2cat.tex 88.1 KB
Newer Older
Leonard Guetta's avatar
Leonard Guetta committed
1
\chapter{Homotopy and homology type of free 2-categories}
2
\chaptermark{Homology of free $2$-categories}
Leonard Guetta's avatar
Leonard Guetta committed
3
\section{Preliminaries: the case of free 1-categories}\label{section:prelimfreecat}
Leonard Guetta's avatar
Leonard Guetta committed
4
In this section, we review some homotopical results on free
Leonard Guetta's avatar
Leonard Guetta committed
5
($1$-)categories that will be of great help in the sequel.
Leonard Guetta's avatar
Leonard Guetta committed
6
\begin{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
7
8
  A \emph{reflexive graph} $G$ consists of the data of two sets $G_0$ and $G_1$
  together with
9
  \begin{itemize}[label=-]
Leonard Guetta's avatar
Leonard Guetta committed
10
  \item a ``source'' map $\src : G_1 \to G_0$,
Leonard Guetta's avatar
Leonard Guetta committed
11
12
13
  \item a ``target'' map $\trgt : G_1 \to G_0$,
  \item a ``unit'' map $1_{(-)} : G_0 \to G_1$,
  \end{itemize}
Leonard Guetta's avatar
Leonard Guetta committed
14
  such that for every $x \in G_0$,
Leonard Guetta's avatar
Leonard Guetta committed
15
  \[
Leonard Guetta's avatar
Leonard Guetta committed
16
17
18
19
20
21
22
23
24
25
26
    \src(1_{x}) = \trgt (1_{x}) = x.
  \]
  The vocabulary of categories is used: elements of $G_0$ are \emph{objects} or
  \emph{$0$-cells}, elements of $G_1$ are \emph{arrows} or \emph{$1$-cells},
  arrows of the form $1_{x}$ with $x$ an object are \emph{units}, etc. A
  \emph{morphism of reflexive graphs} $ f : G \to G'$ consists of maps $f_0 :
  G_0 \to G'_0$ and $f_1 : G_1 \to G'_1$ that commute with sources, targets and
  units in an obvious sense. This defines the category $\Rgrph$ of reflexive
  graphs. Later we will make use of monomorphisms in the category $\Rgrph$; they
  are the morphisms $f : G \to G'$ that are injective on objects and on arrows,
  i.e. such that $f_0 : G_0 \to G_0'$ and $f_1 : G_1 \to G'_1$ are injective.
27
28
  
  There is a ``underlying reflexive graph'' functor
Leonard Guetta's avatar
Leonard Guetta committed
29
  \[
Leonard Guetta's avatar
Leonard Guetta committed
30
    U : \Cat \to \Rgrph,
Leonard Guetta's avatar
Leonard Guetta committed
31
32
33
  \]
  which has a left adjoint
  \[
Leonard Guetta's avatar
Leonard Guetta committed
34
    L : \Rgrph \to \Cat.
Leonard Guetta's avatar
Leonard Guetta committed
35
  \]
Leonard Guetta's avatar
Leonard Guetta committed
36
37
  For a reflexive graph $G$, the objects of $L(G)$ are exactly the objects of
  $G$ and an arrow $f$ of $L(G)$ is a chain
Leonard Guetta's avatar
Leonard Guetta committed
38
  \[
Leonard Guetta's avatar
Leonard Guetta committed
39
40
41
42
    \begin{tikzcd}
      X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1}
      \ar[r,"f_n"]& X_{n}
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
43
  \]
Leonard Guetta's avatar
Leonard Guetta committed
44
45
46
47
  of arrows of $G$, such that \emph{none} of the $f_k$ are units. The integer
  $n$ is referred to as the \emph{length} of $f$ and is denoted by $\ell(f)$.
  Composition is given by concatenation of chains.
\end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
48
\begin{lemma}
Leonard Guetta's avatar
Leonard Guetta committed
49
50
  A category $C$ is free in the sense of \ref{def:freeoocat} if and only if
  there exists a reflexive graph $G$ such that
Leonard Guetta's avatar
Leonard Guetta committed
51
  \[
Leonard Guetta's avatar
Leonard Guetta committed
52
    C \simeq L(G).
Leonard Guetta's avatar
Leonard Guetta committed
53
54
55
  \]
\end{lemma}
\begin{proof}
Leonard Guetta's avatar
Leonard Guetta committed
56
57
58
  If $C$ is free, consider the reflexive graph $G$ such that $G_0 = C_0$ and
  $G_1$ is the subset of $C_1$ whose elements are either generating $1$-cells of
  $C$ or units. It is straightforward to check that $C\simeq L(G)$.
Leonard Guetta's avatar
Leonard Guetta committed
59

Leonard Guetta's avatar
Leonard Guetta committed
60
61
62
  Conversely, if $C \simeq L(G)$ for some reflexive graph $G$, then the
  description of the arrows of $L(G)$ given in the previous paragraph shows that
  $C$ is free and that its set of generating $1$-cells is (isomorphic to) the
63
  set of non unital $1$-cells of $G$.
Leonard Guetta's avatar
Leonard Guetta committed
64
65
\end{proof}
\begin{remark}
Leonard Guetta's avatar
Leonard Guetta committed
66
67
68
69
70
71
  In other words, a category is free on a graph if and only if it is free on a
  reflexive graph. The difference between these two notions is at the level of
  morphisms: there are more morphisms of reflexive graphs because (generating)
  $1$\nbd{}cells may be sent to units. Hence, for a morphism of reflexive graphs
  $f : G \to G'$, the induced functor $L(f)$ is not necessarily rigid in the
  sense of Definition \ref{def:rigidmorphism}.
Leonard Guetta's avatar
Leonard Guetta committed
72
73
\end{remark}
\begin{paragr}
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
74
75
76
  There is another important description of the category $\Rgrph$. Write
  $\Delta_{\leq 1}$ for the full subcategory of $\Delta$ spanned by $[0]$ and $[1]$.
  The category $\Rgrph$ is nothing but $\Psh{\Delta_{\leq 1}}$, the
Leonard Guetta's avatar
Leonard Guetta committed
77
78
79
  category of pre-sheaves on $\Delta_{\leq 1}$. In particular, the canonical
  inclusion $i : \Delta_{\leq 1} \rightarrow \Delta$ induces by pre-composition
  a functor
Leonard Guetta's avatar
blabla    
Leonard Guetta committed
80
  \[
Leonard Guetta's avatar
Leonard Guetta committed
81
    i^* : \Psh{\Delta} \to \Rgrph,
Leonard Guetta's avatar
blabla    
Leonard Guetta committed
82
83
84
  \]
  which, by the usual technique of Kan extensions, has a left adjoint
  \[
Leonard Guetta's avatar
Leonard Guetta committed
85
    i_! : \Rgrph \to \Psh{\Delta}.
Leonard Guetta's avatar
blabla    
Leonard Guetta committed
86
  \]
Leonard Guetta's avatar
Leonard Guetta committed
87
88
  For a graph $G$, the simplicial set $i_!(G)$ has $G_0$ as its set of
  $0$-simplices, $G_1$ as its set of $1$-simplices and all $k$-simplices are
89
  degenerate for $k>1$. For future reference, we put here the following lemma.
Leonard Guetta's avatar
blabla    
Leonard Guetta committed
90
\end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
91
\begin{lemma}\label{lemma:monopreserved}
92
  The functor $i_! : \Rgrph \to \Psh{\Delta}$ preserves monomorphisms.
Leonard Guetta's avatar
blabla    
Leonard Guetta committed
93
94
\end{lemma}
\begin{proof}
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
95
  What we need to show is that, given a morphism of simplicial sets
96
  \[
Leonard Guetta's avatar
Leonard Guetta committed
97
    f : X \to Y,
98
  \]
Leonard Guetta's avatar
Leonard Guetta committed
99
  if $f_0 : X_0 \to Y_0$ and $f_1 : X_1 \to Y_1$ are monomorphisms and if all
100
  $n$\nbd{}simplices of $X$ are degenerate for $n\geq 2$, then $f$ is a
Leonard Guetta's avatar
Leonard Guetta committed
101
  monomorphism. A proof of this assertion is contained in \cite[Paragraph
Leonard Guetta's avatar
Leonard Guetta committed
102
  3.4]{gabriel1967calculus}. The key argument is the Eilenberg--Zilber Lemma
Leonard Guetta's avatar
Leonard Guetta committed
103
  (Proposition 3.1 of op. cit.).
Leonard Guetta's avatar
blabla    
Leonard Guetta committed
104
105
\end{proof}
\begin{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
106
107
108
109
  Let us denote by $N : \Psh{\Delta} \to \Cat$ (instead of $N_1$ as in Paragraph
  \ref{paragr:nerve}) the usual nerve of categories and by $c : \Cat \to
  \Psh{\Delta}$ its left adjoint. Recall that for a (small) category $C$, an
  $n$-simplex of $N(C)$ is a chain
Leonard Guetta's avatar
blabla    
Leonard Guetta committed
110
  \[
Leonard Guetta's avatar
Leonard Guetta committed
111
112
113
114
    \begin{tikzcd}
      X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1}
      \ar[r,"f_n"]& X_{n}
    \end{tikzcd}
Leonard Guetta's avatar
blabla    
Leonard Guetta committed
115
  \]
116
  of arrows of $C$. Such an $n$-simplex is degenerate if and only if at least
Leonard Guetta's avatar
Leonard Guetta committed
117
118
  one of the $f_k$ is a unit. It is straightforward to check that the composite
  of
Leonard Guetta's avatar
blabla    
Leonard Guetta committed
119
  \[
Leonard Guetta's avatar
Leonard Guetta committed
120
    \Cat \overset{N}{\rightarrow} \Psh{\Delta} \overset{i^*}{\rightarrow} \Rgrph
Leonard Guetta's avatar
blabla    
Leonard Guetta committed
121
  \]
Leonard Guetta's avatar
Leonard Guetta committed
122
123
  is nothing but the forgetful functor $U : \Cat \to \Rgrph$. Thus, the functor
  $L : \Rgrph \to \Cat$ is (isomorphic to) the composite of
Leonard Guetta's avatar
blabla    
Leonard Guetta committed
124
  \[
Leonard Guetta's avatar
Leonard Guetta committed
125
126
    \Rgrph \overset{i_!}{\rightarrow} \Psh{\Delta} \overset{c}{\rightarrow}
    \Cat.
Leonard Guetta's avatar
blabla    
Leonard Guetta committed
127
  \]
Leonard Guetta's avatar
Leonard Guetta committed
128

129
  We now review a construction due to Dwyer and Kan
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
130
131
132
  (\cite{dwyer1980simplicial}). Let $G$ be a reflexive graph. For
  every $k\geq 1$, we define the simplicial set $N^k(G)$ as the
  sub-simplicial set of $N(L(G))$ whose $n$-simplices are chains
Leonard Guetta's avatar
Leonard Guetta committed
133
  \[
Leonard Guetta's avatar
Leonard Guetta committed
134
135
136
137
    \begin{tikzcd}
      X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{n-1}
      \ar[r,"f_n"]& X_{n}
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
138
  \]
Leonard Guetta's avatar
Leonard Guetta committed
139
140
  of arrows of $L(G)$ such that
  \[
141
    \sum_{1 \leq i \leq n}\ell(f_i) \leq k.
Leonard Guetta's avatar
Leonard Guetta committed
142
  \] In particular, we have
Leonard Guetta's avatar
Leonard Guetta committed
143
  \[
Leonard Guetta's avatar
Leonard Guetta committed
144
    N^1(G)=i_!(G)
Leonard Guetta's avatar
Leonard Guetta committed
145
146
147
  \]
  and the transfinite composition of
  \[
Leonard Guetta's avatar
Leonard Guetta committed
148
149
    i_!(G) = N^1(G) \hookrightarrow N^2(G) \hookrightarrow \cdots
    \hookrightarrow N^{k}(G) \hookrightarrow N^{k+1}(G) \hookrightarrow \cdots
Leonard Guetta's avatar
Leonard Guetta committed
150
151
152
  \]
  is easily seen to be the map
  \[
Leonard Guetta's avatar
Leonard Guetta committed
153
    \eta_{i_!(G)} : i_!(G) \to Nci_!(G),
Leonard Guetta's avatar
Leonard Guetta committed
154
  \]
Leonard Guetta's avatar
Leonard Guetta committed
155
  where $\eta$ is the unit of the adjunction $c \dashv N$.
Leonard Guetta's avatar
blabla    
Leonard Guetta committed
156
\end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
157
\begin{lemma}[Dwyer--Kan]\label{lemma:dwyerkan}
158
  For every $k\geq 1$, the canonical inclusion map
Leonard Guetta's avatar
Leonard Guetta committed
159
  \[
Leonard Guetta's avatar
Leonard Guetta committed
160
    N^{k}(G) \to N^{k+1}(G)
Leonard Guetta's avatar
Leonard Guetta committed
161
  \]
162
  is a trivial cofibration of simplicial sets.
Leonard Guetta's avatar
Leonard Guetta committed
163
164
\end{lemma}
\begin{proof}
Leonard Guetta's avatar
Leonard Guetta committed
165
166
167
  Let $A_{k+1}=\mathrm{Im}(\partial_0)\cup\mathrm{Im}(\partial_{k+1})$ be the
  union of the first and last face of the standard $(k+1)$-simplex
  $\Delta_{k+1}$. Notice that the canonical inclusion
168
  \[
Leonard Guetta's avatar
Leonard Guetta committed
169
    A_{k+1} \hookrightarrow \Delta_{k+1}
Leonard Guetta's avatar
Leonard Guetta committed
170
171
172
  \]
  is a trivial cofibration. Let $I_{k+1}$ be the set of chains
  \[
Leonard Guetta's avatar
Leonard Guetta committed
173
174
175
176
    \begin{tikzcd}
      f = X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k-1}
      \ar[r,"f_k"]& X_{k}\ar[r,"f_{k+1}"]& X_{k+1}
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
177
178
179
  \]
  of arrows of $L(G)$ such that for every $1 \leq i \leq k+1$
  \[
Leonard Guetta's avatar
Leonard Guetta committed
180
    \ell(f_i)=1,
181
  \]
Leonard Guetta's avatar
Leonard Guetta committed
182
183
  i.e.\ each $f_i$ is a non-unit arrow of $G$. For every $f \in I_{k+1}$, we
  define a morphism $\varphi_f : A_{k+1} \to N^{k}(G)$ in the following fashion:
Leonard Guetta's avatar
Leonard Guetta committed
184
  \begin{itemize}
Leonard Guetta's avatar
Leonard Guetta committed
185
186
  \item[-]$\varphi_{f}\vert_{\mathrm{Im}(\partial_0)}$ is the $k$-simplex of
    $N^{k}(G)$
Leonard Guetta's avatar
Leonard Guetta committed
187
    \[
Leonard Guetta's avatar
Leonard Guetta committed
188
189
190
      \begin{tikzcd}
        X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k} \ar[r,"f_{k+1}"]&
        X_{k+1},
Leonard Guetta's avatar
Leonard Guetta committed
191
192
      \end{tikzcd}
    \]
Leonard Guetta's avatar
Leonard Guetta committed
193
194
195
196
197
198
  \item[-] $\varphi_{f}\vert_{\mathrm{Im}(\partial_{k+1})}$ is the $k$-simplex
    of $N^{k}(G)$
    \[
      \begin{tikzcd}
        X_0 \ar[r,"f_1"]& X_1\ar[r,"f_2"]& X_2 \ar[r]& \cdots \ar[r] &X_{k-1}
        \ar[r,"f_k"]& X_{k}.
Leonard Guetta's avatar
Leonard Guetta committed
199
200
201
      \end{tikzcd}
    \]
  \end{itemize}
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
202
  All in all, we have a cocartesian square
203
  \[
Leonard Guetta's avatar
Leonard Guetta committed
204
205
206
    \begin{tikzcd}
      \displaystyle \coprod_{f \in I_{k+1}}A_{k+1} \ar[d] \ar[r,"(\varphi_f)_f"] & N^{k}(G)\ar[d] \\
      \displaystyle \coprod_{f \in I_{k+1}}\Delta_{k+1} \ar[r] & N^{k+1}(G),
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
207
      \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
Leonard Guetta's avatar
Leonard Guetta committed
208
    \end{tikzcd}
209
  \]
Leonard Guetta's avatar
Leonard Guetta committed
210
  which proves that the right vertical arrow is a trivial cofibration.
Leonard Guetta's avatar
Leonard Guetta committed
211
\end{proof}
Leonard Guetta's avatar
Leonard Guetta committed
212
From this lemma, we deduce the following proposition.
Leonard Guetta's avatar
Leonard Guetta committed
213
214
215
\begin{proposition}
  Let $G$ be a reflexive graph. The map
  \[
Leonard Guetta's avatar
Leonard Guetta committed
216
    \eta_{i_!(G)} : i_!(G) \to Nci_!(G),
Leonard Guetta's avatar
Leonard Guetta committed
217
  \]
Leonard Guetta's avatar
Leonard Guetta committed
218
219
  where $\eta$ is the unit of the adjunction $c \dashv N$, is a trivial
  cofibration of simplicial sets.
Leonard Guetta's avatar
Leonard Guetta committed
220
\end{proposition}
Leonard Guetta's avatar
Leonard Guetta committed
221
\begin{proof}
Leonard Guetta's avatar
Leonard Guetta committed
222
223
  This follows from the fact that trivial cofibrations are stable by transfinite
  composition.
Leonard Guetta's avatar
Leonard Guetta committed
224
225
226
227
228
\end{proof}
From the previous proposition, we deduce the following very useful corollary.
\begin{corollary}\label{cor:hmtpysquaregraph}
  Let
  \[
Leonard Guetta's avatar
Leonard Guetta committed
229
230
231
232
    \begin{tikzcd}
      A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\
      C \ar[r,"\gamma"]& D
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
233
  \]
Leonard Guetta's avatar
Leonard Guetta committed
234
  be a cocartesian square in $\Rgrph$. If either $\alpha$ or $\beta$ is a
Leonard Guetta's avatar
idem    
Leonard Guetta committed
235
  monomorphism, then the induced square of $\Cat$
Leonard Guetta's avatar
Leonard Guetta committed
236
  \[
Leonard Guetta's avatar
Leonard Guetta committed
237
238
239
240
    \begin{tikzcd}
      L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"]& L(B) \ar[d,"L(\delta)"] \\
      L(C) \ar[r,"L(\gamma)"]& L(D)
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
241
  \]
Leonard Guetta's avatar
idem    
Leonard Guetta committed
242
  is a Thomason homotopy cocartesian.
Leonard Guetta's avatar
Leonard Guetta committed
243
244
245
246
\end{corollary}
\begin{proof}
  Since the nerve $N$ induces an equivalence of op-prederivators
  \[
Leonard Guetta's avatar
Leonard Guetta committed
247
    \Ho(\Cat^{\Th}) \to \Ho(\Psh{\Delta}),
Leonard Guetta's avatar
Leonard Guetta committed
248
249
250
  \]
  it suffices to prove that the induced square of simplicial sets
  \[
Leonard Guetta's avatar
Leonard Guetta committed
251
252
253
254
    \begin{tikzcd}
      NL(A) \ar[d,"NL(\alpha)"] \ar[r,"NL(\beta)"]& NL(B) \ar[d,"NL(\delta)"] \\
      NL(C) \ar[r,"NL(\gamma)"]& NL(D)
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
255
  \]
Leonard Guetta's avatar
Leonard Guetta committed
256
257
258
259
260
261
262
263
  is homotopy cocartesian. But, since $L \simeq c\circ i_!$, it follows from
  Lemma \ref{cor:hmtpysquaregraph} that this last square is weakly equivalent to
  the square of simplicial sets
  \[
    \begin{tikzcd}
      i_!(A) \ar[d,"i_!(\alpha)"] \ar[r,"i_!(\beta)"] &i_!(B) \ar[d,"i_!(\delta)"] \\
      i_!(C) \ar[r,"i_!(\gamma)"]& i_!(D).
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
264
  \]
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
265
266
267
268
269
  This square is cocartesian because $i_!$ is a left adjoint. Since
  $i_!$ preserves monomorphisms (Lemma \ref{lemma:monopreserved}), the
  result follows from the fact that the monomorphisms are the
  cofibrations of the standard Quillen model structure on simplicial
  sets and from Lemma \ref{lemma:hmtpycocartesianreedy}.
Leonard Guetta's avatar
Leonard Guetta committed
270
\end{proof}
Leonard Guetta's avatar
Leonard Guetta committed
271
\begin{paragr}
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
272
273
  By working a little more, we obtain the more general result stated
  in the proposition below. Let us say that a morphism of reflexive
274
  graphs $\alpha : A \to B$ is \emph{quasi-injective on arrows} when
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
275
  for all $f$ and $g$ arrows of $A$, if
Leonard Guetta's avatar
Leonard Guetta committed
276
277
278
279
  \[
    \alpha(f)=\alpha(g),
  \]
  then either $f=g$ or $f$ and $g$ are both units. In other words, $\alpha$
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
280
  never sends a non-unit arrow to a unit arrow and $\alpha$ never identifies two
Leonard Guetta's avatar
Leonard Guetta committed
281
282
283
284
  non-unit arrows. It follows that if $\alpha$ is quasi-injective on arrows and
  injective on objects, then it is also injective on arrows and hence, a
  monomorphism of $\Rgrph$.
\end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
285
\begin{proposition}\label{prop:hmtpysquaregraphbetter}
Leonard Guetta's avatar
Leonard Guetta committed
286
287
  Let
  \[
Leonard Guetta's avatar
Leonard Guetta committed
288
289
290
    \begin{tikzcd}
      A \ar[d,"\alpha"] \ar[r,"\beta"] &B \ar[d,"\delta"] \\
      C \ar[r,"\gamma"]& D
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
291
      \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
Leonard Guetta's avatar
Leonard Guetta committed
292
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
293
  \]
Leonard Guetta's avatar
Leonard Guetta committed
294
295
  be a cocartesian square in $\Rgrph$. Suppose that the following two conditions
  are satisfied
Leonard Guetta's avatar
Leonard Guetta committed
296
  \begin{enumerate}[label=\alph*)]
Leonard Guetta's avatar
Leonard Guetta committed
297
  \item Either $\alpha$ or $\beta$ is injective on objects.
Leonard Guetta's avatar
Leonard Guetta committed
298
  \item Either $\alpha$ or $\beta$ is quasi-injective on arrows.
Leonard Guetta's avatar
Leonard Guetta committed
299
  \end{enumerate}
300
  Then, the induced square of $\Cat$
Leonard Guetta's avatar
Leonard Guetta committed
301
  \[
Leonard Guetta's avatar
Leonard Guetta committed
302
303
304
305
    \begin{tikzcd}
      L(A) \ar[d,"L(\alpha)"] \ar[r,"L(\beta)"] &L(B) \ar[d,"L(\delta)"] \\
      L(C) \ar[r,"L(\gamma)"] &L(D)
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
306
  \]
307
  is Thomason homotopy cocartesian square.
Leonard Guetta's avatar
Leonard Guetta committed
308
309
\end{proposition}
\begin{proof}
Leonard Guetta's avatar
Leonard Guetta committed
310
311
312
313
  The case where $\alpha$ or $\beta$ is both injective on objects and
  quasi-injective on arrows is Corollary \ref{cor:hmtpysquaregraph}. Hence, we
  only have to treat the case when $\alpha$ is injective on objects and $\beta$
  is quasi-injective on arrows; the remaining case being symmetric.
Leonard Guetta's avatar
Leonard Guetta committed
314

315
  Let use denote by $E$ the set of objects of $B$ that are in the image of
316
  $\beta$. We consider this set as well as the set $A_0$ of objects of $A$ as discrete reflexive graphs, i.e.\ reflexive graphs
317
  with no non-unit arrows. Now, let $G$ be the reflexive graph defined by the
Leonard Guetta's avatar
Leonard Guetta committed
318
  following cocartesian square
Leonard Guetta's avatar
Leonard Guetta committed
319
  \[
Leonard Guetta's avatar
Leonard Guetta committed
320
    \begin{tikzcd}
321
      A_0\ar[r] \ar[d] & E \ar[d]\\
Leonard Guetta's avatar
Leonard Guetta committed
322
323
      A \ar[r] & G, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
324
  \]
325
326
  where the morphism \[ A_0 \to A\] is the canonical inclusion, and the
  morphism \[A_0 \to E\] is induced by the restriction of $\beta$ on objects. In other words, $G$ is
Leonard Guetta's avatar
Leonard Guetta committed
327
328
  obtained from $A$ by collapsing the objects that are identified through
  $\beta$. It admits the following explicit description: $G_0$ is (isomorphic
329
  to) $E$ and the set of non-unit arrows of $G$ is (isomorphic to) the set of
Leonard Guetta's avatar
Leonard Guetta committed
330
331
  non-unit arrows of $A$; the source (resp.\ target) of a non-unit arrow $f$ of
  $G$ is the source (resp.\ target) of $\beta(f)$. This completely describes $G$.
Leonard Guetta's avatar
Leonard Guetta committed
332
333
  % Notice also for later reference that the morphism \[ \coprod_{x \in E}F_x
  %   \to A\] is a monomorphism, i.e. injective on objects and arrows.
Leonard Guetta's avatar
Leonard Guetta committed
334
335
336

  Now, we have the following solid arrow commutative diagram
  \[
Leonard Guetta's avatar
Leonard Guetta committed
337
    \begin{tikzcd}
338
      A_0 \ar[r] \ar[d] & E  \ar[ddr,bend left]\ar[d]&\\
Leonard Guetta's avatar
Leonard Guetta committed
339
340
341
      A  \ar[drr,bend right,"\beta"'] \ar[r] & G \ar[dr, dotted]&\\
      &&B, \ar[from=1-1,to=2-2,phantom,very near end,"\ulcorner"]
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
342
  \]
Leonard Guetta's avatar
Leonard Guetta committed
343
  where the arrow $E \to B$ is the canonical inclusion. Hence, by universal
344
  property, the dotted arrow exists and makes the whole diagram commute. A
Leonard Guetta's avatar
Leonard Guetta committed
345
346
  thorough verification easily shows that the morphism $G \to B$ is a
  monomorphism of $\Rgrph$.
Leonard Guetta's avatar
Leonard Guetta committed
347

348
349
  By forming successive cocartesian squares and combining with the square
  obtained earlier, we obtain a diagram of three cocartesian squares:
Leonard Guetta's avatar
Leonard Guetta committed
350
351
  \[
    \begin{tikzcd}[row sep = large]
352
      A_0\ar[r] \ar[d] & E \ar[d]&\\
Leonard Guetta's avatar
Leonard Guetta committed
353
354
355
356
357
358
359
360
      A \ar[d,"\alpha"] \ar[r] & G \ar[d] \ar[r] & B \ar[d,"\delta"]\\
      C \ar[r] & H \ar[r] & D. \ar[from=1-1,to=2-2,phantom,"\ulcorner" very near
      end,"\text{\textcircled{\tiny \textbf{1}}}" near start, description]
      \ar[from=2-1,to=3-2,phantom,"\ulcorner" very near
      end,"\text{\textcircled{\tiny \textbf{2}}}", description]
      \ar[from=2-2,to=3-3,phantom,"\ulcorner" very near
      end,"\text{\textcircled{\tiny \textbf{3}}}", description]
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
361
  \]
Leonard Guetta's avatar
Leonard Guetta committed
362
363
364
365
366
367
368
  What we want to prove is that the image by the functor $L$ of the pasting of
  squares \textcircled{\tiny \textbf{2}} and \textcircled{\tiny \textbf{3}} is
  homotopy cocartesian. Since the morphism $G \to B$ is a monomorphism, we
  deduce from Corollary \ref{cor:hmtpysquaregraph} that the image by the functor
  $L$ of square \textcircled{\tiny \textbf{3}} is homotopy cocartesian. Hence,
  in virtue of Lemma \ref{lemma:pastinghmtpycocartesian}, all we have to show is
  that the image by $L$ of square \textcircled{\tiny \textbf{2}} is homotopy
369
  cocartesian. On the other hand, the morphisms
Leonard Guetta's avatar
Leonard Guetta committed
370
  \[
371
    A_0 \to A 
Leonard Guetta's avatar
Leonard Guetta committed
372
  \]
373
374
375
376
377
  and
  \[
    A_0 \to C
  \]
  are monomorphisms and thus, using Corollary
378
379
380
  \ref{cor:hmtpysquaregraph}, we deduce that the image by $L$ of square
  \textcircled{\tiny \textbf{1}} and of the pasting of squares
  \textcircled{\tiny \textbf{1}} and \textcircled{\tiny \textbf{2}}
381
  are homotopy cocartesian. By Lemma \ref{lemma:pastinghmtpycocartesian} again, this proves that the image by $L$ of
382
  square \textcircled{\tiny \textbf{2}} is homotopy cocartesian.
Leonard Guetta's avatar
Leonard Guetta committed
383
\end{proof}
Leonard Guetta's avatar
Leonard Guetta committed
384
385
We now apply Corollary \ref{cor:hmtpysquaregraph} and Proposition
\ref{prop:hmtpysquaregraphbetter} to a few examples.
386
387
\begin{example}[Identifying two objects]\label{example:identifyingobjects}
  Let $C$ be a free category, $A$ and $B$ be two objects of $C$ with $A\neq B$ and let $C'$ be
388
  the category obtained from $C$ by identifying $A$ and $B$, i.e.\ defined by
389
390
391
392
393
  the following cocartesian square
  \[
    \begin{tikzcd}
      \sS_0 \ar[d] \ar[r,"{\langle A,B \rangle}"] & C \ar[d] \\
      \sD_0 \ar[r] & C'.
394
      \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
395
396
397
398
    \end{tikzcd}
  \]
  Then, this square is Thomason homotopy cocartesian. Indeed, it is obviously
  the image by the functor $L$ of a cocartesian square of $\Rgrph$ and the top
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
399
  morphism is a monomorphism. Hence, we can apply Corollary \ref{cor:hmtpysquaregraph}.
400
  \end{example}
Leonard Guetta's avatar
Leonard Guetta committed
401
\begin{example}[Adding a generator]
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
402
  Let $C$ be a free category, $A$ and $B$ two objects of $C$ (possibly equal)
Leonard Guetta's avatar
Leonard Guetta committed
403
  and let $C'$ be the category obtained from $C$ by adding a generator $A \to
404
  B$, i.e.\ defined by the following cocartesian square:
Leonard Guetta's avatar
Leonard Guetta committed
405
  \[
Leonard Guetta's avatar
Leonard Guetta committed
406
    \begin{tikzcd}
407
      \sS_0 \ar[d,"i_1"] \ar[r,"{\langle A, B \rangle}"] & C \ar[d] \\
Leonard Guetta's avatar
Leonard Guetta committed
408
      \sD_1 \ar[r] & C'.
409
      \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
Leonard Guetta's avatar
Leonard Guetta committed
410
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
411
  \]
412
  Then, this square is Thomason homotopy cocartesian. Indeed, it obviously is the image of a square of $\Rgrph$ by
413
414
  the functor $L$ and the morphism $i_1 : \sS_0 \to \sD_1$ comes from
  a monomorphism of $\Rgrph$. Hence, we can apply Corollary
Leonard Guetta's avatar
Leonard Guetta committed
415
  \ref{cor:hmtpysquaregraph}.
Leonard Guetta's avatar
Leonard Guetta committed
416
417
\end{example}
\begin{remark}
418
419
420
  Since $i_1 : \sS_0 \to \sD_1$ is a folk cofibration% , since a Thomason homotopy
  % cocartesian square in $\Cat$ is also so in $\oo\Cat$
  and since every free category is obtained by recursively adding generators
Leonard Guetta's avatar
Leonard Guetta committed
421
  starting from a set of objects (seen as a $0$-category), the previous example
422
  yields another proof that \emph{free} (1\nbd{})categories are \good{} (which we
Leonard Guetta's avatar
Leonard Guetta committed
423
  already knew since we have seen that \emph{all} (1-)categories are \good{}).
Leonard Guetta's avatar
Leonard Guetta committed
424
\end{remark}
Leonard Guetta's avatar
Leonard Guetta committed
425
\begin{example}[Identifying two generators]
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
426
  Let $C$ be a free category and let $f,g : A \to B$ be parallel generating arrows of
Leonard Guetta's avatar
Leonard Guetta committed
427
  $C$ such that $f\neq g$. Now consider the category $C'$ obtained from $C$ by
428
  ``identifying'' $f$ and $g$, i.e. defined by the following cocartesian
Leonard Guetta's avatar
Leonard Guetta committed
429
  square
Leonard Guetta's avatar
Leonard Guetta committed
430
  \[
Leonard Guetta's avatar
Leonard Guetta committed
431
432
433
    \begin{tikzcd}
      \sS_1\ar[d] \ar[r,"{\langle f, g \rangle}"] &C \ar[d] \\
      \sD_1 \ar[r] & C',
434
      \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
Leonard Guetta's avatar
Leonard Guetta committed
435
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
436
  \]
Leonard Guetta's avatar
Leonard Guetta committed
437
438
  where the morphism $\sS_1 \to \sD_1$ is the one that sends the two generating
  arrows of $\sS_1$ to the unique generating arrow of $\sD_1$. Then this square
439
  is Thomason homotopy cocartesian.
Leonard Guetta's avatar
Leonard Guetta committed
440
441
442
443
444
445
446
  Indeed, it is the image by the functor $L$ of a cocartesian square in
  $\Rgrph$, the morphism $\sS_1 \to \sD_1$ is injective on objects and the
  morphism $\sS_1 \to C$ is quasi-injective on arrows. Hence, we can apply
  Proposition \ref{prop:hmtpysquaregraphbetter}. Note that since we did
  \emph{not} suppose that $A\neq B$, the top morphism of the previous square is
  not necessarily a monomorphism and we cannot always apply Corollary
  \ref{cor:hmtpysquaregraph}.
Leonard Guetta's avatar
Leonard Guetta committed
447
\end{example}
Leonard Guetta's avatar
dodo    
Leonard Guetta committed
448
\begin{example}[Killing a generator]\label{example:killinggenerator}
449
  Let $C$ be a free category and let $f : A \to B$ be one of its generating arrows
Leonard Guetta's avatar
Leonard Guetta committed
450
  such that $A \neq B$. Now consider the category $C'$ obtained from $C$ by
451
  ``killing'' $f$, i.e. defined by the following cocartesian square:
Leonard Guetta's avatar
Leonard Guetta committed
452
  \[
Leonard Guetta's avatar
Leonard Guetta committed
453
454
455
    \begin{tikzcd}
      \sD_1 \ar[d] \ar[r,"\langle f \rangle"] & C \ar[d] \\
      \sD_0 \ar[r] & C'.
456
      \ar[from=1-1,to=2-2,phantom,"\ulcorner",very near end]
Leonard Guetta's avatar
Leonard Guetta committed
457
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
458
  \]
459
460
461
462
463
  Then, this above square is Thomason homotopy cocartesian. Indeed, it
  obviously is the image of a cocartesian square in $\Rgrph$ by the
  functor $L$ and since the source and target of $f$ are different,
  the top map comes from a monomorphism of $\Rgrph$. Hence, we can
  apply Corollary \ref{cor:hmtpysquaregraph}.
Leonard Guetta's avatar
Leonard Guetta committed
464
465
\end{example}
\begin{remark}
Leonard Guetta's avatar
Leonard Guetta committed
466
467
468
  Note that in the previous example, we see that it was useful to consider the
  category of reflexive graphs and not only the category of graphs because the
  map $\sD_1 \to \sD_0$ does not come from a morphism in the category of graphs.
Leonard Guetta's avatar
Leonard Guetta committed
469

Leonard Guetta's avatar
Leonard Guetta committed
470
  Note also that the hypothesis that $A\neq B$ was fundamental in the previous
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
471
  example as for $A=B$ the square is \emph{not} Thomason homotopy cocartesian.
Leonard Guetta's avatar
Leonard Guetta committed
472
\end{remark}
Leonard Guetta's avatar
Leonard Guetta committed
473

Leonard Guetta's avatar
Leonard Guetta committed
474
\section{Preliminaries: bisimplicial sets}
Leonard Guetta's avatar
Leonard Guetta committed
475
\begin{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
476
477
478
479
480
481
  A \emph{bisimplicial set} is a presheaf over the category $\Delta \times
  \Delta$,
  \[
    X : \Delta^{\op} \times \Delta^{\op} \to \Set.
  \]
  In a similar fashion as for simplicial sets (\ref{paragr:simpset}), for $n,m
482
  \geq 0$, we use the notations
Leonard Guetta's avatar
Leonard Guetta committed
483
484
485
486
487
488
489
490
491
492
  \begin{align*}
    X_{n,m} &:= X([n],[m]) \\
    \partial_i^h &:=X(\delta^i,\mathrm{id}) : X_{n+1,m} \to X_{n,m}\\
    \partial_j^v &:=X(\mathrm{id},\delta^j) : X_{n,m+1} \to X_{n,m}\\
    s_i^h &:=X(\sigma^i,\mathrm{id}): X_{n,m} \to X_{n+1,m}\\
    s_j^v&:=X(\mathrm{id},\sigma^j) : X_{n,m} \to X_{n,m+1}.
  \end{align*}
  The maps $\partial_i^h$ and $s_i^h$ will be referred to as the
  \emph{horizontal} face and degeneracy operators; and $\partial_i^v$ and
  $s_i^v$ as the \emph{vertical} face and degeneracy operators.
Leonard Guetta's avatar
Leonard Guetta committed
493
 
Leonard Guetta's avatar
Leonard Guetta committed
494
  Note that for every $n\geq 0$, we have simplicial sets
Leonard Guetta's avatar
Leonard Guetta committed
495
  \begin{align*}
Leonard Guetta's avatar
Leonard Guetta committed
496
497
498
499
500
501
502
503
504
    X_{\bullet,n} : \Delta^{\op} &\to \Set \\
    [k] &\mapsto X_{k,n}
  \end{align*}
  and
  \begin{align*}
    X_{n,\bullet} : \Delta^{\op} &\to \Set \\
    [k] &\mapsto X_{n,k}.
  \end{align*}
  The category of bisimplicial sets is denoted by $\Psh{\Delta\times\Delta}$.
Leonard Guetta's avatar
Leonard Guetta committed
505

Leonard Guetta's avatar
Leonard Guetta committed
506
507
  \iffalse Moreover for every $n \geq 0$, if we fix the first variable to $n$,
  we obtain a simplicial set
Leonard Guetta's avatar
Leonard Guetta committed
508
  \begin{align*}
Leonard Guetta's avatar
Leonard Guetta committed
509
    X_{n,\bullet} : \Delta^{\op} &\to \Set \\
Leonard Guetta's avatar
Leonard Guetta committed
510
511
512
513
    [m] &\mapsto X_{n,m}.
  \end{align*}
  Similarly, if we fix the second variable to $n$, we obtain a simplicial
  \begin{align*}
Leonard Guetta's avatar
Leonard Guetta committed
514
    X_{\bullet,n} : \Delta^{\op} &\to \Set \\
Leonard Guetta's avatar
Leonard Guetta committed
515
516
    [m] &\mapsto X_{m,n}.
  \end{align*}
Leonard Guetta's avatar
Leonard Guetta committed
517
  The correspondences
Leonard Guetta's avatar
Leonard Guetta committed
518
  \[
Leonard Guetta's avatar
Leonard Guetta committed
519
    n \mapsto X_{n,\bullet} \,\text{ and }\, n\mapsto X_{\bullet,n}
Leonard Guetta's avatar
Leonard Guetta committed
520
  \]
Leonard Guetta's avatar
Leonard Guetta committed
521
522
  actually define functors $\Delta \to \Psh{\Delta}$. They correspond to the two
  ``currying'' operations
Leonard Guetta's avatar
Leonard Guetta committed
523
  \[
Leonard Guetta's avatar
Leonard Guetta committed
524
    \Psh{\Delta\times\Delta} \to \underline{\Hom}(\Delta^{\op},\Psh{\Delta}),
Leonard Guetta's avatar
Leonard Guetta committed
525
  \]
Leonard Guetta's avatar
Leonard Guetta committed
526
527
528
  which are isomorphisms of categories. In other words, the category of
  bisimplicial sets can be identified with the category of functors
  $\underline{\Hom}(\Delta^{\op},\Psh{\Delta})$ in two canonical ways. \fi
Leonard Guetta's avatar
Leonard Guetta committed
529
530
\end{paragr}
\begin{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
531
532
533
534
535
536
537
  The functor
  \begin{align*}
    \delta : \Delta &\to \Delta\times\Delta \\
    [n] &\mapsto ([n],[n])
  \end{align*}
  induces by pre-composition a functor
  \[
Leonard Guetta's avatar
Leonard Guetta committed
538
539
540
541
542
543
544
    \delta^* : \Psh{\Delta\times\Delta} \to \Psh{\Delta}.
  \]
  By the usual calculus of Kan extensions, $\delta^*$ admits a left adjoint
  $\delta_!$ and a right adjoint $\delta_*$
  \[
    \delta_! \dashv \delta^* \dashv \delta_*.
  \]
545
  We say that a morphism $f : X \to Y$ of bisimplicial sets is a \emph{diagonal
Leonard Guetta's avatar
Leonard Guetta committed
546
    weak equivalence} (resp.\ \emph{diagonal fibration}) when $\delta^*(f)$ is a
547
  weak equivalence (resp.\ fibration) of simplicial sets. By
Leonard Guetta's avatar
Leonard Guetta committed
548
549
550
551
552
553
554
555
  definition, $\delta^*$ induces a morphism of op-prederivators
  \[
    \overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}}) \to
    \Ho(\Psh{\Delta}).
  \]
  Recall from \cite[Proposition 1.2]{moerdijk1989bisimplicial} that the category
  of bisimplicial sets can be equipped with a model structure whose weak
  equivalences are the diagonal weak equivalences and whose fibrations are the
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
556
  diagonal fibrations. We shall refer to this model
Leonard Guetta's avatar
Leonard Guetta committed
557
  structure as the \emph{diagonal model structure}.
Leonard Guetta's avatar
Leonard Guetta committed
558
\end{paragr}
559
\begin{proposition}\label{prop:diageqderivator}
Leonard Guetta's avatar
Leonard Guetta committed
560
561
  Consider that $\Psh{\Delta\times\Delta}$ is equipped with the diagonal model
  structure. Then, the adjunction
Leonard Guetta's avatar
Leonard Guetta committed
562
  \[
Leonard Guetta's avatar
Leonard Guetta committed
563
564
565
    \begin{tikzcd}
      \delta_! : \Psh{\Delta} \ar[r,shift left] & \Psh{\Delta\times\Delta}
      \ar[l,shift left]: \delta^*,
Leonard Guetta's avatar
Leonard Guetta committed
566
567
568
569
570
    \end{tikzcd}
  \]
  is a Quillen equivalence.
\end{proposition}
\begin{proof}
Leonard Guetta's avatar
Leonard Guetta committed
571
572
573
574
  By definition $\delta^*$ preserves weak equivalences and fibrations and thus,
  the adjunction is a Quillen adjunction. The fact that $\delta^*$ induces an
  equivalence at the level of homotopy categories is \cite[Proposition
  1.2]{moerdijk1989bisimplicial}.
Leonard Guetta's avatar
Leonard Guetta committed
575
\end{proof}
Leonard Guetta's avatar
Leonard Guetta committed
576
\begin{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
577
578
579
580
581
582
  In particular, the morphism of op-prederivators
  \[
    \overline{\delta^*} : \Ho(\Psh{\Delta\times \Delta}^{\mathrm{diag}}) \to
    \Ho(\Psh{\Delta})
  \]
  is actually an equivalence of op-prederivators.
Leonard Guetta's avatar
Leonard Guetta committed
583
\end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
584
Diagonal weak equivalences are not the only interesting weak equivalences for
Leonard Guetta's avatar
Leonard Guetta committed
585
bisimplicial sets.
Leonard Guetta's avatar
Leonard Guetta committed
586
\begin{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
587
588
589
  A morphism $f : X \to Y$ of bisimplicial sets is a \emph{vertical (resp.\
    horizontal) weak equivalence} when for every $n \geq 0$, the induced
  morphism of simplicial sets
Leonard Guetta's avatar
Leonard Guetta committed
590
  \[
Leonard Guetta's avatar
Leonard Guetta committed
591
    f_{\bullet,n} : X_{\bullet,n} \to Y_{\bullet,n}
Leonard Guetta's avatar
Leonard Guetta committed
592
593
  \]
  (resp.
Leonard Guetta's avatar
Leonard Guetta committed
594
595
  \[
    f_{n,\bullet} : X_{n,\bullet} \to Y_{n,\bullet})
Leonard Guetta's avatar
Leonard Guetta committed
596
  \]
Leonard Guetta's avatar
Leonard Guetta committed
597
  is a weak equivalence of simplicial sets. Recall now a very useful lemma.
Leonard Guetta's avatar
Leonard Guetta committed
598
599
\end{paragr}
\begin{lemma}\label{bisimpliciallemma}
Leonard Guetta's avatar
Leonard Guetta committed
600
601
  Let $f : X \to Y$ be a morphism of bisimplicial sets. If $f$ is a vertical or
  horizontal weak equivalence then it is a diagonal weak equivalence.
Leonard Guetta's avatar
Leonard Guetta committed
602
603
\end{lemma}
\begin{proof}
Leonard Guetta's avatar
Leonard Guetta committed
604
605
  See for example \cite[Chapter XII,4.3]{bousfield1972homotopy} or
  \cite[Proposition 2.1.7]{cisinski2004localisateur}.
Leonard Guetta's avatar
Leonard Guetta committed
606
607
\end{proof}
\begin{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
608
  In particular, the identity functor of the category of bisimplicial sets
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
609
  induces the morphisms of op-prederivators:
Leonard Guetta's avatar
Leonard Guetta committed
610
611
612
613
  \[
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
  \]
Leonard Guetta's avatar
Leonard Guetta committed
614
  and
Leonard Guetta's avatar
Leonard Guetta committed
615
616
617
618
  \[
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}).
  \]
Leonard Guetta's avatar
Leonard Guetta committed
619
\end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
620
\begin{proposition}\label{prop:bisimplicialcocontinuous}
Leonard Guetta's avatar
Leonard Guetta committed
621
  The morphisms of op-prederivators
Leonard Guetta's avatar
Leonard Guetta committed
622
623
624
625
  \[
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
  \]
Leonard Guetta's avatar
Leonard Guetta committed
626
  and
Leonard Guetta's avatar
Leonard Guetta committed
627
628
629
630
  \[
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to
    \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
  \]
Leonard Guetta's avatar
Leonard Guetta committed
631
632
633
634
635
636
637
  are homotopy cocontinuous.
\end{proposition}
\begin{proof}
  Recall that the category of bisimplicial sets can be equipped with a model
  structure where the weak equivalences are the vertical (resp.\ horizontal)
  weak equivalences and the cofibrations are the monomorphisms (see for example
  \cite[Chapter IV]{goerss2009simplicial} or \cite{cisinski2004localisateur}).
Leonard Guetta's avatar
Leonard Guetta committed
638
639
640
641
642
  We respectively refer to these model structures as the \emph{vertical model
    structure} and \emph{horizontal model structure}. Since the functor
  $\delta^* : \Psh{\Delta\times\Delta} \to \Psh{\Delta}$ preserves
  monomorphisms, it follows from Lemma \ref{bisimpliciallemma} that the
  adjunction
Leonard Guetta's avatar
Leonard Guetta committed
643
644
645
646
  \[
    \begin{tikzcd}
      \delta^* : \Psh{\Delta\times\Delta} \ar[r,shift left] & \ar[l,shift left]
      \Psh{\Delta} : \delta_*
Leonard Guetta's avatar
Leonard Guetta committed
647
648
649
650
651
652
653
654
655
656
657
658
659
660
    \end{tikzcd}
  \]
  is a Quillen adjunction when $\Psh{\Delta\times\Delta}$ is equipped with
  either the vertical model structure or the horizontal model structure. In
  particular, the induced morphisms of op-prederivators
  \[
    \overline{\delta^*} : \Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \to
    \Ho(\Psh{\Delta})
  \]
  and
  \[
    \overline{\delta^*} : \Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \to
    \Ho(\Psh{\Delta})
  \]
Leonard Guetta's avatar
Leonard Guetta committed
661
  are homotopy cocontinuous. Now, the obvious identity
Leonard Guetta's avatar
Leonard Guetta committed
662
  $\delta^*=\delta^* \circ \mathrm{id}_{\Psh{\Delta\times\Delta}}$ implies that
Leonard Guetta's avatar
Leonard Guetta committed
663
  we have commutative triangles
Leonard Guetta's avatar
Leonard Guetta committed
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
  \[
    \begin{tikzcd}
      \Ho(\Psh{\Delta\times\Delta}^{\mathrm{vert}}) \ar[r]
      \ar[rd,"\overline{\delta^*}"']&
      \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
      \ar[d,"\overline{\delta^*}"] \\
      &\Ho(\Psh{\Delta})
    \end{tikzcd}
  \]
  and
  \[
    \begin{tikzcd}
      \Ho(\Psh{\Delta\times\Delta}^{\mathrm{hor}}) \ar[r]
      \ar[rd,"\overline{\delta^*}"']&
      \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}})
      \ar[d,"\overline{\delta^*}"] \\
      &\Ho(\Psh{\Delta}).
    \end{tikzcd}
  \]
  The result follows then from the fact that $\overline{\delta^*} :
  \Ho(\Psh{\Delta\times\Delta}^{\mathrm{diag}}) \to \Ho(\Psh{\Delta})$ is an
  equivalence of op-prederivators.
\end{proof}
In practice, we will use the following corollary.
\begin{corollary}\label{cor:bisimplicialsquare}
  Let
  \[
    \begin{tikzcd}
      A \ar[r,"u"]\ar[d,"f"] & B \ar[d,"g"] \\
      C \ar[r,"v"] & D
    \end{tikzcd}
  \]
696
  be a commutative square in the category of bisimplicial sets satisfying at least one of the two
Leonard Guetta's avatar
Leonard Guetta committed
697
698
699
  following conditions:
  \begin{enumerate}[label=(\alph*)]
  \item For every $n\geq 0$, the square of simplicial sets
Leonard Guetta's avatar
Leonard Guetta committed
700
701
    \[
      \begin{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
702
703
        A_{\bullet,n} \ar[r,"{u_{\bullet,n}}"]\ar[d,"{f_{\bullet,n}}"] & B_{\bullet,n} \ar[d,"{g_{\bullet,n}}"] \\
        C_{\bullet,n} \ar[r,"{v_{\bullet,n}}"] & D_{\bullet,n}
Leonard Guetta's avatar
Leonard Guetta committed
704
705
      \end{tikzcd}
    \]
Leonard Guetta's avatar
Leonard Guetta committed
706
707
708
    is homotopy cocartesian.
  \item For every $n\geq 0$, the square of simplicial sets
    \[
Leonard Guetta's avatar
Leonard Guetta committed
709
      \begin{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
710
711
        A_{n,\bullet} \ar[r,"{u_{n,\bullet}}"]\ar[d,"{f_{n,\bullet}}"] & B_{n,\bullet} \ar[d,"{g_{n,\bullet}}"] \\
        C_{n,\bullet} \ar[r,"{v_{n,\bullet}}"] & D_{n,\bullet}
Leonard Guetta's avatar
Leonard Guetta committed
712
713
      \end{tikzcd}
    \]
Leonard Guetta's avatar
Leonard Guetta committed
714
    is homotopy cocartesian.
Leonard Guetta's avatar
Leonard Guetta committed
715
716
  \end{enumerate}
  Then, the square
Leonard Guetta's avatar
Leonard Guetta committed
717
718
719
720
721
  \[
    \begin{tikzcd}
      \delta^*(A) \ar[r,"\delta^*(u)"]\ar[d,"\delta^*(f)"] & \delta^*(B) \ar[d,"\delta^*(g)"] \\
      \delta^*(C) \ar[r,"\delta^*(v)"] & \delta^*(D)
    \end{tikzcd}
Leonard Guetta's avatar
Leonard Guetta committed
722
723
724
725
  \]
  is a homotopy cocartesian square of simplicial sets.
\end{corollary}
\begin{proof}
Leonard Guetta's avatar
Leonard Guetta committed
726
727
728
  From \cite[Corollary 10.3.10(i)]{groth2013book} we know that the square of
  bisimplicial sets
  \[
Leonard Guetta's avatar
Leonard Guetta committed
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
    \begin{tikzcd}
      A \ar[r,"u"]\ar[d,"f"] & B \ar[d,"g"] \\
      C \ar[r,"v"] & D
    \end{tikzcd}
  \]
  is homotopy cocartesian with respect to the vertical weak equivalences if and
  only if for every $n\geq 0$, the square
  \[
    \begin{tikzcd}
      A_{\bullet,n} \ar[r,"{u_{\bullet,n}}"]\ar[d,"{f_{\bullet,n}}"] & B_{\bullet,n} \ar[d,"{g_{\bullet,n}}"] \\
      C_{\bullet,n} \ar[r,"{v_{\bullet,n}}"] & D_{\bullet,n}
    \end{tikzcd}
  \]
  is a homotopy cocartesian square of simplicial sets and similarly for
  horizontal weak equivalences. The result follows then from Proposition
  \ref{prop:bisimplicialcocontinuous}.
Leonard Guetta's avatar
Leonard Guetta committed
745
\end{proof}
746
\section{Bisimplicial nerve for 2-categories}\label{section:bisimplicialnerve}
Leonard Guetta's avatar
Leonard Guetta committed
747
748
749
We shall now describe a ``nerve'' for $2$-categories with values in bisimplicial
sets and recall a few results that shows that this nerve is, in some sense,
equivalent to the nerve defined in \ref{paragr:nerve}.
Leonard Guetta's avatar
Leonard Guetta committed
750
751
\begin{notation}
  \begin{itemize}
Leonard Guetta's avatar
Leonard Guetta committed
752
  \item[-] Once again, we write $N : \Cat \to \Psh{\Delta}$ instead of $N_1$ for
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
753
    the usual nerve of categories. Moreover, using the usual notation for the
Leonard Guetta's avatar
Leonard Guetta committed
754
    set of $k$-simplices of a simplicial set, if $C$ is a (small) category, then
Leonard Guetta's avatar
Leonard Guetta committed
755
    \[
Leonard Guetta's avatar
Leonard Guetta committed
756
      N(C)_k
Leonard Guetta's avatar
Leonard Guetta committed
757
758
    \]
    is the set of $k$-simplices of the nerve of $C$.
Leonard Guetta's avatar
Leonard Guetta committed
759
760
761
762
763
764
  \item[-] Similarly, we write $N : 2\Cat \to \Psh{\Delta}$ instead of $N_2$ for
    the nerve of $2$-categories. This makes sense since the nerve for categories
    is the restriction of the nerve for $2$-categories.
  \item[-] For $2$-categories, we refer to the $\comp_0$-composition of
    $2$-cells as the \emph{horizontal composition} and the $\comp_1$-composition
    of $2$-cells as the \emph{vertical composition}.
Leonard Guetta's avatar
Leonard Guetta committed
765
766
  \item[-] For a $2$-category $C$ and $x$ and $y$ objects of $C$, we denote by
    \[
Leonard Guetta's avatar
Leonard Guetta committed
767
      C(x,y)
Leonard Guetta's avatar
Leonard Guetta committed
768
    \]
Leonard Guetta's avatar
Leonard Guetta committed
769
770
771
772
    the category whose objects are the $1$-cells of $C$ with $x$ as source and
    $y$ as target, and whose arrows are the $2$-cells of $C$ with $x$ as
    $0$-source and $y$ as $0$-target. Composition is induced by vertical
    composition in $C$.
Leonard Guetta's avatar
Leonard Guetta committed
773
774
775
  \end{itemize}
\end{notation}
\begin{paragr}
Leonard Guetta's avatar
OUF    
Leonard Guetta committed
776
  Every $2$-category $C$ defines a simplicial object in $\Cat$,
Leonard Guetta's avatar
Leonard Guetta committed
777
  \[S(C): \Delta^{\op} \to \Cat,\] where, for each $n \geq 0$,
Leonard Guetta's avatar
Leonard Guetta committed
778
  \[
Leonard Guetta's avatar
Leonard Guetta committed
779
    S_n(C):= \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}C(x_0,x_1)
Leonard Guetta's avatar
Leonard Guetta committed
780
781
    \times \cdots \times C(x_{n-1},x_n).
  \]
782
783
  Note that for $n=0$, the above formula reads $S_0(C)=C_0$. For $n>0$, the face operators  $\partial_i : S_{n}(C) \to S_{n-1}(C)$ are induced by horizontal
  composition for $0 < i <n$ and by projection for $i=0$ or $n$. The degeneracy operators $s_i \colon S_{n}(C) \to S_{n+1}(C)$ are
Leonard Guetta's avatar
Leonard Guetta committed
784
  induced by the units for the horizontal composition.
Leonard Guetta's avatar
Leonard Guetta committed
785

Leonard Guetta's avatar
Leonard Guetta committed
786
  Post-composing $S(C)$ with the nerve functor $N : \Cat \to \Psh{\Delta}$, we
Leonard Guetta's avatar
Leonard Guetta committed
787
  obtain a functor
Leonard Guetta's avatar
Leonard Guetta committed
788
  \[
Leonard Guetta's avatar
Leonard Guetta committed
789
    NS(C) : \Delta^{\op} \to \Psh{\Delta}.
Leonard Guetta's avatar
Leonard Guetta committed
790
791
792
  \]
\end{paragr}
\begin{remark}
Leonard Guetta's avatar
Leonard Guetta committed
793
794
  When $C$ is a $1$-category, the simplicial object $S(C)$ is nothing but the
  usual nerve of $C$ where, for each $n\geq 0$, $S_n(C)$ is seen as a discrete
Leonard Guetta's avatar
Leonard Guetta committed
795
796
  category.
\end{remark}
Leonard Guetta's avatar
Leonard Guetta committed
797
\begin{definition}
Leonard Guetta's avatar
Leonard Guetta committed
798
799
  The \emph{bisimplicial nerve} of a $2$-category $C$ is the bisimplicial set
  $\binerve(C)$ defined as
Leonard Guetta's avatar
Leonard Guetta committed
800
  \[
Leonard Guetta's avatar
Leonard Guetta committed
801
    \binerve(C)_{n,m}:=N(S_n(C))_m,
Leonard Guetta's avatar
Leonard Guetta committed
802
  \]
Leonard Guetta's avatar
Leonard Guetta committed
803
  for all $n,m \geq 0$.
Leonard Guetta's avatar
Leonard Guetta committed
804
\end{definition}
805
\begin{paragr}\label{paragr:formulabisimplicialnerve}
806
  In other words, the bisimplicial nerve of $C$ is obtained by ``un-currying''
Leonard Guetta's avatar
Leonard Guetta committed
807
  the functor $NS(C) : \Delta^{op} \to \Psh{\Delta}$.
808
  
Leonard Guetta's avatar
Leonard Guetta committed
809
  Since the nerve $N$ commutes with products and sums, we obtain the formula
Leonard Guetta's avatar
Leonard Guetta committed
810
  \begin{equation}\label{fomulabinerve}
Leonard Guetta's avatar
Leonard Guetta committed
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
    \binerve(C)_{n,m} = \coprod_{(x_0,\cdots,x_n)\in \Ob(C)^{\times (n+1)}}N(C(x_0,x_1))_m \times \cdots \times N(C(x_{n-1},x_n))_m.
  \end{equation}
  More intuitively, an element of $\binerve(C)_{n,m}$ consists of a ``pasting
  scheme'' in $C$ that looks like
  \[
    m \underbrace{\left\{\begin{tikzcd}[column sep=huge,ampersand
          replacement=\&] \bullet \ar[r,"\vdots"]\ar[r,bend
          left=90,looseness=1.4,""{name=A,below}] \ar[r,bend
          left=35,""{name=B,above}] \ar[r,bend
          right=35,"\vdots",""{name=G,below}]\ar[r,bend
          right=90,looseness=1.4,""{name=H,above}] \&
          \bullet\ar[r,"\vdots"]\ar[r,bend
          left=90,looseness=1.4,""{name=C,below}] \ar[r,bend
          left=35,""{name=D,above}] \ar[r,bend
          right=35,"\vdots",""{name=I,below}]\ar[r,bend
          right=90,looseness=1.4,""{name=J,above}]
          \&\bullet\ar[r,phantom,description,"\cdots"]\&\bullet\ar[r,"\vdots"]\ar[r,bend
          left=90,looseness=1.4,""{name=E,below}] \ar[r,bend
          left=35,""{name=F,above}] \ar[r,bend
          right=35,"\vdots",""{name=K,below}]\ar[r,bend
          right=90,looseness=1.4,""{name=L,above}] \&\bullet
          \ar[from=A,to=B,Rightarrow] \ar[from=C,to=D,Rightarrow]
          \ar[from=E,to=F,Rightarrow] \ar[from=G,to=H,Rightarrow]
          \ar[from=I,to=J,Rightarrow] \ar[from=K,to=L,Rightarrow]
        \end{tikzcd}\right.}_{ n }.
  \]
Leonard Guetta's avatar
Leonard Guetta committed
837
\end{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
838
In the definition of the bisimplicial nerve of a $2$\nbd{}category we gave, one
Leonard Guetta's avatar
Leonard Guetta committed
839
direction of the bisimplicial set is privileged over the other. We now give
Leonard Guetta's avatar
Leonard Guetta committed
840
an equivalent definition of the bisimplicial nerve which uses the other direction.
Leonard Guetta's avatar
Leonard Guetta committed
841
\begin{paragr}
Leonard Guetta's avatar
Leonard Guetta committed
842
  Let $C$ be a $2$\nbd{}category. For every $k \geq 1$, we define a
Leonard Guetta's avatar
Leonard Guetta committed
843
  $1$\nbd{}category $V_k(C)$ in the following fashion:
Leonard Guetta's avatar
Leonard Guetta committed
844
  \begin{itemize}[label=-]
Leonard Guetta's avatar
Leonard Guetta committed
845
  \item The objects of $V_k(C)$ are the objects of $C$.
Leonard Guetta's avatar
Leonard Guetta committed
846
847
  \item A morphism $\alpha$ is a sequence
    \[
Leonard Guetta's avatar
Leonard Guetta committed
848
      \alpha=(\alpha_1,\alpha_2,\cdots,\alpha_k)
Leonard Guetta's avatar
Leonard Guetta committed
849
    \]
Leonard Guetta's avatar
Leonard Guetta committed
850
851
    of vertically composable $2$-cells of $C$, i.e.\ such that for
    every $1 \leq i \leq k-1$, we have
Leonard Guetta's avatar
Leonard Guetta committed
852
    \[
Leonard Guetta's avatar
Leonard Guetta committed
853
      \src(\alpha_i)=\trgt(\alpha_{i+1}).
Leonard Guetta's avatar
Leonard Guetta committed
854
    \]
855
    The source and target of $\alpha$ are given by
Leonard Guetta's avatar
Leonard Guetta committed
856
    \[
Leonard Guetta's avatar
Leonard Guetta committed
857
858
      \src(\alpha):=\src_0(\alpha_1)\text{ and
      }\trgt(\alpha):=\trgt_0(\alpha_1).
Leonard Guetta's avatar
Leonard Guetta committed
859
    \]
Leonard Guetta's avatar
Leonard Guetta committed
860
861
    (Note that we could have used any of the $\alpha_i$ instead of $\alpha_1$
    since they all have the same $0$\nbd{}source and $0$\nbd{}target.)
Leonard Guetta's avatar
Leonard Guetta committed
862
863
  \item Composition is given by
    \[
Leonard Guetta's avatar
Leonard Guetta committed
864
      (\alpha_1,\alpha_2,\cdots,\alpha_k)\circ(\beta_1,\beta_2,\cdots,\beta_k):=(\alpha_1\comp_0\beta_1,\alpha_2\comp_0\beta_2,\cdots,\alpha_k\comp_0\beta_k)
Leonard Guetta's avatar
Leonard Guetta committed
865
866
867
    \]
    and the unit on an object $x$ is the sequence
    \[
Leonard Guetta's avatar
Leonard Guetta committed
868
      (1^2_x,\cdots, 1^2_x).