Combinatorial Descent Data for Gerbes

0.56 MB·

by  Amnon Yekutieli

COMBINATORIAL DESCENT DATA FOR GERBES
AMNONYEKUTIELI
Abstract. Weconsiderdescentdataincosimplicialcrossedgroupoids. This
isacombinatorialabstractionofthedescentdataforgerbesinalgebraicgeom-
etry. Themainresultisthis: aweakequivalencebetweencosimplicialcrossed
groupoidsinducesabijectionongaugeequivalenceclassesofdescentdata.
2
1
0
0. Introduction
2
n For a cosimplicial crossed groupoid G={Gp}p∈N we denote by Desc(G) the set
a of gauge equivalence classes of descent data. The purpose of this note is to prove:
J

5 Theorem 0.1. Let F :G→H be a weak equivalence between cosimplicial crossed
1 groupoids. Then the function

Desc(F):Desc(G)→Desc(H)
]
T
is bijective.
K
. The various notions involved are recalled or defined in Section 1. The theorem
h
is repeated as Theorem 2.4 in Section 2, and proved there. Connections with other
t
a papers, and several remarks, are in Section 3.
m
Theorem0.1playsacrucialroleinthenewversionofourpaper[Ye1]ontwisted
[ deformation quantization of algebraic varieties. This is explained in Remark 3.2.

2 Acknowledgments. I wish to thank Matan Prezma, Ronald Brown, Sharon Hol-
v lander,VladimirHinich,BehrangNoohiandLawrenceBreenforusefuldiscussions.
9
1 1. Combinatorial Descent Data
9
1 We begin with a quick review of cosimplicial theory. Let ∆ denote the simplex
9. category. The set of objects of ∆ is the set N of natural numbers. Given p,q ∈N,
0 the morphisms α:p→q in ∆ are order preserving functions
1
α:{0,...,p}→{0,...,q}.
1
v: We denote this set of morphisms by ∆qp. An element of ∆qp may be thought of
i as a sequence i = (i ,...,i ) of integers with 0 ≤ i ≤ ··· ≤ i ≤ q. We call
X 0 p 0 p
∆q := {∆qp}p∈N the q-dimensional combinatorial simplex, and an element i ∈ ∆qp
ar is a p-dimensional face of ∆q.
Let C be some category. A cosimplicial object in C is a functor C : ∆ → C.
We shall usually write Cp := C(p) ∈ Ob(C), and leave the morphisms C(α) :
C(p) → C(q), for α ∈ ∆q, implicit. Thus we shall refer to the cosimplicial object
p
C as {Cp}p∈N. The category of cosimplicial objects in C, where the morphisms are
natural transformations of functors ∆→C, is denoted by ∆(C).
If C is a category of sets with structure (i.e. there is a faithful functor C→Set),
then an object C ∈ Ob(C) has elements c ∈ Cp. Let {Cp}p∈N be a cosimplicial
Date:15Jan2012.
Key words and phrases. Cosimplicialcrossedgroupoids,descent,gerbes.
Mathematics Subject Classification2000. Primary: 18G50;Secondary: 18G30,20L05.
ThisresearchwassupportedbytheIsraelScienceFoundation.
1
2 AMNONYEKUTIELI
object of C. Given a face i ∈ ∆q and an element c ∈ Cp, it will be convenient to
p
write
(1.1) c :=C(i)(c)∈Cq.
i
Thepicturetokeepinmindisof“theelementcpushedtothefaceiofthesimplex
∆q”. See Figure 1 for an illustration.
For a groupoid G and x ∈ Ob(G) we write G(x) := G(x,x), the automorphism
group of x. Suppose N is another groupoid, such that Ob(N)=Ob(G). An action
’
Ψ of G on N is a collection of group isomorphisms Ψ(g) : N(x) −→ N(y) for all
x,y ∈Ob(G) and g ∈G(x,y), such that Ψ(h◦g)=Ψ(h)◦Ψ(g) whenever g and h
arecomposable,andΨ(1 )=1 . Theprototypicalexampleistheadjointaction
x N(x)
Ad of G on itself, namely
G
Ad (g)(h):=g◦h◦g−1.
G
Definition 1.2. A crossed groupoid is a structure
G=(G1,G2,AdG1(cid:121)G2,D)
consisting of:
• Groupoids G and G , such that G is totally disconnected, and Ob(G )=
1 2 2 1
Ob(G ). We write Ob(G):=Ob(G ).
2 1
• An action AdG1(cid:121)G2 of G1 on G2, called the twisting.
• A morphism of groupoids (i.e. a functor) D:G →G called the feedback,
2 1
which is the identity on objects.
These are the conditions:
(i) The morphism D is G1-equivariant with respect to the actions AdG1(cid:121)G2
and Ad . Namely
G1
D(AdG1(cid:121)G2(g)(a))=AdG1(g)(D(a))
in the group G (y), for any x,y ∈Ob(G), g ∈G (x,y) and a∈G (x).
1 1 2
(ii) For any x∈Ob(G) and a∈G (x) there is equality
2
AdG1(cid:121)G2(D(a))=AdG2(x)(a),
as automorphisms of the group G (x).
2
We sometimes refer to the morphisms in the groupoid G as 1-morphisms, and
1
to the morphisms in G as 2-morphisms.
2
Remark 1.3. A crossed groupoid is better known as a crossed module over a
groupoid, orasa2-truncated crossed complex; see[Bw]. WhenOb(G)isasingleton
then G is just a crossed module (or a crossed group). More on this in Remark 3.3.
Suppose H = (H1,H2,AdH1(cid:121)H2,D) is another crossed groupoid. A morphism
of crossed groupoids F : G → H is a pair of morphisms of groupoids F : G →
i
H , i = 1,2, that respect the twistings and the feedbacks. We denote by CrGrpd
i
the category consisting of crossed groupoids and morphisms between them. (It is
isomorphic to the category 2Grpd of strict 2-groupoids.)
We shall be interested in cosimplicial crossed groupoids, i.e. in objects of the
category ∆(CrGrpd). A cosimplicial crossed groupoid G = {Gp}p∈N has in each
simplicial dimension p a 2-groupoid Gp. The morphisms G(i) : Gp → Gq, for
i∈∆q, are implicit, and we use notation (1.1).
p
Letusfixp∈N. Thenforanyx∈Ob(Gp)thereisagrouphomomorphism(the
feedback)
D:Gp(x)→Gp(x).
2 1
COMBINATORIAL DESCENT DATA 3
Figure 1. Illustration of a combinatorial descent datum (x,g,a)
in the cosimplicial crossed groupoid G={Gp}p∈N.
Andforeverymorphismg :x→yinGpthereisagroupisomorphism(thetwisting)
1
Ad(g)=AdGp(cid:121)Gp(g):Gp2(x)→Gp2(y).
1 2
NotethatweareusingtheexpressionAd(g)tomeanbothAdGp(cid:121)Gp(g)andAdGp(g);
1 2 1
hopefully that will not cause confusion.
Definition 1.4. Let G={Gp}p∈N be a cosimplicial crossed groupoid. A combina-
torial descent datum in G is a triple (x,g,a) of elements of the following sorts:
(0) x∈Ob(G0).
(1) g ∈G1(x ,x ), where x ,x ∈Ob(G1) are the objects corresponding
1 (0) (1) (0) (1)
to the vertices (0) and (1) of ∆1.
(2) a∈G2(x ),wherex ∈Ob(G2)istheobjectcorrespondingtothevertex
2 (0) (0)
(0) of ∆2.
The conditions are as follows:
(i) (Failure of 1-cocycle)
g−1 ◦g ◦g =D(a)
(0,2) (1,2) (0,1)
in the group G2(x ). Here x ∈Ob(G2) and g ∈G2(x ,x ) corre-
1 (0) (i) (i,j) 1 (i) (j)
spond to the faces (i) and (i,j) respectively of ∆2.
(ii) (Twisted 2-cocycle)
a−1 ◦a ◦a =Ad(g−1 )(a )
(0,1,3) (0,2,3) (0,1,2) (0,1) (1,2,3)
in the group G3(x ). Here x ∈ Ob(G3), g ∈ G3(x ,x ) and
2 (0) (i) (i,j) 1 (i) (j)
a ∈G3(x ) correspond to the faces (i), (i,j) and (i,j,k) respectively
(i,j,k) 2 (i)
of ∆3.
We denote by Desc(G) the set of all descent data in G.
See Figure 1 for an illustration.
Definition 1.5. Let (x,g,a) and (x0,g0,a0) be descent data in the cosimplicial
crossedgroupoidG. Agaugetransformation(x,g,a)→(x0,g0,a0)isapair(f,c)of
elements of the following sorts:
(0) f ∈G0(x,x0).
1
4 AMNONYEKUTIELI
Figure 2. Illustration of a gauge transformation (f,c) :
(x,g,a)→(x0,g0,a0) between descent data.
(1) c∈G1(x ),wherex ∈Ob(G1)istheobjectcorrespondingtothevertex
2 (0) (0)
(0) of ∆1.
These two conditions must hold:
(i)
g0 =f ◦g◦D(c)◦f−1
(1) (0)
in the set G1(x0 ,x0 ).
1 (0) (1)
(ii)
(cid:16) (cid:17)
a0 =Ad(f ) c−1 ◦a◦Ad(g−1 )(c )◦c
(0) (0,2) (0,1) (1,2) (0,1)
in the group G2(x0 ).
2 (0)
This is illustrated in Figure 2.
Proposition 1.6. Let G be a cosimplicial crossed groupoid. The gauge transfor-
mations form an equivalence relation on the set Desc(G).
We call this relation gauge equivalence.
Proof. Givenadescentdatum(x,g,a),thepair(1 ,1 )isagaugetransformation
x 1x
from (x,g,a) to itself. Next let
(f,c):(x,g,a)→(x0,g0,a0)
and
(f0,c0):(x0,g0,a0)→(x00,g00,a00)
be gauge transformations between descent data. Then
(cid:0)f0◦f, c◦f−1(c0)(cid:1):(x,g,a)→(x00,g00,a00)
(0)
is a gauge transformation. And
(cid:0)f−1,f (c−1)(cid:1):(x0,g0,a0)→(x,g,a)
(0)
is a gauge transformation. (cid:3)
COMBINATORIAL DESCENT DATA 5
LetF :G→Hbeamorphismofcosimplicialcrossedgroupoids. Givenadescent
datum (x,g,a)∈Desc(G), the triple
F(x,g,a):=(F(x),F(g),F(a))
is a descent datum in H. The resulting function
Desc(F):Desc(G)→Desc(H)
respects the gauge equivalence relations.
Definition 1.7. For a cosimplicial crossed groupoid G we write
Desc(G)
Desc(G):= .
gauge equivalence
For a morphism F :G→H of cosimplicial crossed groupoids, we denote by
Desc(F):Desc(G)→Desc(H)
the induced function.
2. The Main Theorem
RecallthatforagroupoidG,thesetofisomorphismclassesofobjectsisdenoted
by π (G).
0
Definition 2.1. Let G = (G1,G2,AdG1(cid:121)G2,D) be a crossed groupoid. We define
the homotopy set
π (G):=π (G ),
0 0 1
and the homotopy groups
(cid:0) (cid:1)
π (G,x):=Coker D:G (x)→G (x)
1 2 1
and
(cid:0) (cid:1)
π (G,x):=Ker D:G (x)→G (x)
2 2 1
for x∈Ob(G).
The set π (G) and the groups π (G,x) are functorial in G. The group π (G,x)
0 i 2
is central in G (x), and in particular it is abelian.
2
Definition 2.2. A morphism of crossed groupoids F : G → H is called a weak
equivalence if the function
π (F):π (G)→π (H)
0 0 0
is bijective, and the group homomorphisms
π (F,x):π (G,x)→π (H,F(x))
i i i
are bijective for all x∈Ob(G) and i∈{1,2}.
Definition 2.3. A morphism F : G → H of cosimplicial crossed groupoids is
calledaweak equivalenceifineverysimplicialdimensionpthemorphismofcrossed
groupoids Fp :Gp →Hp is a weak equivalence.
Theorem 2.4. Let F :G→H be a weak equivalence between cosimplicial crossed
groupoids. Then the function
Desc(F):Desc(G)→Desc(H)
from Definition 1.7 is bijective.
Weneedacoupleofauxiliaryresultsfirst. ApartialdescentdatuminGisapair
(x,g) of elements x∈Ob(G0) and g ∈G1(x ,x ) (cf. Definition 1.4). Let (x,g)
1 (0) (1)
and(x0,g0)bepartialdescentdata. Apartialgaugetransformation(x,g)→(x0,g0)
is a pair (f,c) of elements as in Definition 1.5, that satisfies condition (i) of that
definition.
6 AMNONYEKUTIELI
Lemma 2.5. Let (x,g,a) be a descent datum in the cosimplicial crossed groupoid
G, let(x0,g0)beapartialdescentdatuminG, andlet(f,c)beapartialgaugetrans-
formation (x,g)→(x0,g0). Then there is a unique element a0 ∈G2(x0 ) such that
2 (0)
the triple (x0,g0,a0) is a descent datum in G, and (f,c) is a gauge transformation
(x,g,a)→(x0,g0,a0).
Proof. Define
(cid:16) (cid:17)
a0 :=Ad(f ) c−1 ◦a◦Ad(g−1 )(c )◦c ∈G2(x0 ).
(0) (0,2) (0,1) (1,2) (0,1) 2 (0)
Then a0 satisfies condition (ii) of Definition 1.5, and moreover it is unique.
We have to show that the triple (x0,g0,a0) is a descent datum. Let us check
condition (i) of Definition 1.4. We have
(g0 )−1◦g0 ◦g0
(0,2) (1,2) (0,1)
=(cid:77) (cid:16)f ◦g ◦D(c )◦f−1(cid:17)−1◦(cid:16)f ◦g ◦D(c )◦f−1(cid:17)
(2) (0,2) (0,2) (0) (2) (1,2) (1,2) (1)
(cid:16) (cid:17)
◦ f ◦g ◦D(c )◦f−1
(1) (0,1) (0,1) (0)
=♦ f ◦D(c )−1◦g−1 ◦g ◦D(c )◦g ◦D(c )◦f−1
(0) (0,2) (0,2) (1,2) (1,2) (0,1) (0,1) (0)
(cid:16) (cid:17)
=♥ Ad(f ) D(c−1 )◦D(a)◦g−1 ◦D(c )◦g ◦D(c )
(0) (0,2) (0,1) (1,2) (0,1) (0,1)
(cid:16) (cid:17)
=? Ad(f ) D(c−1 )◦D(a)◦Ad(g−1 )(D(c ))◦D(c )
(0) (0,2) (0,1) (1,2) (0,1)
=(cid:3) D(cid:16)Ad(f )(cid:16)c−1 ◦a◦Ad(g−1 )(c )◦c (cid:17)(cid:17)=(cid:79) D(a0).
(0) (0,2) (0,1) (1,2) (0,1)
(cid:77)
The equality marked = is true because of condition (i) of Definition 1.5, applied
♦
to the elements g0 . The equality marked = is true because of cancellation. The
(i,j)
♥
equality marked = is because condition (i) of Definition 1.4 holds for (x,g,a),
?
and by the definition of Ad(f ). The equality marked = is by the definition of
(0)
Ad(g−1 ). Theequalitymarked=(cid:3) isbecauseDisG2-equivariant(thisiscondition
(0,1) 1
(cid:79)
(i) of Definition 1.2). And the equality marked = holds by definition of a0.
Finallywehavetocheckthatcondition(ii)ofDefinition1.4holdsfor(x0,g0,a0).
Namely, letting
(2.6) u0 :=(a0 )−1◦a0 ◦a0 ◦Ad(cid:0)(g0 )−1(cid:1)(a0 )−1,
(0,1,3) (0,2,3) (0,1,2) (0,1) (1,2,3)
we have to show that u0 =1.
From the definition of a0 we get
Ad(cid:0)(g0 )−1(cid:1)(a0 )
(0,1) (1,2,3)
=Ad(cid:0)(g0 )−1◦f (cid:1)(cid:16)c−1 ◦a ◦Ad(g−1 )(c )◦c (cid:17)
(0,1) (1) (1,3) (1,2,3) (1,2) (2,3) (1,2)
=♥ Ad(cid:0)f ◦D(c−1 )◦(g )−1(cid:1)
(0) (0,1) (0,1)
(cid:16) (cid:17)
c−1 ◦a ◦Ad(g−1 )(c )◦c
(1,3) (1,2,3) (1,2) (2,3) (1,2)
(2.7)
=♦ (cid:0)Ad(f )◦Ad(D(c−1 ))◦Ad(g )−1(cid:1)
(0) (0,1) (0,1)
(cid:16) (cid:17)
c−1 ◦a ◦Ad(g−1 )(c )◦c
(1,3) (1,2,3) (1,2) (2,3) (1,2)
=(cid:3) Ad(cid:0)f (cid:1)(cid:16)c−1 ◦Ad(cid:0)g−1 (cid:1)(c−1 )◦Ad(cid:0)g−1 (cid:1)(a )
(0) (0,1) (0,1) (1,3) (0,1) (1,2,3)
◦Ad(cid:0)g−1 ◦g−1 (cid:1)(c )◦Ad(cid:0)g−1 (cid:1)(c )◦c (cid:17).
(0,1) (1,2) (2,3) (0,1) (1,2) (0,1)
COMBINATORIAL DESCENT DATA 7
♥
The equality marked = is true because
(g0 )−1◦f =f ◦D(c−1 )◦(g )−1;
(0,1) (1) (0) (0,1) (0,1)
♦
this is from condition (i) of Definition 1.5. The equality marked = is because Ad
is a group homomorphism. And =(cid:3) is because Ad(D(c−1 )) = Ad(c−1 ), which is
(0,1) (0,1)
an instance of condition (ii) of Definition 1.2.
A consequence of condition (ii) of Definition 1.4 and condition (ii) of Definition
1.2 is that
a−1 ◦c◦a =Ad(a−1 )(c)
(0,1,2) (0,1,2) (0,1,2)
=Ad(D(a−1 ))(c)=Ad(g−1 ◦g−1 ◦g )(c)
(0,1,2) (0,1) (1,2) (0,2)
for any c∈G2(x ). Therefore, taking c:=Ad(g−1 )(c ), we get
2 (0) (0,2) (2,3)
(2.8) Ad(g−1 )(c )◦a =a ◦Ad(g−1 ◦g−1 )(c ).
(0,2) (2,3) (0,1,2) (0,1,2) (0,1) (1,2) (2,3)
By the definition of a0 and by formula (2.7) we have
(cid:16) (cid:17)−1
u0 =Ad(f ) c−1 ◦a ◦Ad(g−1 )(c )◦c
(0) (0,3) (0,1,3) (0,1) (1,3) (0,1)
(cid:16) (cid:17)
◦Ad(f ) c−1 ◦a ◦Ad(g−1 )(c )◦c
(0) (0,3) (0,2,3) (0,2) (2,3) (0,2)
(cid:16) (cid:17)
◦Ad(f ) c−1 ◦a ◦Ad(g−1 )(c )◦c
(0) (0,2) (0,1,2) (0,1) (1,2) (0,1)
◦Ad(cid:0)f (cid:1)(cid:16)c−1 ◦Ad(cid:0)g−1 (cid:1)(c−1 )◦Ad(cid:0)g−1 (cid:1)(a )
(0) (0,1) (0,1) (1,3) (0,1) (1,2,3)
◦Ad(cid:0)g−1 ◦g−1 (cid:1)(c )◦Ad(cid:0)g−1 (cid:1)(c )◦c (cid:17)−1 .
(0,1) (1,2) (2,3) (0,1) (1,2) (0,1)
Canceling adjacent inverse terms we get
u0 =Ad(f )(cid:16)c−1 ◦Ad(g−1 )(c−1 )◦v0◦Ad(cid:0)g−1 (cid:1)(c )◦c (cid:17) ,
(0) (0,1) (0,1) (1,3) (0,1) (1,3) (0,1)
where
v0 :=a−1 ◦a ◦Ad(g−1 ◦g−1 )(c )◦a
(0,1,3) (0,2,3) (0,1) (1,2) (2,3) (0,1,2)
◦Ad(cid:0)g−1 ◦g−1 (cid:1)(c−1 )◦Ad(cid:0)g−1 (cid:1)(a−1 ) .
(0,1) (1,2) (2,3) (0,1) (1,2,3)
It suffices to prove that v0 =1. Using formula (2.8) we have
v0 =a−1 ◦a ◦a ◦Ad(g−1 ◦g−1 )(c )
(0,1,3) (0,2,3) (0,1,2) (0,1) (1,2) (2,3)
◦Ad(cid:0)g−1 ◦g−1 (cid:1)(c−1 )◦Ad(cid:0)g−1 (cid:1)(a−1 ) .
(0,1) (1,2) (2,3) (0,1) (1,2,3)
We now cancel two adjacent inverse terms, and use the fact that condition (ii) of
Definition 1.4 holds for (x,g,a), to conclude that v0 =1. (cid:3)
Suppose G is a crossed groupoid and x,x0 ∈ Ob(G). There is a right action of
the group G (x) on the set G (x,x0), namely g 7→ g◦D(a) for g ∈ G (x,x0) and
2 1 1
a∈G (x). The quotient set is
2
(2.9) π (G,x,x0):=G (x,x0)/G (x).
1 1 2
Given g,g0 ∈G (x,x0) let us define
1
(2.10) G (x)(g,g0):={a∈G (x)|g0 =g◦D(a)}.
2 2
Soπ (G,x,x)=π (G,x)andG (x)(1 ,1 )=π (G,x)inthenotationofDefinition
1 1 2 x x 2
2.1.
8 AMNONYEKUTIELI
Lemma 2.11. Let F : G → H be a weak equivalence between crossed groupoids.
Then the induced functions
π (F,x,x0):π (G,x,x0)→π (cid:0)H,F(x),F(x0)(cid:1)
1 1 1
and
F :G (x)(g,g0)→H (cid:0)F(x)(cid:1)(cid:0)F(g),F(g0)(cid:1)
2 2
are bijective for all x,x0 ∈Ob(G) and f,f0 ∈G (x,x0).
1
Proof. This is the same as the usual proof for 2-groupoids (cf. [MS, Lemma 1.1]).
(cid:3)
Proof of Theorem 2.4. Theproofisa“nonabeliandiagramchasing”,madepossible
by Lemma 2.5.
We begin by proving that the function Desc(F) is surjective. Given a descent
datum (y,h,b) ∈ Desc(H), we have to find a descent datum (x,g,a) ∈ Desc(G),
and a gauge transformation (f,c):(y,h,b)→F(x,g,a) in H.
Since the function π (F0) : π (G0) → π (H0) is surjective, there is an object
0 0 0
x∈Ob(G0), and a 1-morphism f ∈H0(y,y0), where y0 :=F(x)∈Ob(H0). Define
1
h00 :=f ◦h◦f−1 ∈H0(y0 ,y0 )
(1) (0) 1 (0) (1)
and c00 := 1 ∈ H1(y ). Then (y0,h00) is a partial descent datum in H, and
1y(0) 2 (0)
(f,c00) : (y,h) → (y0,h00) is a partial gauge transformation. According to Lemma
2.5thereisauniqueelementb00 ∈H2(y0 )suchthat(y0,h00,b00)isadescentdatum
2 (0)
in H, and (f,c00):(y,h,b)→(y0,h00,b00) is a gauge transformation.
Now by Lemma 2.11 the function
π (F1,x ,x ):π (G1,x ,x )→π (H1,y0 ,y0 )
1 (0) (1) 1 (0) (1) 1 (0) (1)
is surjective. Hence there are elements g ∈ G1(x ,x ) and c0 ∈ H1(y0 ) such
1 (0) (1) 2 (0)
that, letting h0 := F(g) ∈ H1(y0 ,y0 ), we have h00 = h0 ◦ D(c0). Consider
1 (0) (1)
the partial gauge transformation (1 ,c0) : (y0,h00) → (y0,h0). Lemma 2.5 there
y0
is a unique element b0 ∈ H2(y0 ) such that (y0,h0,b0) is a descent datum in
2 (0)
H, and (1 ,c0) : (y0,h00,b00) → (y0,h0,b0) is a gauge transformation. Let c :=
y0
Ad(f−1)(c0)−1 ∈H1(y ). Then
(0) 2 (0)
(f,c):(y,h,b)→(y0,h0,b0)
is a gauge transformation in H, and (y0,h0)=F(x,g).
By Lemma 2.11 the function
F2 :G2(x )(cid:0)1 ,g−1 ◦g ◦g (cid:1)
2 (0) x(0) (0,2) (1,2) (0,1)
→H2(y0 )(cid:0)1 ,(h0 )−1◦h0 ◦h0 (cid:1)
2 (0) y0 (0,2) (1,2) (0,1)
(0)
is bijective. Let a∈G2(x ) be the unique element such that
2 (0)
D(a)=g−1 ◦g ◦g
(0,2) (1,2) (0,1)
and F(a)=b0. Then the triple of elements (x,g,a) satisfies condition (i) of Defini-
tion 1.4, and F(x,g,a)=(y0,h0,b0). Now the element
u:=a−1 ◦a ◦a ◦Ad(g−1 )(a )−1 ∈G3(x )
(0,1,3) (0,2,3) (0,1,2) (0,1) (1,2,3) 2 (0)
satisfies D(u)=1, so it belongs to the subgroup π (G3,x )⊂G3(x ). Since the
2 (0) 2 (0)
group homomorphism
π (F3,x ):π (G3,x )→π (H3,y0 )
2 (0) 2 (0) 2 (0)
COMBINATORIAL DESCENT DATA 9
is injective, and since
π (F3,x )(u)=F3(u)=
2 (0)
(b0 )−1◦b0 ◦b0 ◦Ad(h0 )−1(b0 )−1 =1,
(0,1,3) (0,2,3) (0,1,2) (0,1) (1,2,3)
weconcludethatu=1. Thusthetriple(x,g,a)satisfiescondition(ii)ofDefinition
1.4, so it is a descent datum in G.
Now we prove that the function Desc(F) is injective. Given
(x,g,a),(x0,g0,a0)∈Desc(G),
define (y,h,b) := F(x,g,a) and (y0,h0,b0) := F(x0,g0,a0). Assume we are given a
gauge transformation
(f,c):(y,h,b)→(y0,h0,b0)
in H. We have to produce a gauge transformation
(e,d):(x,g,a)→(x0,g0,a0)
in G.
We know that the function
π (F0,x,x0):π (G0,x,x0)→π (cid:0)H0,y,y0(cid:1)
1 1 1
is surjective. Therefore there is a 1-morphism e ∈ G0(x,x0), and a 2-morphism
1
v ∈H0(y), such that F(e)=f ◦D(v). Let
2
f˜:=f ◦D(v)∈H0(y,y0)
1
and
c˜:=Ad(h−1)(v−1)◦c◦v ∈H0(y ).
(1) (0) 2 (0)
A simple calculation shows that
(f˜,c˜):(y,h,b)→(y0,h0,b0)
is also a gauge transformation.
Recall that F(e)=f˜, so
F(g−1◦e−1◦g0◦e )=h−1◦f˜−1◦h0◦f˜ =D(c˜).
(1) (0) (1) (0)
This is condition (i) of Definition 1.5 for the gauge transformation (f˜,c˜). Because
the group homomorphism
π (F1,x ):π (G1,x )→π (H1,y )
1 (0) 1 (0) 1 (0)
is injective, it follows that there is an element d0 ∈G1(x ) such that
2 (0)
g−1◦e−1◦g0◦e =D(d0).
(1) (0)
Consider the element w := c˜◦F(d0)−1 ∈ H1(y ). It satisfies D(w) = 1, so it
2 (0)
belongs to the subgroup π (H1,y ). But the homomorphism
2 (0)
π (F1,x ):π (G1,x )→π (H1,y )
2 (0) 2 (0) 2 (0)
is bijective, so there is a unique element v ∈ G1(x ) satisfying D(v) = 1 and
2 (0)
F(v)=w. Let d:=v◦d0 ∈G1(x ). Then F(d)=c˜and
2 (0)
D(d)=D(d0)=g−1◦e−1◦g0◦e .
(1) (0)
Thus the pair (e,d) is a partial gauge transformation (x,g)→(x0,g0).
The last thing to check is that condition (ii) of Definition 1.5 holds for (e,d).
Let
u:=Ad(e−1)(a0)−1◦d−1 ◦a◦Ad(g−1 )(d )◦d ∈G2(x ).
(0) (0,2) (0,1) (1,2) (0,1) 2 (0)
10 AMNONYEKUTIELI
A direct calculation shows that D(u) = 1 , so u ∈ π (G2,x ). We know that
x(0) 2 (0)
F(u)=1 , and that the homomorphism
y(0)
π (F2,x ):π (G2,x )→π (H2,y )
2 (0) 2 (0) 2 (0)
is injective. It follows that u=1 , which is what we had to check. (cid:3)
1x(0)
3. Some Remarks
To finish the note, here are a few remarks and clarifications, to help place our
work in context.
Remark 3.1. Cosimplicial crossed groupoids arise naturally in the geometry of
gerbes. This is well-known – see [Gi, Br1, Br2, BGNT]. Let us quickly indicate
how this occurs. Let G be a sheaf of groups on a topological space X, and let
Aut(G) be its sheaf of automorphism groups. There is an obvious sheaf of crossed
groups
(cid:0) (cid:1)
G = D:G →Aut(G)
onX. ConsideranopencoveringU ={U }ofX. TheČechconstructiongivesrise
k
toacosimplicialcrossedgroupG:=C(U,G). ThesetDesc(G)isanapproximation
of the set of equivalence classes of G-gerbes on X (the discrepancy is because we
may have to use refinements and hypercoverings).
IfeveryG-gerbetotallytrivializesonU (see[Ye1,Definition9.16]),thenDesc(G)
actually classifies G-gerbes. If V is another such covering, and V →U is a refine-
ment, then there is an induced weak equivalence of cosimplicial crossed groupoids
C(U,G)→C(V,G).
More general gerbes (not G-gerbes as above) can sometimes be classified by a
sheafofcrossedgroupoidsG –seenextremark. Forthefullgeneralityisitnecessary
to invoke more complicated combinatorics – see Remark 3.6.
Remark 3.2. Theorem 0.1 is used to prove twisted deformation quantization in
our paper [Ye1] (see also the survey article [Ye3]). We look at a smooth algebraic
variety X over a field of characteristic 0. There are sheaves of crossed groupoids
G and H on X, that are obtained from certain sheaves of DG Lie algebras by the
Deligne crossed groupoid construction. Let U be a finite affine open covering of X,
and consider the cosimplicial crossed groupoids G := C(U,G) and H := C(U,H).
The sets Desc(G) and Desc(H) classify twisted Poisson deformations and twisted
associativedeformationsofO ,respectively. Thesetwisteddeformationsarestacky
X
versionsofusualdeformations(similartothestacksofalgebroidsin[Ko,KS]). The
fact that twisted deformations totally trivialize on any affine open covering relies
on our paper [Ye4].
There is a weak equivalence F : G → H coming from the Kontsevich Formality
Theorem. Due to Theorem 0.1 we know that Desc(F) is a bijection, and the
resulting map on twisted deformations is called the twisted quantization map.
The quantization map Desc(F) does not come from a refinement or any similar
local operation. Indeed, it is conjectured that in some cases (e.g. Calabi-Yau sur-
faces)thequantizationwouldsendasheaftoastack,thusdestroyingthegeometry.
Remark 3.3. Acrossedgroupoid(i.e.crossedmoduleoveragroupoid)isthesame
as a strict 2-groupoid. The homotopy set π (G) and groups π (G,x) are the same
0 i
in both incarnations. A crossed groupoid is also the same as a category with inner
gaugegroups(P,IG,ig)wherePisagroupoid–see[Ye1,Section5andProposition
10.4].
Traditionally papers used 2-groupoid language to discuss descent for gerbes (cf.
[BGNT]). In[Ye1]werealizedthatthecrossedgroupoidlanguageismoreeffective: