contractible.tex 33.3 KB
Newer Older
1
2
3
\chapter{Homology of contractible \texorpdfstring{$\oo$}{ω}-categories and its
  consequences}
\chaptermark{Contractible $\omega$-categories and consequences}
Leonard Guetta's avatar
Leonard Guetta committed
4
\section{Contractible \texorpdfstring{$\oo$}{ω}-categories}
5
Recall that for every $\oo$\nbd{}category $C$, we write $p_C : C \to \sD_0$ for the canonical morphism to the terminal object of $\sD_0$.
Leonard Guetta's avatar
Leonard Guetta committed
6
\begin{definition}\label{def:contractible}
Leonard Guetta's avatar
Leonard Guetta committed
7
  An $\oo$\nbd{}category $C$ is \emph{oplax contractible} when the canonical morphism $p_C : C \to \sD_0$ is an oplax homotopy equivalence (Definition \ref{def:oplaxhmtpyequiv}).
Leonard Guetta's avatar
Leonard Guetta committed
8
9
10
11
12
13
14
15
16

  %% there exists an object $x_0$ of $X$ and an oplax transformation
  %% \[
  %% \begin{tikzcd}
  %%   X \ar[r,"p_X"] \ar[rd,"1_X"',""{name=A,above}]& \sD_0 \ar[d,"\langle x_0 \rangle"]\\
  %%   &X.
  %%   \ar[from=A,to=1-2,"\alpha",Rightarrow]
  %% \end{tikzcd}
  %% \]
17
\end{definition}
Leonard Guetta's avatar
Leonard Guetta committed
18
19
20


%% \begin{paragr}
21
%%   In other words, an $\oo$\nbd{}category $X$ is contractible when there exists an object $x_0$ of $X$ such that $\langle x_0 \rangle : \sD_0 \to X$ is a deformation retract (Paragraph \ref{paragr:defrtract}). It follows from Lemma \ref{lemma:oplaxloc} that $p_X : X \to \sD_0$ is a Thomason equivalence. In particular, we have the following lemma.
Leonard Guetta's avatar
Leonard Guetta committed
22
23
24
%% \end{paragr}

%% \begin{lemma}\label{lemma:hmlgycontractible} 
Leonard Guetta's avatar
Leonard Guetta committed
25
%%   Let $X$ be a contractible $\oo$\nbd{}category. The morphism of $\ho(\Ch)$
Leonard Guetta's avatar
Leonard Guetta committed
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
%%   \[
%%   \sH(X) \to \sH(\sD_0)
%%   \]
%%   induced by the canonical morphism $p_X : X \to \sD_0$ is an isomorphism.
%% \end{lemma}
%% \begin{paragr}
%%   In addition to the previous result, it is immediate to check that $\sH(\sD_0)$ is nothing but $\mathbb{Z}$ seen as an object of $\ho(\Ch)$ concentrated in degree $0$. 
%% \end{paragr}
%% \begin{corollary}
%%   Let $u : X \to Y$ be a morphism of $\oo\Cat$. If $u$ is a homotopy equivalence (Paragraph \ref{paragr:hmtpyequiv}), then
%%   \[
%%   \sH^{\pol}(\gamma_{\folk}(u))
%%   \]
%%   is an isomorphism.
%% \end{corollary}
%% \todo{Expliquer p-e pourquoi le corollaire précédent est important.}
%% \begin{corollary}\label{cor:hmlgypolcontractible}
%%   Let $X$ be an $\oo$-category. If $X$ is contractible, then the morphism
%%   \[
%%   \sH^{\pol}(\gamma_{\folk}(p_X)) : \sH^{\pol}(X) \to \sH^{\pol}(\sD_0)
%%   \]
%%   is an isomorphism of $\ho(\Ch)$.
%% \end{corollary}
%% We can now prove the main result of this section.

\begin{proposition}\label{prop:contractibleisgood}
Leonard Guetta's avatar
Leonard Guetta committed
52
  Every oplax contractible $\oo$\nbd{}category $C$ is \good{} and we have
Leonard Guetta's avatar
Leonard Guetta committed
53
  \[
Leonard Guetta's avatar
Leonard Guetta committed
54
  \sH^{\pol}(C)\simeq \sH^{\sing}(C)\simeq \mathbb{Z}
Leonard Guetta's avatar
Leonard Guetta committed
55
  \]
Leonard Guetta's avatar
Leonard Guetta committed
56
  where $\mathbb{Z}$ is seen as an object of $\ho(\Ch)$ concentrated in degree $0$.
Leonard Guetta's avatar
Leonard Guetta committed
57
58
\end{proposition}
\begin{proof}
Leonard Guetta's avatar
Leonard Guetta committed
59
Consider the commutative square
Leonard Guetta's avatar
Leonard Guetta committed
60
61
  \[
  \begin{tikzcd}
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
62
63
    \sH^{\pol}(C) \ar[d,"\sH^{\pol}(p_C)"] \ar[r,"\pi_C"] & \sH^{\sing}(C) \ar[d,"\sH^{\sing}(p_C)"] \\
    \sH^{\pol}(\sD_0) \ar[r,"\pi_{\sS_0}"] & \sH^{\sing}(\sD_0).
Leonard Guetta's avatar
Leonard Guetta committed
64
65
  \end{tikzcd}
  \]
66
67
68
69
70
71
72
  It follows respectively from Proposition \ref{prop:oplaxhmtpyisthom} and
  Proposition \ref{prop:oplaxhmtpypolhmlgy} that the right and left vertical
  morphisms of the above square are isomorphisms. Then, an immediate computation
  left to the reader shows that $\sD_0$ is \good{} and that
  $\sH^{\pol}(\sD_0)\simeq \sH^{\sing}(\sD_0)\simeq \mathbb{Z}$. By a 2-out-of-3
  property, we deduce that $\pi_C : \sH^{\sing}(C)\to \sH^{\pol}(C)$ is an
  isomorphism and $\sH^{\pol}(C)\simeq \sH^{\sing}(C)\simeq \mathbb{Z}$.
Leonard Guetta's avatar
Leonard Guetta committed
73
\end{proof}
Leonard Guetta's avatar
Leonard Guetta committed
74
\begin{remark}
Leonard Guetta's avatar
Leonard Guetta committed
75
  Definition \ref{def:contractible} admits an obvious ``lax'' variation and Proposition \ref{prop:contractibleisgood} is also true for lax contractible $\oo$\nbd{}categories.
Leonard Guetta's avatar
Leonard Guetta committed
76
  \end{remark}
77
We end this section with an important result on slice $\oo$\nbd{}categories (Paragraph \ref{paragr:slices}).
Leonard Guetta's avatar
Leonard Guetta committed
78
\begin{proposition}\label{prop:slicecontractible}
Leonard Guetta's avatar
Leonard Guetta committed
79
  Let $A$ be an $\oo$\nbd{}category and $a_0$ an object of $A$. The $\oo$\nbd{}category $A/a_0$ is oplax contractible. 
Leonard Guetta's avatar
Leonard Guetta committed
80
81
   \end{proposition}
\begin{proof}
Leonard Guetta's avatar
Leonard Guetta committed
82
  This follows from the dual of \cite[Proposition 5.22]{ara2020theoreme}.
Leonard Guetta's avatar
Leonard Guetta committed
83
84
  \end{proof}
\section{Homology of globes and spheres}
85
\begin{lemma}\label{lemma:globescontractible}
86
  For every $n \in \mathbb{N}$, the $\oo$\nbd{}category $\sD_n$ is oplax contractible.
Leonard Guetta's avatar
Leonard Guetta committed
87
88
\end{lemma}
\begin{proof}
Leonard Guetta's avatar
Leonard Guetta committed
89
  Recall that we write $e_n$ for the unique non-trivial $n$\nbd{}cell of $\sD_n$ and that by definition $\sD_n$ has exactly two non-trivial $k$\nbd{}cells for every $k$ such that $0\leq k<n$. These two $k$\nbd{}cells are parallel and are given by $\src_k(e_n)$ and $\trgt_k(e_n)$.
90

Leonard Guetta's avatar
OUF    
Leonard Guetta committed
91
  Let $r : \sD_0 \to \sD_n$ be the $\oo$\nbd{}functor that points to $\trgt_0(e_n)$ (which means that $r=\langle \trgt_0(e_n) \rangle$ with the notations of \ref{paragr:defglobe}). For every $k$\nbd{}cell $x$ of $\sD_n$, we have
92
93
94
  \[
  r(p(x))=\1^k_{\trgt_0(e_n)},
  \]
95
  where we write $p$ for the unique $\oo$\nbd{}functor $\sD_n \to \sD_0$.
96
97
98
99
100

  Now for $0 \leq k <n$, we define $\alpha_{\src_k(e_n)}$ and $\alpha_{\trgt_k(e_n)}$ as
  \[
  \alpha_{\src_k(e_n)}=\begin{cases}\trgt_{k+1}(e_n), \text{ if } k<n-1 \\ e_n, \text{ if } k=n-1\end{cases} \text{ and }  \alpha_{\trgt_k(e_n)}=\begin{cases}1_{\trgt_k(e_n)}, \text{ if } k<n-1 \\ e_n, \text{ if } k=n-1\end{cases}.
  \]
101
  It is straightforward to check that this data define an oplax transformation $\alpha : \mathrm{id}_{\sD_n} \Rightarrow r\circ p$ (see \ref{paragr:formulasoplax} and Example \ref{example:natisoplax}), which proves the result. 
102
103
104
105
106
107
108
  %% \[
  %% \alpha_{\src_k(e_n)}=\trgt_{k+1}(e_n) \text{ and } \alpha_{\trgt_k(e_n)}=1_{\trgt_k(e_n)},
  %% \]
  %% and also
  %% \[
  %% \alpha_{\src_{n-1}(e_n)}=e_n=\alpha_{\src_{n-1}(e_n)}.
  %% \]
Leonard Guetta's avatar
Leonard Guetta committed
109
\end{proof}
110
111
112
113
114
115
116
117
In particular, for every $n \in \mathbb{N}$, $\sD_n$ is \good{}. Recall from \ref{paragr:inclusionsphereglobe} that for every $n \geq 0$, we have a cocartesian square
\[
\begin{tikzcd}
     \sS_{n-1} \ar[r,"i_n"] \ar[d,"i_n"] & \sD_n \ar[d,"j_n^+"]\\
    \sD_n \ar[r,"j_n^-"'] & \sS_{n}.
    \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
  \end{tikzcd}
\]
Leonard Guetta's avatar
Leonard Guetta committed
118
\begin{lemma}\label{lemma:squarenerve}
119
  For every $n \geq 0$, the commutative square of simplicial sets
Leonard Guetta's avatar
Leonard Guetta committed
120
121
  \[
  \begin{tikzcd}
122
123
    N_{\oo}(\sS_{n-1}) \ar[r,"N_{\oo}(i_n)"] \ar[d,"N_{\oo}(i_n)"] & N_{\oo}(\sD_{n}) \ar[d,"N_{\oo}(j_n^+)"] \\
    N_{\oo}(\sD_{n}) \ar[r,"N_{\oo}(j_n^-)"] & N_{\oo}(\sS_{n})
Leonard Guetta's avatar
Leonard Guetta committed
124
125
126
127
128
  \end{tikzcd}
  \]
  is cocartesian.
\end{lemma}
\begin{proof}
129
  Since colimits in presheaf categories are computed pointwise, what we need
Leonard Guetta's avatar
Leonard Guetta committed
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
  to show is that for every $k\geq 0$, the following commutative square is
  cocartesian
  \begin{equation}\label{squarenervesphere}
    \begin{tikzcd}[column sep=huge,row sep=huge]
    \Hom_{\oo\Cat}(\Or_k,\sS_{n-1}) \ar[r,"{\Hom_{\oo\Cat}(\Or_k,i_n)}"] \ar[d,"{\Hom_{\oo\Cat}(\Or_k,i_n)}"'] & \Hom_{\oo\Cat}(\Or_k,\sD_{n}) \ar[d,"{\Hom_{\oo\Cat}(\Or_k,j_n^+)}"] \\
    \Hom_{\oo\Cat}(\Or_k,\sD_{n}) \ar[r,"{\Hom_{\oo\Cat}(\Or_k,j_n^-)}"'] & \Hom_{\oo\Cat}(\Or_k,\sS_{n}).
    \end{tikzcd}
  \end{equation}
  Notice first that the square
  \[
    \begin{tikzcd}
      \sS_{n-1} \ar[r,"i_n"] \ar[d,"i_n"] & \sD_n \ar[d,"j_n^+"]\\
      \sD_n \ar[r,"j_n^-"'] & \sS_{n}.
    \end{tikzcd}
  \]
Leonard Guetta's avatar
Leonard Guetta committed
145
  is cartesian and all four morphisms are monomorphisms. Since the
Leonard Guetta's avatar
idem    
Leonard Guetta committed
146
  functor \[\Hom_{\oo\Cat}(\Or_k,-):\oo\Cat \to \Set \] preserves limits, the square
147
148
149
150
151
152
153
  \eqref{squarenervesphere} is a cartesian square of $\Set$ all of
  whose four morphisms are monomorphisms.  Hence, in order to prove
  that square \eqref{squarenervesphere} is cocartesian, we only need
  to show that for every $k \geq 0$ and every $\oo$\nbd{}functor
  $\varphi : \Or_k \to \sS_{n}$, there exists an $\oo$\nbd{}functor
  $\varphi' : \Or_k \to \sD_n$ such that either $j_n^+ \circ \varphi '
  = \varphi$ or $j_n^- \circ \varphi' = \varphi$.
Leonard Guetta's avatar
Leonard Guetta committed
154
  
Leonard Guetta's avatar
Leonard Guetta committed
155
156
157
158
159
160
161
162
163
164
165
166
167
  %% Notice now that the morphisms $j_n^+$ and $j_n^-$ trivially satisfy the following
  %% properties:
  %% \begin{enumerate}[label=(\roman*)]
  %% \item Every $k$\nbd{}cell of $\sS_n$ with $0 \leq k <n$ is the image by
  %%   $j^+_n$ of a (unique) $k$\nbd{}cell of $\sD_n$ \emph{and} the image by
  %%   $j^-_n$ of a (unique) $k$\nbd{}cell of $\sD_n$
  %% \item Every non-trivial $n$\nbd{}cell of $\sS_n$ (there are only two of them) is the image either by 
  %%   $j^+_n$ or by $j^-_n$ of a non-trivial $n$\nbd{}cell of $\sD_n$.
  %% \end{enumerate}
  For convenience, let us write $h_n^+$ (resp.\ $h_n^-$) for the only generating $n$\nbd{}cell of $\sS_n$ contained in the image of $j^+_n$ (resp.\ $j_n^-$). The cells $h_n^+$ and $h_n^-$ are the only non-trivial $n$\nbd{}cells of $\sS_n$. We also write $\alpha_k$ for the principal cell of $\Or_k$ (see \ref{paragr:orientals}). This is the only non-trivial $k$\nbd{}cell of $\Or_k$.

  Now, let $\varphi : \Or_k \to \sS_n$ be an $\oo$\nbd{}functor. There are several cases to distinguish.
  \begin{description}
168
169
  \item[Case $k<n$:] Since every generating cell of $\gamma$ of $\Or_k$ is of
    dimension not greater than $k$, the cell $\varphi(\gamma)$ is of dimension strictly lower than $n$. Since all cells of dimension strictly lower than $n$ are both in the image of $j^+_n$ and in the image of $j^-_n$, $\varphi$ obviously factors through $j^+_n$ (and $j^-_n$).
Leonard Guetta's avatar
Leonard Guetta committed
170
  \item[Case $k=n$:] The image of $\alpha_n$ is either a non-trivial $n$\nbd{}cell of $\sS_n$ or a unit on a strictly lower dimensional cell. In the second situation, everything works like the case $k<n$. Now suppose for example that $\varphi(\alpha_n)$ is $h^+_n$, which is in the image of $j^+_n$. Since all of the other generating cells of $\Or_n$ are of dimension strictly lower than $n$, their images by $\varphi$ are also of dimension strictly lower than $n$ and hence, are all contained in the image of $j^+_n$. Altogether this proves that $\varphi$ factors through $j^+_n$. The case where $\varphi(\alpha_n)=h^-_n$ is symmetric.
Leonard Guetta's avatar
Leonard Guetta committed
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
  \item[Case $k>n$:] Since $\sS_n$ is an $n$\nbd{}category, the image of
    $\alpha_k$ is necessarily of the form $\varphi(\alpha_k)=\1^k_{x}$ with $x$
    a cell of $\sS_n$ of dimension non-greater than $n$. If $x$ is a unit on a
    cell whose dimension is strictly lower than $n$, then everything works like
    in the case $k<n$. If not, this means that $x$ is a non-trivial
    $n$\nbd{}cell of $\sS_n$. Suppose for example that $x=h^+_n$. Now let
    $\gamma$ be a generator of $\Or_k$ of dimension $k-1$. We have
    $\varphi(\gamma)=\1^{k-1}_y$ with $y$ which is either a unit on a cell of
    dimension strictly lower than $n$, or a non-degenerate $n$\nbd{}cell of
    $\sS_n$ (if $k-1=n$, we use the convention that $\1^{k-1}_y=y$). In the
    first situation, $y$ is in the image of $j^+_n$ as in the case $k<n$, and
    thus, so is $\1^{k-1}_y$. In the second situation, this means \emph{a
      priori} that either $y=h_n^+$ or $y=h_n^-$. But we know that $\gamma$ is
    part of a composition that is equal to either the source or the target of
    $\alpha_k$ (see \ref{paragr:orientals}) and thus, $y$ is part of a composition that is equal to either the source or the target of $x=h^+_n$. Since no composition involving $h^-_n$ can be equal to $h^+_n$ (one could invoke the function introduced in \ref{prop:countingfunction}), this implies that $y=h_n^+$ and hence, $f(\gamma)$ is in the image of $j^+_n$. This goes for all generating cells of dimension $k-1$ of $\Or_k$ and we can recursively apply the same reasoning for generating cells of dimension $k-2$, then $k-3$ and so forth. Altogether, this proves that $\varphi$ factorizes through $j^+_n$. The case where $x=h^-_n$ and $\varphi$ factorizes through $j^-_n$ is symmetric.
Leonard Guetta's avatar
Leonard Guetta committed
186
    \end{description}
Leonard Guetta's avatar
Leonard Guetta committed
187
188
\end{proof}
From these two lemmas, follows the important proposition below.
Leonard Guetta's avatar
Leonard Guetta committed
189
\begin{proposition}\label{prop:spheresaregood}
Leonard Guetta's avatar
Leonard Guetta committed
190
  For every $n \geq -1$, the $\oo$\nbd{}category $\sS_n$ is \good{}.
Leonard Guetta's avatar
Leonard Guetta committed
191
192
\end{proposition}
\begin{proof}
193
194
195
196
  %% We proceed by induction on $n$. When $n=-1$, it is trivial to check that the
  %% empty $\oo$\nbd{}category is \good{}.
  Recall that the cofibrations of simplicial sets are exactly the monomorphisms, and in particular all simplicial sets are cofibrant. Since $i_n : \sS_{n-1} \to \sD_n$ is a monomorphism for every $n \geq 0$ and since $N_{\oo}$ preserves monomorphisms (as a right adjoint), it follows from Lemma \ref{lemma:squarenerve} and Lemma \ref{lemma:hmtpycocartesianreedy} that the square
\[
Leonard Guetta's avatar
Leonard Guetta committed
197
  \begin{tikzcd}
198
199
    N_{\oo}(\sS_{n-1}) \ar[r,"N_{\oo}(i_n)"] \ar[d,"N_{\oo}(i_n)"] & N_{\oo}(\sD_{n}) \ar[d,"N_{\oo}(j_n^+)"] \\
    N_{\oo}(\sD_{n}) \ar[r,"N_{\oo}(j_n^-)"] & N_{\oo}(\sS_{n})
Leonard Guetta's avatar
Leonard Guetta committed
200
  \end{tikzcd}
201
202
  \]
is a homotopy cocartesian square of simplicial sets. Since $N_{\oo}$ induces an
203
  equivalence of op-prederivators $\Ho(\oo\Cat^{\Th}) \to \Ho(\Psh{\Delta})$
204
205
206
207
208
209
210
211
212
  (Theorem \ref{thm:gagna}), it follows that the square of $\oo\Cat$
  \[
    \begin{tikzcd}
    \sS_{n-1} \ar[r,"i_n"] \ar[d,"i_n"] & \sD_{n} \ar[d,"j_n^+"] \\
    \sD_{n} \ar[r,"j_n^-"] & \sS_{n}
    \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
    \end{tikzcd}
    \]
  is Thomason homotopy cocartesian for every $n\geq 0$. Finally, since $i_n : \sS_{n-1} \to \sD_{n}$ is
213
  a folk cofibration and $\sS_{n-1}$ and $\sD_{n}$ are folk cofibrant for every $n\geq0$, we deduce the desired result from Corollary \ref{cor:usefulcriterion} and an immediate induction. The base case being simply that $\sS_{-1}=\emptyset$ is obviously \good{}.
Leonard Guetta's avatar
Leonard Guetta committed
214
215
\end{proof}
\begin{paragr}
216
  The previous proposition implies what we claimed in Paragraph \ref{paragr:prelimcriteriongoodcat}, which is that the morphism of op-prederivators
Leonard Guetta's avatar
Leonard Guetta committed
217
218
219
  \[
  \J : \Ho(\oo\Cat^{\folk}) \to \Ho(\oo\Cat^{\Th})
  \]
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
220
  induced by the identity functor of $\oo\Cat$ is \emph{not} homotopy cocontinuous.
221
222
223
  Indeed, recall from Paragraph \ref{paragr:bubble} that the category
  $B^2\mathbb{N}$ is \emph{not} \good{}; but on the other hand we have a
  cocartesian square
Leonard Guetta's avatar
Leonard Guetta committed
224
225
  \[
  \begin{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
226
    \sS_1 \ar[r] \ar[d,"i_2"] & \sD_0 \ar[d] \\
227
228
    \sD_2 \ar[r] & B^2\mathbb{N},
    \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
Leonard Guetta's avatar
Leonard Guetta committed
229
230
  \end{tikzcd}
  \]
231
232
233
234
235
236
237
238
  where the map $\sD_2 \to B^2\mathbb{N}$ points the unique generating
  $2$-cell of $B^2\mathbb{N}$ and $\sD_0 \to B^2\mathbb{N}$ points to
  the only object of $B^2\mathbb{N}$. Since $\sS_1$, $\sD_0$ and $\sD_2$ are
  free and $i_2$ is a folk cofibration,
  the square is also folk homotopy cocartesian. If $\J$ was homotopy
  cocontinuous, then this square would also be Thomason homotopy
  cocartesian. Since we know that $\sS_1$, $\sD_0$ and $\sD_2$ are
  \good{}, this would imply that $B^2\mathbb{N}$ is \good{}.
Leonard Guetta's avatar
Leonard Guetta committed
239

240
  From Proposition \ref{prop:spheresaregood}, we also deduce the proposition
241
  below which gives a criterion to detect \good{} $\oo$\nbd{}categories when we
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
242
  already know that they are free. 
Leonard Guetta's avatar
Leonard Guetta committed
243
244
\end{paragr}
\begin{proposition}
Leonard Guetta's avatar
Leonard Guetta committed
245
246
  Let $C$ be a free $\oo$\nbd{}category and for every $k \in \mathbb{N}$ let
  $\Sigma_k$ be its $k$\nbd{}basis. If for every $k \in \mathbb{N}$, the following
247
  cocartesian square (see \ref{def:nbasis})
Leonard Guetta's avatar
Leonard Guetta committed
248
249
  \[
  \begin{tikzcd}[column sep=large]
250
251
252
253
   \displaystyle \coprod_{x \in \Sigma_k} \sS_{k-1} \ar[d,"\coprod i_k"'] \ar[r,"{\langle s(x),t(x) \rangle}"]& \sk_{k-1}(C) \ar[d,hook] \\
    \displaystyle \coprod_{x \in \Sigma_k} \sD_{k} \ar[r,"\langle x \rangle"] &
    \sk_k(C)
    \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
Leonard Guetta's avatar
Leonard Guetta committed
254
255
  \end{tikzcd}
  \]
Leonard Guetta's avatar
Leonard Guetta committed
256
  is Thomason homotopy cocartesian, then $C$ is \good{}.
Leonard Guetta's avatar
Leonard Guetta committed
257
258
\end{proposition}
\begin{proof}
259
  Since the morphisms $i_k$ are folk cofibrations and the
260
261
262
  $\oo$\nbd{}categories $\sS_{k-1}$ and $\sD_{k}$ are folk cofibrant
  and \good{}, it follows from Corollary \ref{cor:usefulcriterion} and
  an immediate induction that all $\sk_k(C)$ are \good{}. The result
Leonard Guetta's avatar
Leonard Guetta committed
263
  follows then from Lemma \ref{lemma:filtration}, Corollary \ref{cor:cofprojms} and Proposition
264
  \ref{prop:sequentialhmtpycolimit}.
Leonard Guetta's avatar
Leonard Guetta committed
265
266
\end{proof}
\section{The miraculous case of 1-categories}
267
268
269
270
Recall that the terms \emph{1-category} and \emph{(small) category} are
synonymous. While we have used the latter one more often so far, in this section
we will mostly use the former one. As usual, the canonical functor $\iota_1 :
\Cat \to \oo\Cat$ is treated as an inclusion functor and hence we always consider
271
$1$\nbd{}categories as particular cases of $\oo$\nbd{}categories.
Leonard Guetta's avatar
Leonard Guetta committed
272

273
274
The goal of what follows is to show that every $1$\nbd{}category is \good{}. In order to do that, we will prove that every 1-category
is a canonical colimit of contractible $1$\nbd{}categories and that this colimit is
275
homotopic both
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
276
with respect to the folk weak equivalences and with respect to the Thomason equivalences.
277
We call the reader's attention to an important subtlety here: even though the
Leonard Guetta's avatar
Leonard Guetta committed
278
279
280
desired result only refers to $1$\nbd{}categories, we have to work in the setting
of $\oo$\nbd{}categories. This can be explained from the fact that if we take a
cofibrant resolution of a $1$\nbd{}category $C$ in the folk model structure on $\oo\Cat$
Leonard Guetta's avatar
Leonard Guetta committed
281
282
283
\[
P \to C,
\]
Leonard Guetta's avatar
Leonard Guetta committed
284
285
then $P$ is not necessarily a $1$\nbd{}category. In particular, polygraphic
homology groups of a $1$\nbd{}category need \emph{not} be trivial in dimension
286
higher than $1$.
Leonard Guetta's avatar
Leonard Guetta committed
287
\begin{paragr}
288
289
290
 Let $A$ be a $1$\nbd{}category and $a$ an object of $A$. Recall that we write $A/a$ for the slice $1$\nbd{}category of $A$ over $a$, that is the $1$\nbd{}category whose description is as follows:
  \begin{itemize}[label=-]
  \item an object of $A/a$ is a pair $(a', p : a' \to a)$ where $a'$ is an object of $A$ and $p$ is an arrow of $A$,
291
292
293
  \item an arrow of $A/a$ is a pair $(q,p : a' \to a)$ where $p$ is an arrow of
    $A$ and $q$ is an arrow of $A$ of the form $q : a'' \to a'$. The target of
    $(q,p)$ is given by $(a',p)$ and the source by $(a'',p\circ q)$. % $ q : a' \to a''$ of $A$ such that $p'\circ q = p$.
Leonard Guetta's avatar
Leonard Guetta committed
294
  \end{itemize}
295
  We write $\pi_a$ for the canonical forgetful functor
296
   \[
Leonard Guetta's avatar
Leonard Guetta committed
297
  \begin{aligned}
298
299
    \pi_{a} : A/a &\to A \\
    (a',p) &\mapsto a'.
Leonard Guetta's avatar
Leonard Guetta committed
300
301
  \end{aligned}
  \]
302
  This is a special case of the more general notion of slice $\oo$\nbd{}category introduced in \ref{paragr:slices}. In particular, given an $\oo$\nbd{}category $X$ and an $\oo$\nbd{}functor $f : X \to A$, we have defined the $\oo$\nbd{}category $X/a$ and the $\oo$\nbd{}functor
Leonard Guetta's avatar
Leonard Guetta committed
303
  \[
304
  f/a : X/a \to A/a
Leonard Guetta's avatar
Leonard Guetta committed
305
306
  \]
  as the following pullback
Leonard Guetta's avatar
Leonard Guetta committed
307
308
  \[
  \begin{tikzcd}
309
310
    X/a \ar[r]\ar[dr, phantom, "\lrcorner", very near start] \ar[d,"f/a"'] & X \ar[d,"f"] \\
    A/a \ar[r,"\pi_{a}"] &A.
Leonard Guetta's avatar
Leonard Guetta committed
311
312
  \end{tikzcd}
  \]
Leonard Guetta's avatar
Leonard Guetta committed
313
314
315
  More explicitly, the $n$\nbd{}cells of $X/a$ can be described as pairs $(x,p)$
  where $x$ is an $n$\nbd{}cell of $X$ and $p$ is an arrow of $A$ of the form
  \[
316
    p : f(x)\to a \text{ if }n=0
Leonard Guetta's avatar
Leonard Guetta committed
317
318
319
320
321
322
323
324
325
  \]
  and
  \[
    p : f(\trgt_0(x)) \to a \text{ if }n>0.
  \]
  \emph{From now on, let us use the convention that $\trgt_0(x)=x$ when $x$ is a
    $0$\nbd{}cell of $X$}.

  When $n>0$, the source and target of an $n$\nbd{}cell $(x,p)$ of $X/a$ are given by
Leonard Guetta's avatar
Leonard Guetta committed
326
  \[
327
  \src((x,p))=(\src(x),p) \text{ and } \trgt((x,p))=(\trgt(x),p).
Leonard Guetta's avatar
Leonard Guetta committed
328
329
330
\]
%% LA DESCRIPTION DU BUT AU DESSUS N'EST PAS BONNE POUR LA DIMENSION 1
%% A CORRIGER !!
331
  Moreover, the $\oo$\nbd{}functor $f/a$ is described as 
Leonard Guetta's avatar
Leonard Guetta committed
332
333
334
  \[
  (x,p) \mapsto (f(x),p),
  \]
335
  and the canonical $\oo$\nbd{}functor $X/a \to X$ as
Leonard Guetta's avatar
Leonard Guetta committed
336
337
338
  \[
  (x,p) \mapsto x.
  \]
339
\end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
340
\begin{paragr}\label{paragr:unfolding}
341
  Let $f : X \to A$ be an $\oo$\nbd{}functor with $A$ a $1$\nbd{}category. Every arrow $\beta : a \to a'$ of $A$ induces an $\oo$\nbd{}functor
342
  \begin{align*}
343
    X/\beta : X/a &\to X/{a'} \\
Leonard Guetta's avatar
Leonard Guetta committed
344
    (x,p) & \mapsto (x,\beta \circ p),
345
  \end{align*}
346
  which takes part in a commutative triangle
Leonard Guetta's avatar
Leonard Guetta committed
347
348
  \[
  \begin{tikzcd}[column sep=tiny]
349
    X/{a} \ar[rr,"X/{\beta}"] \ar[dr] && X/{a'} \ar[dl] \\
Leonard Guetta's avatar
Leonard Guetta committed
350
351
    &X&.
  \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
352
353
354
  \]
  This defines a functor
  \begin{align*}
355
    X/{-} : A &\to \oo\Cat\\
356
    a &\mapsto X/a
Leonard Guetta's avatar
Leonard Guetta committed
357
  \end{align*}
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
358
  and a canonical $\oo$\nbd{}functor
Leonard Guetta's avatar
Leonard Guetta committed
359
  \[
360
  \colim_{a \in A} (X/{a}) \to X.
Leonard Guetta's avatar
Leonard Guetta committed
361
  \]
Leonard Guetta's avatar
Leonard Guetta committed
362
  Let $f' : X' \to A$ be another $\oo$\nbd{}functor and let
Leonard Guetta's avatar
Leonard Guetta committed
363
364
  \[
  \begin{tikzcd}
365
    X \ar[rr,"g"] \ar[dr,"f"'] && X' \ar[dl,"f'"] \\
Leonard Guetta's avatar
Leonard Guetta committed
366
367
368
    &A&
  \end{tikzcd}
  \]
369
  be a commutative triangle in $\oo\Cat$. Recall from \ref{paragr:slices} that for every object $a$ of $A$, there is an $\oo$\nbd{}functor $g/a$ defined as
Leonard Guetta's avatar
Leonard Guetta committed
370
  \begin{align*}
371
372
    g/a : X/a &\to X'/a \\
    (x,p) &\mapsto (g(x),p).
Leonard Guetta's avatar
Leonard Guetta committed
373
  \end{align*}
374
375
376
  This defines a natural transformation
  \[g/- : X/- \Rightarrow X'/-,\]
  and thus induces an $\oo$\nbd{}functor
Leonard Guetta's avatar
Leonard Guetta committed
377
  \[
378
  \colim_{a \in A}(X/a) \to \colim_{a \in A}(X'/a).
Leonard Guetta's avatar
Leonard Guetta committed
379
  \]
380
  Furthermore, it is immediate to check that the square
Leonard Guetta's avatar
Leonard Guetta committed
381
382
  \[
  \begin{tikzcd}
383
384
  \displaystyle \colim_{a \in A}(X/a) \ar[d] \ar[r] & X \ar[d,"g"] \\
  \displaystyle\colim_{a \in A}(X'/a) \ar[r] & X',
Leonard Guetta's avatar
Leonard Guetta committed
385
  \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
386
387
388
  \]
  is commutative.
\end{paragr}
389
\begin{lemma}\label{lemma:colimslice}
390
  Let $f : X \to A$ be an $\oo$\nbd{}functor such that $A$ is a $1$\nbd{}category. The canonical $\oo$\nbd{}functor
Leonard Guetta's avatar
Leonard Guetta committed
391
  \[
392
  \colim_{a \in A}(X/a) \to X
Leonard Guetta's avatar
Leonard Guetta committed
393
394
395
396
  \]
  is an isomorphism.
\end{lemma}
\begin{proof}
397
398
 We have to show that the cocone
 \[
399
   (X/a \to X)_{a \in \Ob(A)}
400
401
402
403
 \]
 is colimiting.
 Let
 \[
404
   (\phi_{a} : X/a \to C)_{a \in \Ob(A)}
405
 \]
Leonard Guetta's avatar
Leonard Guetta committed
406
 be another cocone and let $x$ be a $n$\nbd{}arrow of $X$. Notice that the pair
407
408
409
 \[
   (x,1_{f(\trgt_0(x))})
 \]
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
410
 is a $n$\nbd{}arrow of $X/f(\trgt_0(x))$. We leave it to the reader to check that the formula
411
412
\begin{align*}
     \phi : X &\to C \\
413
     x &\mapsto \phi_{f(\trgt_0(x))}(x,1_{f(\trgt_0(x))}).
414
   \end{align*}
415
 defines an $\oo$\nbd{}functor and it is straightforward to check that for every object $a$ of $A$ the triangle
416
417
 \[
   \begin{tikzcd}
418
     X/a\ar[dr,"\phi_{a}"']\ar[r] & X \ar[d,"\phi"] \\
419
420
421
     & C
   \end{tikzcd}
 \]
422
423
 is commutative. This proves the existence part of the universal property.
 
Leonard Guetta's avatar
Leonard Guetta committed
424
  Now let $\phi' : X \to C$ be another $\oo$\nbd{}functor that makes the previous triangles commute for every object $a$ of $A$ and let $x$ be an $n$\nbd{}cell of $X$. Since the triangle
425
426
 \[
   \begin{tikzcd}
427
     X/f(\trgt_0(x)) \ar[dr,"\phi_{f(\trgt_0(x))}"']\ar[r] & X \ar[d,"\phi'"] \\
428
429
430
431
432
     & C
   \end{tikzcd}
 \]
 is commutative, we necessarily have
 \[
433
   \phi'(x)=\phi_{f(\trgt_0(x))}(x,1_{f(\trgt_0(x))})
434
435
 \]
 which proves that $\phi'=\phi$.
Leonard Guetta's avatar
Leonard Guetta committed
436
437
\end{proof}
\begin{paragr}
438
  In particular, when we apply the previous lemma to $\mathrm{id}_A : A \to A$,
439
  we obtain that every $1$\nbd{}category $A$ is (canonically isomorphic to) the colimit
Leonard Guetta's avatar
Leonard Guetta committed
440
  \[
441
  \colim_{a \in A} (A/a).
442
443
444
\]
% In other words, this simply say that the colimit of the Yoneda embedding $A \to
% \Psh{A}$ is the terminal presheaves
445
  We now proceed to prove that this colimit is homotopic with respect to
446
  the folk weak equivalences.
Leonard Guetta's avatar
Leonard Guetta committed
447
\end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
448
Up to Lemma \ref{lemma:basisofslice}, we fix once and for all an $\oo$\nbd{}functor $f : X \to A$ with $A$ a $1$\nbd{}category.
449
\begin{lemma}\label{lemma:sliceisfree}
450
451
452
  If $X$ is free, then for every object $a$ of $A$, the $\oo$\nbd{}category
  $X/a$ is free. More precisely, if $\Sigma^X_n$ is the $n$\nbd{}basis of $X$,
  then the $n$\nbd{}basis of $X/a$ is the set
Leonard Guetta's avatar
Leonard Guetta committed
453
  \[
454
  \Sigma^{X/a}_n := \{(x,p) \in (X/a)_n \vert x \in \Sigma^X_n \}.
Leonard Guetta's avatar
Leonard Guetta committed
455
456
457
  \]
\end{lemma}
\begin{proof}
458
  It is immediate to check that for every object $a$ of $A$, the canonical
459
  forgetful functor $\pi_{a} : A/a \to A$ is a discrete Conduché functor (see Section
460
  \ref{section:conduche}). Hence, from Lemma \ref{lemma:pullbackconduche} we
461
  know that $X/a \to X$ is a discrete Conduché $\oo$\nbd{}functor. The result follows
462
  then from Theorem \ref{thm:conduche}.
Leonard Guetta's avatar
Leonard Guetta committed
463
464
\end{proof}
\begin{paragr}
465
  When $X$ is free, every arrow $\beta : a \to a'$ of $A$ induces a map
Leonard Guetta's avatar
Leonard Guetta committed
466
  \begin{align*}
467
    \Sigma^{X/a}_n &\to \Sigma^{X/a'}_n \\
Leonard Guetta's avatar
Leonard Guetta committed
468
469
470
471
472
    (x,p) &\mapsto (x,\beta\circ p).
  \end{align*}
  This defines a functor
  \begin{align*}
    \Sigma^{X/{\shortminus}}_n : A &\to \Set \\
473
    a &\mapsto \Sigma^{X/a}_n.
Leonard Guetta's avatar
Leonard Guetta committed
474
475
  \end{align*}
\end{paragr}
476
\begin{lemma}\label{lemma:basisofslice}
Leonard Guetta's avatar
Leonard Guetta committed
477
  If $X$ is free, then there is an isomorphism of functors 
Leonard Guetta's avatar
Leonard Guetta committed
478
  \[
479
  \Sigma^{X/\shortminus}_n \simeq \coprod_{x \in \Sigma^X_n}\Hom_A\left(f(\trgt_0(x)),\shortminus\right).
Leonard Guetta's avatar
Leonard Guetta committed
480
481
482
  \]
\end{lemma}
\begin{proof}
483
  For every object $a$ of $A$ and every $x \in \Sigma_n^X$, we have a canonical map
Leonard Guetta's avatar
Leonard Guetta committed
484
  \begin{align*}
485
    \Hom_A\left(f(\trgt_0(x)),a\right) &\to \Sigma^{X/a}_n \\
Leonard Guetta's avatar
Leonard Guetta committed
486
487
488
489
    p &\mapsto (x,p).
  \end{align*}
  By universal property, this induces a map
  \[
490
  \coprod_{x \in \Sigma^X_n}\Hom_A\left(f(\trgt_0(x)),a\right) \to  \Sigma^{X/a}_n,
Leonard Guetta's avatar
Leonard Guetta committed
491
  \]
492
  which is natural in $a$. A simple verification shows that it is a bijection.
Leonard Guetta's avatar
Leonard Guetta committed
493
\end{proof}
494
\begin{proposition}\label{prop:sliceiscofibrant}
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
495
  Let $A$ be a $1$\nbd{}category, $X$ be a free $\oo$\nbd{}category and $f : X \to A$ be an $\oo$\nbd{}functor. The functor
496
497
  \begin{align*}
    A &\to \oo\Cat \\
498
    a &\mapsto X/a
499
  \end{align*}
500
501
  is a cofibrant object for the projective model structure on $\oo\Cat(A)$
  induced by the folk model structure on $\oo\Cat$ (\ref{paragr:projmod}).
Leonard Guetta's avatar
Leonard Guetta committed
502
\end{proposition}
503
\begin{proof}
504
505
506
507
  Recall that the set
  \[
    \{i_n: \sS_{n-1} \to \sD_n \vert n \in \mathbb{N} \}
  \]
508
  is a set of generating folk cofibrations.
509
  From Lemmas \ref{lemma:sliceisfree} and \ref{lemma:basisofslice} we deduce
510
  that for every object $a$ of $A$ and every $n \in \mathbb{N}$, the canonical square
511
512
  \[
    \begin{tikzcd}
513
514
      \displaystyle\coprod_{x \in \Sigma^X_n}\coprod_{\Hom_A(f(\trgt_0(x)),a)}\sS_{n-1} \ar[r] \ar[d] & \sk_{n-1}(X/a) \ar[d]\\
      \displaystyle\coprod_{x \in \Sigma^X_n}\coprod_{\Hom_A(f(\trgt_0(x)),a)}\sD_n \ar[r]& \sk_n{(X/a)}
515
516
517
    \end{tikzcd}
  \]
  is cocartesian. It is straightforward to check that this square is natural in
518
  $a$ in an obvious sense, which means that we have a cocartesian square in $\oo\Cat(A)$:
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
519
520
  \[
      \begin{tikzcd}
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
521
522
      \displaystyle\coprod_{x \in \Sigma^X_n}\sS_{n-1}\otimes f(\trgt_0(x)) \ar[r] \ar[d] & \sk_{n-1}(X/-) \ar[d]\\
      \displaystyle\coprod_{x \in \Sigma^X_n}\sD_n\otimes f(\trgt_0(x)) \ar[r]& \sk_n{(X/-)}
Leonard Guetta's avatar
Leonard Guetta committed
523
      \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
524
525
    \end{tikzcd}
  \]
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
526
  (see \ref{paragr:cofprojms} for notations). From the second part of Proposition \ref{prop:modprs}, we deduce that for every $n\geq 0$, \[\sk_{n-1}(X/-) \to \sk_{n}(X/-)\] is a cofibration for the
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
527
528
  projective model structure on $\oo\Cat(A)$. Thus, the transfinite composition
  \[
Leonard Guetta's avatar
Leonard Guetta committed
529
  \emptyset \to \sk_{0}(X/-) \to \sk_{1}(X/) \to \cdots \to \sk_{n}(X/-) \to \cdots,
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
530
  \]
Leonard Guetta's avatar
Leonard Guetta committed
531
  which is canonically isomorphic to $\emptyset \to X/-$ (see Lemma \ref{lemma:filtration}), is also a cofibration for the projective model structure. 
532
\end{proof}
Leonard Guetta's avatar
Leonard Guetta committed
533
\begin{corollary}\label{cor:folkhmtpycol}
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
534
  Let $A$ be a $1$\nbd{}category and $f : X \to A$ be an $\oo$\nbd{}functor. The canonical arrow of $\ho(\oo\Cat^{\folk})$
535
  \[
536
  \hocolim^{\folk}_{a \in A}(X/a) \to X,
537
  \]
538
  induced by the co-cone $(X/a \to X)_{a \in \Ob(A)}$, is an isomorphism.
539
540
541
\end{corollary}
Beware that in the previous corollary, we did \emph{not} suppose that $X$ was free.
\begin{proof}
542
  Let $P$ be a free $\omega$-category and $g : P \to X$ a folk trivial fibration
543
  and consider the following commutative diagram of $\ho(\oo\Cat^{\folk})$ 
544
545
  \begin{equation}\label{comsquare}
    \begin{tikzcd}
546
547
      \displaystyle\hocolim^{\folk}_{a \in A}(P/a) \ar[d] \ar[r] & \displaystyle\colim_{a \in A}(P/a) \ar[d] \ar[r] & P \ar[d]\\
      \displaystyle\hocolim^{\folk}_{a \in A}(X/a)  \ar[r] & \displaystyle\colim_{a \in A}(X/a) \ar[r] & X
548
549
    \end{tikzcd}
  \end{equation}
550
  where the middle and most left vertical arrows are induced by the arrows
551
  \[
552
    g/a : P/a \to X/a,
553
  \]
554
555
556
  and the most right vertical arrow is induced by $g$.
  Since trivial fibrations are stable by pullback, $g/a$ is a trivial fibration.
  This proves that the most left vertical arrow of diagram \eqref{comsquare} is an isomorphism.
557
  
Leonard Guetta's avatar
Leonard Guetta committed
558
  Now, from Proposition \ref{prop:sliceiscofibrant} and Corollary
559
560
561
562
  \ref{cor:cofprojms}, we deduce that the arrow \[\hocolim_{a \in
    A}^{\folk}(P/a)\to \colim_{a \in A}(P/a)\] is an isomorphism. Moreover, from Lemma \ref{lemma:colimslice}, we know that the arrows
  \[\colim_{a \in A}(P/a)\to P\] and \[\colim_{a \in A}(X/a)\to X\] are
  isomorphisms. 
563

564
565
566
567
568
  Finally, since $g$ is a folk weak equivalence, the most right vertical arrow of diagram \eqref{comsquare} is an
  isomorphism and by an immediate 2-out-of-3 property this proves
  that all arrows of \eqref{comsquare} are isomorphisms. In particular, so is
  the composition of the two bottom horizontal arrows, which is what we desired
  to show.
569
\end{proof}
570
We now move on to the next step needed to prove that every $1$\nbd{}category is \good{}. For that purpose, let us recall a construction commonly referred to as the ``Grothendieck construction''.
571
\begin{paragr}
572
  Let $A$ be a $1$\nbd{}category and $F : A \to \Cat$ a functor. We denote by $\int F$ or $\int_{a \in A}F(a)$ the category such that:
573
  \begin{itemize}[label=-]
574
575
576
577
578
579
580
581
582
583
  \item An object of $\int F$ is a pair $(a,x)$ where $a$ is an object of $A$ and $x$ is an object of $F(a)$.
  \item An arrow $(a,x) \to (a',x')$ of $\int F$ is a pair $(f,k)$ where
    \[
    f : a \to a'
    \]
    is an arrow of $A$, and
    \[
    k : F(f)(x) \to x'.
    \]
  \end{itemize}
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
584
  The unit on $(a,x)$ is the pair $(1_a,1_x)$ and the composition of $(f,k) : (a,x) \to (a',x')$ and $(f',k') : (a',x') \to (a'',x'')$ is given by:
585
586
587
  \[
  (f',k')\circ(f,k)=(f'\circ f,k'\circ F(f')(k)).
  \]
588
  Every natural transformation
589
590
  \[
  \begin{tikzcd}
591
    A \ar[r,bend left,"F",""{name=A,below},pos=19/30]\ar[r,bend right,"G"',""{name=B, above},pos=11/20] & \Cat \ar[from=A,to=B,Rightarrow,"\alpha",pos=9/20]
592
593
594
595
596
597
598
  \end{tikzcd}
  \]
  induces a functor
  \begin{align*}
    \int \alpha : \int F &\to \int G\\
    (a,x) &\mapsto (a,\alpha_a(x)).
  \end{align*}
599
  Altogether, this defines a functor
600
601
602
603
604
605
  \begin{align*}
    \int : \Cat(A)&\to \Cat \\
   F&\mapsto \int F,
  \end{align*}
  where $\Cat(A)$ is the category of functors from $A$ to $\Cat$. 
\end{paragr}
606
We now recall an important theorem due to Thomason.
607
\begin{theorem}[Thomason]\label{thm:Thomason}
608
  The functor $\int : \Cat(A) \to \Cat$ sends the pointwise Thomason equivalences (\ref{paragr:homder}) to Thomason equivalences and the induced functor
609
610
611
612
613
  \[
  \overline{\int} : \ho(\Cat^{\Th}(A)) \to \ho(\Cat^{\Th})
  \]
  is canonically isomorphic to the homotopy colimit functor
  \[
Leonard Guetta's avatar
Leonard Guetta committed
614
  \hocolim^{\Th}_A : \ho(\Cat^{\Th}(A)) \to \ho(\Cat^{\Th}).
615
616
617
618
619
620
  \]
\end{theorem}
\begin{proof}
  The original source for this Theorem is \cite{thomason1979homotopy}. However, the definition of homotopy colimit used by Thomason, albeit equivalent, is not the same as the one we used in this dissertation and is slightly outdated. A more modern proof of the theorem can be found in \cite[Proposition 2.3.1 and Théorème 1.3.7]{maltsiniotis2005theorie}.
\end{proof}
\begin{corollary}\label{cor:thomhmtpycol}
621
  Let $A$ be a $1$\nbd{}category. The canonical map
622
  \[
Leonard Guetta's avatar
Leonard Guetta committed
623
  \hocolim^{\Th}_{a \in A}(A/a) \to A
624
625
626
627
628
629
630
631
  \]
  induced by the co-cone $(A/a \to A)_{a \in \Ob(A)}$, is an isomorphism of $\ho(\Cat^{\Th})$. 
\end{corollary}
\begin{proof}
  For every object $a$ of $A$, the canonical map to the terminal category
  \[
  A/a \to \sD_0
  \]
632
  is a Thomason equivalence. This comes from the fact that $A/a$ is oplax contractible (Proposition \ref{prop:slicecontractible}), or from \cite[Section 1, Corollary 2]{quillen1973higher} and the fact that $A/a$ has a terminal object.
633
634
635
636
637

  In particular, the morphism of functors
  \[
  A/(-) \Rightarrow k_{\sD_0},
  \]
638
  where $k_{\sD_0}$ is the constant functor $A \to \Cat$ with value the terminal category $\sD_0$, is a pointwise Thomason equivalence. It follows from the first part of Theorem \ref{thm:Thomason} that
639
640
641
  \[
  \int_{a \in A}A/a \to \int_{a \in A}k_{\sD_0}
  \]
Leonard Guetta's avatar
Leonard Guetta committed
642
  is a Thomason equivalence and an immediate computation shows that \[\int_{a \in A}k_{\sD_0} \simeq A.\] From the second part of Theorem \ref{thm:Thomason}, we have that
643
  \[
Leonard Guetta's avatar
Leonard Guetta committed
644
  \hocolim^{\Th}_{a \in A}(A/a) \simeq A.
645
  \]
646
  A thorough analysis of all the isomorphisms involved shows that this last isomorphism is indeed induced by the co-cone $(A/a \to A)_{a \in \Ob(A)}$.
647
648
\end{proof}
\begin{remark}
Leonard Guetta's avatar
Leonard Guetta committed
649
  It is possible to extend the previous corollary to prove that for every functor $f : X \to A$ ($X$ and $A$ being $1$\nbd{}categories), we have \[\hocolim^{\Th}_{a \in A} (X/a) \simeq X.\] However, to prove that it is also the case when $X$ is an $\oo$\nbd{}category and $f$ an $\oo$\nbd{}functor, as in Corollary \ref{cor:folkhmtpycol}, one would need to extend the Grothendieck construction to functors with value in $\oo\Cat$ and to prove an $\oo$\nbd{}categorical analogue of Theorem \ref{thm:Thomason}. Such results, while being highly plausible, go beyond the scope of this dissertation.
650
651
\end{remark}
Putting all the pieces together, we are now able to prove the awaited Theorem.
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
652
\begin{theorem}\label{thm:categoriesaregood}
653
  Every $1$\nbd{}category is \good{}.
654
655
656
\end{theorem}
\begin{proof}
  All the arguments of the proof have already been given and we sum them up here essentially for the sake of clarity.
657
  Let $A$ be a $1$\nbd{}category. Consider the diagram
658
659
660
661
662
663
664
665
666
667
668
  \begin{align*}
    A &\to \oo\Cat\\
    a &\mapsto A/a
  \end{align*}
  and the co-cone
  \[
  (A/a \to A)_{a \in \Ob(A)}.
  \]
  \begin{itemize}[label=-]
  \item The canonical map of $\ho(\oo\Cat^{\folk})$
    \[
669
    \hocolim_{a \in A}^{\folk} (A/a) \to A
670
671
672
673
    \]
    is an isomorphism thanks to Corollary \ref{cor:folkhmtpycol} applied to $\mathrm{id}_A : A \to A$.
  \item The canonical map of $\ho(\oo\Cat^{\Th})$
    \[
674
    \hocolim_{a \in A}^{\Th} (A/a) \to A
675
    \]
Leonard Guetta's avatar
Leonard Guetta committed
676
677
678
    is an isomorphism thanks to Corollary \ref{cor:thomhmtpycol} and the fact
    that the canonical morphisms of op-prederivators $\Ho(\Cat^{\Th}) \to
    \Ho(\oo\Cat^{\Th})$ is homotopy cocontinuous (see \ref{paragr:thomhmtpycol}).
679
    \item Every $A/a$ is \good{} thanks to Proposition \ref{prop:contractibleisgood} and Proposition \ref{prop:slicecontractible}.
680
681
682
  \end{itemize}
  Thus, Proposition \ref{prop:criteriongoodcat} applies and this proves that $A$ is \good{}.
\end{proof}
683
684
685
686
687

%%% Local Variables:
%%% mode: latex
%%% TeX-master: "main"
%%% End: