\W_n^{\Th})$ and
Leonard Guetta's avatar
resto    
Leonard Guetta committed
177
178
179
    \[
    \gamma^{\Th} : n\Cat \to \Ho(n\Cat^{\Th}) 
    \]
Leonard Guetta's avatar
Leonard Guetta committed
180
    for the localization morphism. The reason is to avoid confusion with other weak equivalences on $n\Cat$ that we will introduce later.
Leonard Guetta's avatar
resto    
Leonard Guetta committed
181
  \end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
182
  \begin{paragr}
183
184
185
186
187
188
    By definition, the nerve functor induces a morphism of localizers

    \[
      {N_n : (n\Cat,\W_n^{\Th}) \to (\Psh{\Delta},\W_{\Delta})}
    \]
    and hence a morphism of op\nbd{}prederivators
Leonard Guetta's avatar
Leonard Guetta committed
189
190
191
192
193
    \[
    \overline{N_n} : \Ho(n\Cat^{\Th}) \to \Ho(\Psh{\Delta}).
    \]
  \end{paragr}
  \begin{theorem}[Gagna]\label{thm:gagna}
Leonard Guetta's avatar
Leonard Guetta committed
194
    For every $1 \leq n \leq \oo$, the morphism \[{\overline{N}_n : \Ho(n\Cat^\Th) \to \Ho(\Psh{\Delta})}\] is an equivalence of op\nbd{}prederivators.
Leonard Guetta's avatar
Leonard Guetta committed
195
196
  \end{theorem}
  \begin{proof}
197
    Recall from \ref{paragr:nerve} that $c_n : \Psh{\Delta} \to n\Cat$ denotes
Leonard Guetta's avatar
idem    
Leonard Guetta committed
198
    the left adjoint of the nerve functor $N_n$. In \cite{gagna2018strict}, Gagna proves that there exists a functor $Q : \Psh{\Delta} \to \Psh{\Delta}$, as well as a zigzag of morphisms of functors
Leonard Guetta's avatar
Leonard Guetta committed
199
200
201
    \[
    N_{n}c_{n}Q \overset{\alpha}{\longleftarrow} Q \overset{\gamma}{\longrightarrow} \mathrm{id}_{\Psh{\Delta}}
    \]
Leonard Guetta's avatar
Leonard Guetta committed
202
    and a morphism of functors
Leonard Guetta's avatar
Leonard Guetta committed
203
204
205
206
207
208
209
    \[
    c_{n}Q N_{n} \overset{\beta}{\longrightarrow} \mathrm{id}_{\Psh{\Delta}},
    \]
    such that $c_{n}Q$ preserves weak equivalences and $\alpha$, $\beta$ and $\gamma$ are weak equivalences argument by argument. This easily implies that
       \[
    \overline{c_{n}Q} : \Ho(\Psh{\Delta})\to \Ho(n\Cat^{\Th})
    \]
Leonard Guetta's avatar
Leonard Guetta committed
210
    is a quasi-inverse (\ref{paragr:prederequivadjun}) of
Leonard Guetta's avatar
Leonard Guetta committed
211
    \[
Leonard Guetta's avatar
Leonard Guetta committed
212
    \overline{N_n} : \Ho(n\Cat^{\Th}) \to \Ho(\Psh{\Delta}).\qedhere
Leonard Guetta's avatar
Leonard Guetta committed
213
214
    \]
  \end{proof}
Leonard Guetta's avatar
Leonard Guetta committed
215
  From Lemma \ref{lemma:dereq}, we obtain the following corollary.
Leonard Guetta's avatar
Leonard Guetta committed
216
  \begin{corollary}\label{cor:thomhmtpycocomplete}
Leonard Guetta's avatar
Leonard Guetta committed
217
    For every $1 \leq n \leq \oo$, the localizer $(n\Cat^{\Th},\W_n^{\Th})$ is homotopy cocomplete (Definition \ref{def:cocompletelocalizer}).
Leonard Guetta's avatar
Leonard Guetta committed
218
  \end{corollary}
Leonard Guetta's avatar
Leonard Guetta committed
219
  We will speak of ``Thomason homotopy colimits'' and ``Thomason homotopy
220
  cocartesian squares'' for homotopy colimits and homotopy cocartesian squares in
221
  the localizer $(n\Cat^{\Th},\W_n^{\Th})$. (See also \ref{paragr:thomhmtpycol} below.)
Leonard Guetta's avatar
Leonard Guetta committed
222
223
224

  Another consequence of Gagna's theorem is the following
  corollary.
Leonard Guetta's avatar
Leonard Guetta committed
225
226
227
228
  \begin{corollary}\label{cor:thomsaturated}
    For every $1 \leq n \leq \oo$, the class $\W_n^{\Th}$ is saturated (\ref{paragr:loc}).
  \end{corollary}
  \begin{proof}
Leonard Guetta's avatar
Leonard Guetta committed
229
    This follows immediately from the fact that $\overline{N_n} : \ho(n\Cat^{\Th}) \to \ho(\Psh{\Delta})$ is an equivalence of categories and the fact that weak equivalences of simplicial sets are saturated (because they are the weak equivalences of a model structure). 
Leonard Guetta's avatar
Leonard Guetta committed
230
    \end{proof}
Leonard Guetta's avatar
Leonard Guetta committed
231
  \begin{remark}
232
    Corollaries \ref{cor:thomhmtpycocomplete} and \ref{cor:thomsaturated} would also follow from the existence of a model structure on $n\Cat$ with $\W^{\Th}_n$ as the weak equivalences. For $n=1$, this was established by Thomason \cite{thomason1980cat}, and for $n=2$, by Ara and Maltsiniotis \cite{ara2014vers}. For $n>3$, the existence of such a model structure is conjectured but not yet established.
Leonard Guetta's avatar
Leonard Guetta committed
233
  \end{remark}
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 \[\Ho(n\Cat^\Th) \to \Ho(m\Cat^{\Th}).\]
235
  \begin{proposition}\label{prop:nthomeqder}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
240
  is an equivalence of op\nbd{}prederivators.
Leonard Guetta's avatar
Leonard Guetta committed
241
  \end{proposition}
Leonard Guetta's avatar
Leonard Guetta committed
242
243
244
245
  \begin{proof}
    This follows from Theorem \ref{thm:gagna} and the commutativity of the triangle
    \[
    \begin{tikzcd}[column sep=tiny]
Leonard Guetta's avatar
idem    
Leonard Guetta committed
246
      \Ho(n\Cat^{\Th}) \ar[rr] \ar[rd,"\overline{N_n}"'] & & \Ho(m\Cat^{\Th}) \ar[dl,"\overline{N_m}"] \\
Leonard Guetta's avatar
Leonard Guetta committed
247
248
      &\Ho(\Psh{\Delta})&.
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
249
250
    \]
  \end{proof}
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's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
268
     \begin{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
270
       \[
Leonard Guetta's avatar
Leonard Guetta committed
271
       \underline{\hom}_{\mathrm{oplax}}(-,-),\, \underline{\hom}_{\mathrm{lax}}(-,-) : \oo\Cat^{\op}\times\oo\Cat \to \oo\Cat
Leonard Guetta's avatar
Leonard Guetta committed
272
       \]
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
279
280
281
       \[
       \Hom_{\oo\Cat}(Y,Z)\simeq \Hom_{\oo\Cat}(\sD_0,\underline{\hom}_{\mathrm{oplax}}(Y,Z)).
       \]
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
283
284
     \end{paragr}
     \begin{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
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 \[
Leonard Guetta's avatar
Leonard Guetta committed
286
287
288
289
290
291
292
293
294
       \alpha : u \Rightarrow v
       \]
       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's avatar
Leonard Guetta committed
295
         \item As an $\oo$\nbd{}functor $\alpha : \sD_1\otimes X \to Y$ such that the following diagram
Leonard Guetta's avatar
Leonard Guetta committed
296
297
298
       \[
       \begin{tikzcd}
         X\ar[rd,"u"] \ar[d,"i_0^X"']& \\
Leonard Guetta's avatar
Leonard Guetta committed
299
300
301
         \sD_1\otimes X \ar[r,"\alpha"] & Y, \\
         X \ar[ru,"v"'] \ar[u,"i_1^X"]&
       \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
302
       \]
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
305
306
307
308
       \[
       \begin{tikzcd}
         & Y \\
         X \ar[ru,"u"] \ar[r,"\alpha"] \ar[rd,"v"']& \homlax(\sD_1,Y) \ar[u,"\pi_0^Y"'] \ar[d,"\pi_1^Y"] \\
309
         & Y,
Leonard Guetta's avatar
Leonard Guetta committed
310
311
       \end{tikzcd}
       \]
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
313
       \end{itemize}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
315
316
317
     \end{paragr}
     

Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
319

Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
326
      \item for every $n$\nbd{}cell of $x$ of $X$ with $n>0$, an $(n+1)$-cell of $Y$
Leonard Guetta's avatar
Leonard Guetta committed
327
    \[
Leonard Guetta's avatar
Leonard Guetta committed
328
    \alpha_x : \alpha_{\trgt_{n-1}(x)}\comp_{n-1}\cdots\comp_1\alpha_{\trgt_0(x)}\comp_0u(x) \to v(x)\comp_0\alpha_{\src_0(x)}\comp_1\cdots\comp_{n-1}\alpha_{\src_{n-1}(x)}
Leonard Guetta's avatar
Leonard Guetta committed
329
330
331
    \]
    subject to the following axioms:
    \begin{enumerate}
Leonard Guetta's avatar
Leonard Guetta committed
332
    \item for every $n$\nbd{}cell $x$ of $X$,
Leonard Guetta's avatar
Leonard Guetta committed
333
      \[\alpha_{1_x}=1_{\alpha_x},\]
Leonard Guetta's avatar
Leonard Guetta committed
334
    \item for all $0\leq k < n$, for all $n$\nbd{}cells $x$ and $y$ of $X$ that are $k$-composable,
Leonard Guetta's avatar
Leonard Guetta committed
335
336
      \[
      \begin{multlined}
337
338
      \alpha_{x \comp_k y}={\left(v(\trgt_{k+1}(x))\comp_0\alpha_{\src_0(y)}\comp_1\cdots\comp_{k-1}\alpha_{\src_{k-1}(y)}\comp_k\alpha_y\right)}\\
            {\comp_{k+1}\left(\alpha_x \comp_k\alpha_{\trgt_{k-1}(x)}\comp_{k-1}\cdots\comp_1\alpha_{\trgt_0(x)}\comp_0u(\src_{k+1}(y))\right)}.
Leonard Guetta's avatar
Leonard Guetta committed
339
340
341
342
343
344
            \end{multlined}
      \]
      \end{enumerate}
   
       \end{itemize}
     \end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
345
     \begin{example}\label{example:natisoplax}
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's avatar
Leonard Guetta committed
351
352
     \end{example}
     \begin{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
354
355
356
       \[
       (1_u)_{x}:=1_{u(x)}
       \]
Leonard Guetta's avatar
Leonard Guetta committed
357
       for every cell $x$ of $C$. More abstractly, this oplax transformation corresponds to the $\oo$\nbd{}functor
Leonard Guetta's avatar
Leonard Guetta committed
358
       \[
359
       \sD_1 \otimes C \overset{p\otimes u}{\longrightarrow} \sD_0 \otimes D \simeq D,
Leonard Guetta's avatar
Leonard Guetta committed
360
       \]
Leonard Guetta's avatar
Leonard Guetta committed
361
       where $p$ is the only $\oo$\nbd{}functor $\sD_1\to \sD_0$.
Leonard Guetta's avatar
Leonard Guetta committed
362
       \end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
380

Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
389
  \end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
390
      \begin{remark}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
393
      \end{remark}
Leonard Guetta's avatar
Leonard Guetta committed
394
      \section{Homotopy equivalences and deformation retracts}
Leonard Guetta's avatar
Leonard Guetta committed
395
      \begin{paragr}\label{paragr:hmtpyequiv}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
397
398
      \end{paragr}
      \begin{definition}\label{def:oplaxhmtpyequiv}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
400
      \end{definition}
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's avatar
Leonard Guetta committed
402
      \begin{lemma}\label{lemma:oplaxloc}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
404
405
      \end{lemma}
      \begin{proof}
Leonard Guetta's avatar
Leonard Guetta committed
406
        This follows immediately from \cite[Théorème B.11]{ara2020theoreme}.
Leonard Guetta's avatar
Leonard Guetta committed
407
      \end{proof}
Leonard Guetta's avatar
Leonard Guetta committed
408
      From this lemma and the fact that the Thomason equivalences are saturated (Corollary \ref{cor:thomsaturated}), we deduce the following proposition.
Leonard Guetta's avatar
Leonard Guetta committed
409
410
      \begin{proposition}\label{prop:oplaxhmtpyisthom}
        Every oplax homotopy equivalence is a Thomason equivalence.
Leonard Guetta's avatar
Leonard Guetta committed
411
      \end{proposition}
Leonard Guetta's avatar
Leonard Guetta committed
412
      \begin{paragr}\label{paragr:defrtract}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
414
        \begin{enumerate}[label=(\alph*)]
Leonard Guetta's avatar
Leonard Guetta committed
415
        \item $r\circ i=\mathrm{id}_C$,
Leonard Guetta's avatar
Leonard Guetta committed
416
        \item there exists an oplax transformation $\alpha : \mathrm{id}_D \Rightarrow i\circ r$.
Leonard Guetta's avatar
Leonard Guetta committed
417
        \end{enumerate}
Leonard Guetta's avatar
Leonard Guetta committed
418
419
        Furthermore, $i$ is a \emph{strong oplax deformation retract} if $\alpha$ can be chosen such that:
        \begin{enumerate}[label=(\alph*),resume]
420
          \item $\alpha \star i = 1_i$.
Leonard Guetta's avatar
Leonard Guetta committed
421
        \end{enumerate}
Leonard Guetta's avatar
Leonard Guetta committed
422
        An oplax deformation retract is a particular case of homotopy equivalence and thus of Thomason equivalence. 
Leonard Guetta's avatar
Leonard Guetta committed
423
      \end{paragr}
424
      \begin{lemma}\label{lemma:pushoutstrngdefrtract}
Leonard Guetta's avatar
Leonard Guetta committed
425
       The pushout of a strong oplax deformation retract is a strong oplax deformation retract.
Leonard Guetta's avatar
Leonard Guetta committed
426
427
      \end{lemma}
      \begin{proof}
428
429
        Let $i : A \to B$ be a strong oplax deformation retract and
        \begin{equation}\label{cocartsquareretract}\tag{i}
Leonard Guetta's avatar
Leonard Guetta committed
430
431
        \begin{tikzcd}
          A \ar[d,"i"] \ar[r,"u"] & A' \ar[d,"i'"] \\
432
433
434
          B \ar[r,"v"] & B'\ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
        \end{tikzcd}
        \end{equation}
Leonard Guetta's avatar
Leonard Guetta committed
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
436
437
438
        \begin{equation}\label{diagramtransf}\tag{ii}
       \begin{tikzcd}
         B\ar[rd,"\mathrm{id}_B"] \ar[d,"i_0^B"']& \\
Leonard Guetta's avatar
Leonard Guetta committed
439
440
441
         \sD_1\otimes B \ar[r,"\alpha"] & B, \\
         B \ar[ru,"i\circ r"'] \ar[u,"i_1^B"]&
       \end{tikzcd}
442
443
444
445
        \end{equation}
        and
        \begin{equation}\label{diagramstrong}\tag{iii}
          \begin{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
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}
448
449
450
451
452
453
        \end{equation}
        where $p$ is the unique morphism $\sD_1 \to \sD_0$, are commutative.

        From the commutativity of the following solid arrow diagram
        \[
        \begin{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
454
          A \ar[r,"u"] \ar[d,"i"] & A' \ar[d,"i'"] \ar[dd,bend left=75,"\mathrm{id}_{A'}"] \\
455
          B \ar[d,"r"] \ar[r,"v"] & B' \ar[d,"r'",dashed ] \\
Leonard Guetta's avatar
Leonard Guetta committed
456
          A \ar[r,"u"] & A',
457
          \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
Leonard Guetta's avatar
Leonard Guetta committed
458
        \end{tikzcd}
459
        \]
460
        we deduce the existence of $r' : B' \to A'$ that makes the whole diagram commute. In particular, we have $r' \circ i' = \mathrm{id}_{A'}$.
461
462
463
464
465
466

        From the commutativity of (\ref{diagramstrong}), we easily deduce the commutativity of the following solid arrow diagram
        \[
        \begin{tikzcd}
          \sD_1\otimes A \ar[r,"\sD_1\otimes u"] \ar[d,"\sD_1\otimes i"] & \sD_1 \otimes A' \ar[d,"\sD_1 \otimes i'"] \ar[dd,bend left=75,"p\otimes i'"] \\
         \sD_1\otimes B \ar[d,"\alpha"] \ar[r,"\sD_1 \otimes v"] & \sD_1 \otimes B' \ar[d,"\alpha'",dashed ] \\
467
         B \ar[r,"v"] &  B'.
Leonard Guetta's avatar
Leonard Guetta committed
468
        \end{tikzcd}
469
        \]
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 \[\alpha' \circ (\sD_1 \otimes i') = p \otimes i'.\]
471
        
Leonard Guetta's avatar
Leonard Guetta committed
472
        Now, notice that for every $\oo$\nbd{}category $C$, the maps
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
        \[
        \begin{tikzcd}
        B \ar[r,"i_{\epsilon}^{B}"] \ar[d,"v"] & \sD_1 \otimes B \ar[d,"\sD_1 \otimes v"] \ar[r,"\alpha"] & B \ar[d,"v"] \\
        B' \ar[r,"i_{\epsilon}^{B'}"] & \sD_1 \otimes B' \ar[r,"\alpha'"] & B'
Leonard Guetta's avatar
Leonard Guetta committed
493
494
        \end{tikzcd}
        \]
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's avatar
Leonard Guetta committed
518
      \end{proof}
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's avatar
Leonard Guetta committed
521
        Every strong oplax deformation retract is a co-universal Thomason equivalence.
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}
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's avatar
idem    
Leonard Guetta committed
533
\sectionmark{The folk model structure}
Leonard Guetta's avatar
Leonard Guetta committed
534
\begin{paragr}\label{paragr:ooequivalence}
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's avatar
Leonard Guetta committed
538
    \item[-] there exist $r, s \in C_{n+1}$ such that $r : x \to y$, $s : y \to x$,
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's avatar
Leonard Guetta committed
551
  Let $x$ and $y$ be two $0$-cells of an $n$\nbd{}category $C$.
552
553
  \begin{itemize}[label=-]
  \item When $n=1$, $x \sim_{\omega} y$ means that $x$ and $y$ are isomorphic.
Leonard Guetta's avatar
Leonard Guetta committed
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 $1_x$.
555
    \end{itemize}
Leonard Guetta's avatar
Leonard Guetta committed
556
\end{example}
Leonard Guetta's avatar
Leonard Guetta committed
557
For later reference, we put here the following trivial but important lemma, whose proof is omitted.
Leonard Guetta's avatar
Leonard Guetta committed
558
\begin{lemma}\label{lemma:ooequivalenceisfunctorial}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
560
  \end{lemma}
561
  \begin{definition}\label{def:eqomegacat}
Leonard Guetta's avatar
Leonard Guetta committed
562
    An $\omega$-functor $F : C \to D$ is an \emph{equivalence of $\oo$\nbd{}categories} when:
563
    \begin{itemize}
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
564
      \item[-] for every $y \in D_0$, there exists $x \in C_0$ such that
Leonard Guetta's avatar
Leonard Guetta committed
565
      \[F(x)\sim_{\omega}y,\]
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
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
567
568
569
      \[\alpha : x \to y
      \]
      and
Leonard Guetta's avatar
Leonard Guetta committed
570
      \[F(\alpha)\sim_{\omega}\beta.\]
571
572
573
      \end{itemize}
  \end{definition}
  \begin{example}\label{example:equivalencecategories}
Leonard Guetta's avatar
Leonard Guetta committed
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. 
575
  \end{example}
Leonard Guetta's avatar
Leonard Guetta committed
576

577
  \begin{theorem}\label{thm:folkms}
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}).
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's avatar
Leonard Guetta committed
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

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's avatar
Leonard Guetta committed
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's avatar
dodo    
Leonard Guetta committed
602
603
 \end{paragr}
 \begin{paragr}\label{paragr:folktrivialfib}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
dodo    
Leonard Guetta committed
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's avatar
Leonard Guetta committed
622
     This characterization of folk trivial fibrations is to be compared with Definition \ref{def:eqomegacat} of equivalences of $\oo$\nbd{}categories. 
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
623
624
     \end{itemize}
 \end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
625
  \begin{proposition}\label{prop:freeiscofibrant}
626
627
628
    An $\omega$-category is cofibrant for the folk model structure if and only if it is free.
  \end{proposition}
  \begin{proof}
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})
630
    \[
Leonard Guetta's avatar
Leonard Guetta committed
631
     \sk_{0}(C) \to \sk_{1}(C) \to \cdots \to \sk_n(C) \to \sk_{n+1}(C) \to \cdots
632
633
634
    \]
    For the converse, see \cite{metayer2008cofibrant}.
  \end{proof}
Leonard Guetta's avatar
Leonard Guetta committed
635
   \iffalse
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
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}
651
    \begin{lemma}\label{lemma:nervehomotopical}
652
The nerve functor $N_{\omega} : \omega\Cat \to \Psh{\Delta}$ sends the equivalences of $\oo$\nbd{}categories to weak equivalences of simplicial sets.    
653
654
  \end{lemma}
  \begin{proof}
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.
659

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
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
    \[
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
668
    \sk_{n-1}(\Or_n) \to \Or_n.
669
    \]
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
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.
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's avatar
Leonard Guetta committed
674
    Every equivalence of $\oo$\nbd{}categories is a Thomason equivalence. 
675
676
  \end{proposition}
  \begin{remark}
Leonard Guetta's avatar
Leonard Guetta committed
677
    The converse of the above proposition is false. For example, the unique $\oo$\nbd{}functor
678
679
680
    \[
    \sD_1 \to \sD_0
    \]
Leonard Guetta's avatar
Leonard Guetta committed
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}).
682
683
    \end{remark}
  \begin{paragr}\label{paragr:compweakeq}
Leonard Guetta's avatar
Leonard Guetta committed
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
685
686
   % \begin{equation}\label{cantoTh}
    \[
Leonard Guetta's avatar
Leonard Guetta committed
687
    \mathcal{J} : \ho(\oo\Cat^{\folk}) \to \ho(\oo\Cat^{\Th}).
688
689
    \]
    %\end{equation}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
695
    This functor cannot be an equivalence since this would imply that every Thomason equivalence is an equivalence of $\oo$\nbd{}categories. 
696
  \end{paragr}
697
   \section{{Slice \texorpdfstring{$\oo$}{ω}-categories and folk Theorem~A}}
Leonard Guetta's avatar
Leonard Guetta committed
698
  \begin{paragr}\label{paragr:slices}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
707
    We also define an $\oo$\nbd{}functor $\pi : A/a_0 \to A$ as the following composition
Leonard Guetta's avatar
Leonard Guetta committed
708
709
710
    \[
    \pi : A/a_0 \to \homlax(\sD_1,A) \overset{\pi^A_0}{\longrightarrow} A.
    \]
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
712
    \begin{itemize}[label=-]
Leonard Guetta's avatar
Leonard Guetta committed
713
    \item An $n$\nbd{}cell of $A/a_0$ is a table 
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
730
731
      \end{tabular}
      
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}
753
    \item The source and target of the $n$\nbd{}cell $(a,x)$ are given by the matrices:
Leonard Guetta's avatar
Leonard Guetta committed
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
      \[
      s(x,a)=\begin{pmatrix}
      \begin{matrix}
      (x_0,a_1) & (x_1,a_2) & \cdots & (x_{n-2},a_{n-1}) \\[0.5em]
        (x_0',a_1') & (x_1',a_2') & \cdots & (x_{n-2}',a_{n-1}')
      \end{matrix}
      & (x_{n-1},a_{n})
      \end{pmatrix}
      \]
      \[
      t(x,a)=\begin{pmatrix}
      \begin{matrix}
      (x_0,a_1) & (x_1,a_2) & \cdots & (x_{n-2},a_{n-1}) \\[0.5em]
        (x_0',a_1') & (x_1',a_2') & \cdots & (x_{n-2}',a_{n-1}')
      \end{matrix}
769
770
      & (x'_{n-1},a'_{n})
      \end{pmatrix}.
Leonard Guetta's avatar
Leonard Guetta committed
771
772
      \]
 %     It is understood that when $n=1$, the source is simply $(x_0,a_1)$ and the target $(x_0,a_1')$
Leonard Guetta's avatar
Leonard Guetta committed
773
    \item The unit of the $n$\nbd{}cell $(a,x)$ is given by the table:
Leonard Guetta's avatar
Leonard Guetta committed
774
775
776
777
778
779
780
      \[
      1_{(x,a)}=\begin{pmatrix}
      \begin{matrix}
      (x_0,a_1) & (x_1,a_2) & \cdots & (x_{n-1},a_n) & (x_n,a_{n+1}) \\[0.5em]
        (x_0',a_1') & (x_1',a_2') & \cdots & (x_{n-1}',a_n') & (x_n,a_{n+1})
      \end{matrix}
      & (1_{x_n},1_{a_{n+1}})
781
      \end{pmatrix}.
Leonard Guetta's avatar
Leonard Guetta committed
782
      \]
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
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]
798
799
        $c_i=a_i\comp_{k+1} b_i \comp_{k} a'_{k} \comp_{k-1} a'_{k-1} \comp_{k-2} \cdots \comp_{1} a'_1\comp_0 x_k$,&for every $k+2 \leq i \leq n+1$, \\[0.75em]
        $c'_i=a'_i\comp_{k+1} b'_i \comp_{k} a'_{k} \comp_{k-1} a'_{k-1} \comp_{k-2} \cdots \comp_{1} a'_1\comp_0 x'_k$,&for every $k+2 \leq i \leq n$.\\
Leonard Guetta's avatar
Leonard Guetta committed
800
801
      \end{tabular}
    \end{itemize}
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's avatar
Leonard Guetta committed
803
804
805
806
        \begin{align*}
      A/a_0 &\to A \\
      (x,a) &\mapsto x_n.
        \end{align*}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
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}
812
  \begin{example}\label{example:slicecategories}
Leonard Guetta's avatar
Leonard Guetta committed
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 $a_0$.
814
  \end{example}
Leonard Guetta's avatar
Leonard Guetta committed
815
  \begin{paragr}\label{paragr:comma}
Leonard Guetta's avatar
Leonard Guetta committed
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's avatar
Leonard Guetta committed
817
818
    \[
    \begin{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
819
820
    A/b_0 \ar[d,"u/b_0"'] \ar[r] & A \ar[d,"u"] \\
    B/b_0 \ar[r,"\pi"'] & B.
Leonard Guetta's avatar
Leonard Guetta committed
821
    \ar[from=1-1,to=2-2,phantom,description,very near start,"\lrcorner"]
Leonard Guetta's avatar
Leonard Guetta committed
822
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
823
    \]
Leonard Guetta's avatar
Leonard Guetta committed
824
    More explicitly, an $n$\nbd{}cell $(x,b)$ of $A/b_0$ is a table
Leonard Guetta's avatar
Leonard Guetta committed
825
826
827
828
829
830
831
832
833
    \[
    (x,b)=\begin{pmatrix}
    \begin{matrix}
      (x_0,b_1) & (x_1,b_2) & \cdots & (x_{n-1},b_n) \\[0.5em]
      (x_0',b_1') & (x_1',b_2') & \cdots & (x_{n-1}',b_n')
    \end{matrix}
    & (x_n,b_{n+1})
    \end{pmatrix}
    \]
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's avatar
Leonard Guetta committed
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'