G-gerbes, principal 2-group bundles and characteristic classes

0.45 MB·

by  Gregory Ginot

G
-gerbes, prin
ipal 2-group bundles
and
hara
teristi

lasses
∗ †‡
Grégory Ginot Mathieu Stiénon
8
0
Abstra
t
0
2
We give an expli
it des
ription of a 1-1
orresponden
e between Morita
[G → Aut(G)]
n equivalen
e
lasses of, on the one hand, prin
ipal 2-group -
G
a bundles over Lie groupoids and, on the other hand, -extensions of Lie
J [G → Aut(G)]
2 grouGpoids (i.e. between -bundles over di(cid:27)erentiable sta
ks
and -gerbes over di(cid:27)erentiable sta
ks). We also introdu
e universal
har-
2 G
a
teristi

lasses for 2-group bundles. For groupoid
entral -extensions,

] we provethat the universal
hara
teristi

lasses
oin
ide with the Diximer
T
Douady
lasses that
an be
omputed from
onne
tion-type data.
A
.
h
t Contents
a
m
[ 1 Introdu
tion 2

2
v 2 Generalized morphisms and prin
ipal Lie 2-group bundles 4
8
3 2.1 Lie 2-groupoids, Crossed modules and Morita morphisms . . . . . . 4
2
1 2.2 Generalized morphisms of Lie 2-groupoids . . . . . . . . . . . . . . 7
.
1 2.3 2-group bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
0
8 G
0 3 Groupoid -extensions 11
: G [G → Aut(G)]
v
3.1 From groupoid -extensions to -bundles . . . . . . . 12
i
X [G → Aut(G)] G
3.2 From -bundles to groupoid -extensions . . . . . . . 14
r
a
3.3 Proof of Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Universal
hara
teristi
maps and Dixmier-Douady
lasses 16
4.1 Cohomology of Lie 2-groupoids . . . . . . . . . . . . . . . . . . . . 16
4.2 Cohomology
hara
teristi
map for 2-group bundles . . . . . . . . . 18
G
4.3 DD
lasses for groupoid
entral -extensions . . . . . . . . . . . . 19
4.4 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
S1
4.5 The
ase of
entral -extensions . . . . . . . . . . . . . . . . . . . 22
4.6 Proof of Theorem 4.14 . . . . . . . . . . . . . . . . . . . . . . . . . 24
∗
UPMC Paris 6, Institut Mathématique de Jussieu, Équipe Analyse Algébrique, Case 82,
4 p†la
e Jussieu, 75252 Paris, Fran
e ginotmath.jussieu.fr
Pennsylvania State University, Department of Mathemati
s, 109 M
Allister Building,
Uni‡versityPark, PA 16802, U.S.A. stienonmath.psu.edu
Resear
hsupportedbytheEuropeanUnionthroughtheFP6MarieCurieR.T.N.ENIGMA
(Contra
t numberMRTN-CT-2004-5652).
1
1 Introdu
tion
G
This paper is devoted to the study of the relation between groupoid -extensions
and prin
ipal Lie 2-group bundles, and of their
hara
teristi

lasses.
Γ ⇉ Γ Γ
2 1 1
A Lie 2-group is a Lie groupoid , whose spa
es of obje
ts and of mor-
Γ
2
phisms are Lie grouρps and all of whose stru
ture mapsρare group morphisms.
(G −→ H) G −→ H
A
rossed module is a Lie group morphism together with an
H G
a
tion of on satisfying suitable
ompatibility
onditions. It is standardρthat
[G −→ H]
Lie 2-groups are in bije
tion with
rossed modules [3,12℄. In thρis paper,
(G −→ H)
denotes the 2-group
orresponding to the
rossed module .
Lie2-groupsarisenaturallyinmathemati
alphysi
s. Forinstan
e,inhighergauge
theory [2,4℄, Lie 2-group bundles provide a well suited framework for des
ribing
theparalleltransportofstrings[1,4,29℄. Severalre
entworkshaveapproa
hedthe

on
ept of bundles with a (cid:16)stru
ture Lie 2-group(cid:17) over a manifold from various
perspe
tives [1,4,6,36℄. Here we take an alternative point of view and give a
de(cid:28)nition of prin
ipal Lie 2-group bundles of a global nature (i.e. not resorting
to a des
ription expli
itly involving lo
al
harts and
o
y
les) and whi
h allows
for the base spa
e to be a Lie groupoid. In other words, we
onsider 2-group
prin
ipal bundles over di(cid:27)erentiable sta
ks [7℄. Our approa
h immediately leads
to a natural
onstru
tion of (cid:16)universal
hara
teristi

lasses(cid:17) for prin
ipal 2-group
bundles.
G P M
Let us start with Lie (1-)groups. A prin
ipal -bundle over a manifold
M

anoni
ally determines a homotopy
lass of maps from to the
lassifying spa
e
BG G G
of the group . In fa
t, the set of isomorphism
lasses of -prin
ipal bundles
f
M M −→ BG
over is in bije
tion with the set of homotopy
lasses of maps [14,
H∗(BG)
38,39℄. Pulling ba
k the generators of (the universal
lasses) through
f P M
, one obtains
hara
teristi

lasses of the prin
ipal bundle over . These

hara
teristi

lasses
oin
ide with those obtained from a
onne
tion by applying
the Chern-Weil
onstru
tion [17,32℄.
There is an analogue but mu
h less known, di(cid:27)erential geometri
rather than
G M
purely topologi
al, point of view: a prin
ipal -bundle over a manifold
an
be thought of as a (cid:16)generalized morphism(cid:17) (in the sense of Hilsum-Skandalis [24℄)
M G
from the manifold to the Lie group both
onsidered as 1-groupoids. To see
G
this, re
all that a prin
ipal -bundle
an be de(cid:28)ned as a
olle
tion of transition
g : U → G
ij ij
fun
tions on the double interse
tions of some open
overing, satis-
g g = g
ij jk ik
fying the
o
y
le
ondition . These transition fun
tions
onstitute a
U ⇉ U
ij i
morphism of groupoids from the ƒe
h groupoid ` ` asso
iated to the
{U } G⇉ ∗
i i∈I
open
overing to the Lie group . Hen
e we have a diagram
(M ⇉ M)←∼− ( U ⇉ U )→ (G ⇉ ∗)
a ij a i
in the
ategory of Lie groupoids and their morphisms whose leftward arrow is a
M
Morita equivalen
e, in other words a generalized morphism from the manifold
G
to the Lie group .
This se
ond point of view, or more pre
isely its generalization to the 2-groupoid

ontext,
onstitutes the foundations on whi
h our approa
h is built. The general-
izationofthe
on
eptof(cid:16)generalizedmorphism(cid:17) to2-groupoidsisstraightforward:
Γ ∆ Γ ←−φ E −→f ∆
a generalized morphism of Lie 2-groupoids is a diagram ∼
2Gpd φ
in the
ategory of Lie 2-groupoids and their morphisms, where is a
2
Morita equivalen
e (a (cid:16)smooth(cid:17) equivalen
e of 2-groupoids). It is sometimes use-
ful to think of two Morita equivalent Lie 2-groupoids as two di(cid:27)erent
hoi
es of
an atlas (or open
over) on the same geoρmetri
obje
t (whi
h is a di(cid:27)erentiable
[G −→ H] Γ
2-sta
k [7,11℄). Wede(cid:28)ne aprin
ipal ρ -bundle overaLiegroupoid to be
Γ [G −→ H]
a generalized morphism from to (up to equivalen
e). See Se
tion 2.3.
The
on
ept of (geometri
) nerve of Lie groupoids extends to the 2-
ategori
al

ontext as a fun
tor from the
ategory of Lie 2-groupoids to the
ategory of
simpli
ial manifolds [40℄. By
onvention, the
ohomology of a 2-groupoid is the

ohomology of its nerve, whi
h
an be
omputed via a double
omplex (for in-
stan
e, see [20℄). Cru
ially, Morita equivalen
es indu
e isomorphisms in
oho-
Γ F [G → H]
mology. Therefore, any generalized morphism of 2-groupoids
[G → H] B Γ
de(cid:28)ning a prin
ipal -bundle over the groupoid yields a pullba
k
F∗ : H•([G → H]) → H•(Γ)
homorphism in
ohomology, whi
h is
alled the

ohomology
hara
teristi
map (
hara
teristi
map for short). The
ohomology
H•([G → H])

lasses in [20℄ should be viewed as universal
hara
teristi

lasses
F∗ B
and their images by as the
hara
teristi

lasses of .
Lie 2-group prin
ipal bundles are
losely related to non-abelian gerbes. Ge-
G
ometri
ally, non-abelian -gerbes over di(cid:27)erentiable sta
ks
an be
onsidered
G G
as groupoid -extensions modulo Morita equivalen
e [26℄. By a groupoid -
1 → M × G −→i Γ˜ −→φ
extension, we mean a short exa
t sequen
e of groupoids
Γ → 1 M × G
, where is a bundle of groups. Here we establish an expli
it 1-1
G G

orresponden
e between groupoid -extensions up to Morita equivalen
e (i.e. -
[G → Aut(G)]
gerbes over di(cid:27)erentiable sta
ks) and prin
ipal -bundles over Lie
[G → Aut(G)]
groupoids modulo Morita equivalen
e (i.e. -prin
ipal bundles over
di(cid:27)erentiable sta
ks). This is Theorem 3.4. Note that a restri
ted version of this

orresponden
e is highlighted in [27, Theorem 4℄.
H2(X,G)
It is known that Giraud's se
ond non abelian
ohomology group
lassi-
G X H1(X,[G →
(cid:28)es the -gerbes over a di(cid:27)erentiable sta
k [21℄ while Dede
kers'
Aut(G)]) [G → Aut(G)]

lassi(cid:28)es the prin
ipal -bundles [15,16℄. In [9,10℄, Breen
showed that thesetwo
ohomology groups areisomorphi
. Insomesense, our the-
orem above
an be
onsidered as an expli
it geometri
proof of Breen's theorem
in the smooth
ontext. Indeed, one of the main motivations behind the present
G
paperistherelation between -extensionsand2-group prin
ipal bundles. Webe-
lieve our result throws a bridge between the groupoid extension approa
h to the
G
di(cid:27)erential geometry of -gerbes developed in [26℄ and the one based on higher
gauge theory due to Baez-S
hreiber [4℄. This will be investigated somewhere else.
G G
Animportant
lassof -extensionsisformedbytheso
alled
entral -extensions
[G → Aut(G)]
[26℄, those for whi
h the stru
ture 2-group redu
es to the 2-group
[Z(G) → 1] Z(G) G G
(where stands for the
enter of ). They
orrespond to -
G
gerbes with trivial band or -bound gerbes [26℄. Ea
h su
h extension determines
[Z(G) → 1] Γ
a prin
ipal -bundle over the base groupoid . In [7℄, Behrend-Xu
H3(Γ) S1
gave a natural
onstru
tion asso
iating a
lass in to a
entral -extension
Γ
of a Lie groupoid . When the base Lie groupoid is Morita equivalent to a
S1
smooth manifold (viewed as a trivial 2-groupoid), a
entral -extension is what
has been studied by Murray and Hit
hin under the name bundle gerbe [25,35℄.
The Behrend-Xu
lass of a bundle gerbe
oin
ides with its Dixmier-Douady
lass,
3
whi
h
an be des
ribed by the -
urvature. In the present paper, we extend
DD ∈
(α)
the
onstru
tion of Behrend-Xu and de(cid:28)ne a Dixmier-Douady
lass
H3(Γ)⊗Z(g) G G
for any
entral -extension, where is a
onne
ted redu
tive Lie
G [Z(G) → 1]
group. Sin
e a
entral -extension indu
es a -prin
ipal bundle over
3
Γ H3([Z(G) → 1]) → H3(Γ)
, there is also a
harateristi
map . Dualizing, one
CC ∈ H3(Γ)⊗Z(g)
φ
obtains a
lass . We prove that the Dixmier-Douady
lass
DD CC
(α) φ

oin
ides with the
hara
teristi

lass . In a
ertain sense, this is the
gerbe analogue of the Chern-Weil isomorphism for prin
ipal bundles [17,32℄.
The paper is organized as follows. Se
tion 2 is
on
erned with generalized mor-
phisms of Lie 2-groupoids and with 2-group bundles and re
alls some standard
material on Lie 2-groupoids. The main feature of Se
tion 3 is Theorem 3.4 on the
Ad
G [G −−→ Aut(G)]
equivalen
e of groupoids -extensions and prin
ipal -bundles. In
Se
tion4wede(cid:28)nethe
hara
teristi
map/
lassesofprin
ipalLie2-groupbundles,
G
we present the
onstru
tion of the Dixmier-Douady
lasses of groupoid
entral -
G
extensions and we prove that the Dixmier-Douady
lass of a
entral -extension
[Z(G) → 1]

oin
ides with the universal
hara
teristi

lassof theindu
ed -bundle
(cid:22) see Theorem 4.14.
G G
Note that, when is dis
rete, the relation between groupoid -extensions and
2-group prin
ipal bundles was also independently studied by Hae(cid:29)iger [22℄.
Some of the results of the present paper are related to results announ
ed by Baez
Stevenson [5℄. Re
ently, Sati, Stashe(cid:27) and S
hreiber have studied
hara
teristi
L
∞

lasses for 2-group bundles by the mean of -algebras [36℄. It would be very
L
∞
interesting to relate their
onstru
tion to ours using integration of -algebras
as in [19,23℄.
A
knowledgments The authors are grateful to André Hae(cid:29)iger, Jim Stashe(cid:27),
Urs S
hreiber and Ping Xu for many useful dis
ussions and suggestions.
2 Generalized morphisms and prin
ipal Lie 2-group
bundles
2.1 Lie 2-groupoids, Crossed modules and Morita morphisms
This se
tion is
on
erned with Lie 2-groupoids and Morita equivalen
es. The
material is rather standard. For instan
e, see [31,34℄ for the general theory of
Lie groupoids and [3,42℄ for Lie 2-groupoids. A Lie 2-groupoid is a double Lie
groupoid
s
Γ2 //// Γ0
t
u l id id
(1)
(cid:15)(cid:15) (cid:15)(cid:15) s (cid:15)(cid:15) (cid:15)(cid:15)
Γ1 //// Γ0
t
id
Γ0 ////Γ0
in the sense of [12℄, where the right
olumn id denotes the trivial
Γ
0
groupoid asso
iated to the smooth manifold . It makes sense to use the sym-
s t Γ ⇉ Γ
2 0
bols and to denote the sour
e and target maps of the groupoid sin
e
s◦l = s◦u t◦l = t◦u
and .
Remark 2.1. A Lie 2-groupoid isthus a small 2-
ategory in whi
h all arrows are
invertible, the sets of obje
ts, 1-arrows and 2-arrows are smooth manifolds, all
stru
turemapsaresmoothandthesour
esandtargetsaresurje
tivesubmersions.
4
l s
Γ2 ////Γ1 ////Γ0
In the sequel, the 2-groupoid (1) will be denoted u t . The so
l
Γ2 ////Γ1

alledverti
al(resp. horizontal)multipli
ationinthegroupoid u (resp.
s
Γ2 //// Γ0 ⋆ ∗
t ) will be denoted by (resp. )
id s
Γ1 //// Γ1 //// Γ0
Clearly, a Lie groupoid
an be seen as a Lie 2-groupoid id t .
Γ ∗
0
A Lie 2-groupoid where is the one-point spa
e is known as a Lie 2-group.
There is a well-known equivalen
e between Lie 2-groupoids and
rossed modules
of groupoids [12℄. A
rossed module of groupoids is a morphism of groupoids
ρ
X //Γ
1 1
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
X //Γ
0 = 0
X = Γ X ⇉ X
0 0 1 0
whi
h is the identity on the base spa
es (i.e. ) and where is
a family of groups (i.e. sour
e and target maps
oin
ide), together with a right
(γ,x) 7→ xγ Γ X
a
tion by automorphisms of on satisfying:
ρ(xγ) = γ−1ρ(x)γ ∀(x,γ)∈ X × Γ ,
1 Γ0 1 (2)
xρ(y) = y−1xy ∀(x,y)∈ X × X .
1 Γ0 1 (3)
X
1
Note that the equalities (2) and (3) make sense be
ause is a family of groups.
G
Example 2.2. Given any Lie group , we obtain a
rossed module by setting
N = G Γ = Aut(G) Γ = ∗ ρ(g) = Ad g
1 1 0 g
, , and (the
onjugation by ).
Γ ⇉ Γ
1 0
Example 2.3. ALiegroupoid indu
esa
rossedmoduleinthefollowing
S = {x ∈ Γ /s(x) = t(x)} Γ S
Γ 1 1 Γ
way. Let be the set of
losed loops in . Then
Γ Γ S
0 1 Γ
is a family of groups over and a
ts by
onjugation on . Therefore, we
obtain a
rossed module
i
S //Γ
Γ 1
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
Γ //Γ
0 = 0
i
where is the in
lusion map.
l s
Γ2 ////Γ1 ////Γ0
A 2-groupoid u t determines a
rossed module of groupoids
(G −→ρ H) H = Γ ⇉ Γ G = {g ∈ Γ |l(g) ∈ Γ ⊂ Γ } ρ(g) =
1 0 1 2 0 1
as follows. Here , ,
u(g) H = Γ G ⊂ Γ
1 1 1 2
and the a
tion of on is by
onjugation. More pre
isely, if
l
1h h Γ2 //// Γ1 gh = 1h−1∗g∗1h
istheunitoveranobje
t inthegroupoid u ,then .
ρ
X −→ Γ
Conversely, givena
rossedmoduleofgroupoids ,onegetsaLie2-groupoid
X1⋉Γ1 l //// Γ1 s //// Γ0 X1 ⋉Γ1 ⇉ Γ1
u t , where is the transformation groupoid
X ⋉Γ ⇉ Γ
1 1 0
and is the semi-dire
t produ
t of groupoids. More pre
isely, for all
x,x′ ∈ X γ,γ′ ∈ Γ
1 1
and , the stru
tures maps are de(cid:28)ned by
l(x,γ) = γ, (x′,γ′)∗(x,γ) = (x′xγ′−1,γ′γ),
u(x,γ) = ρ(x)γ, (x′,ρ(x)γ)⋆(x,γ) = (x′x,γ).
5
In thρe sequel, weρwill denote the Lie 2-groupoid asso
iated to the
rossed module
(G −→ H) [G −→ H]
by .
Ad
G −−→ Aut(G)
Example 2.4. The
rossedmoduleofgroups(cid:0) (cid:1)yieldsthe2-group
G⋉Aut(G) l ////Aut(G) ////∗
u with stru
ture maps
l(g,ϕ) = ϕ u(g,φ) = Ad ◦ϕ
g
(g ,Ad ◦ϕ )⋆(g ,ϕ ) = (g g ,ϕ )
1 g2 2 2 2 1 2 2
(g ,ϕ )∗(g ,ϕ )= (g ϕ (g ),ϕ ◦ϕ )
1 1 2 2 1 1 2 1 2
φ
Γ −→ ∆ (φ ,φ ,φ )
0 1 2
A (stri
t) morphism of Lie 2-groupoids is a triple of smooth
φ : Γ → ∆ i = 0,1,2
i i i
maps ( )
ommuting with all stru
ture maps. Morphisms
of
rossed modules are de(cid:28)ned similarly.
∆ f : M → ∆
0
Let be a Lie 2-groupoid. Given a surje
tive submersion , we
an
l s
∆[M] : ∆2[M] ////∆1[M] ////M
form the pullba
k Lie 2-groupoid u t , where
∆ [M] = {(m,γ,n) ∈ M ×∆ ×M s(γ) = f(m), t(γ)= f(n)}, i= 1,2.
i i
s.t.
s,t
The maps are the proje
tions on the (cid:28)rst and last fa
tor respe
tively. The
u,l
maps , the horizontal and verti
al multipli
ations are indu
ed by the ones on
∆
as follows:
u(m,γ,n) = (m,u(γ),n), (m,γ,n)∗(n,γ′,p)= (m,γ ∗γ′,p),
l(m,γ,n) = (m,l(γ),n), (m,γ,n)⋆(m,γ′,n)= (m,γ ⋆γ′,n).
∆[M] → ∆ m 7→ f(m)
There is a natural map of groupoids de(cid:28)ned by and
(m,γ,n) 7→ γ
.
Pullba
k of 2-groupoids yield a
onvenient de(cid:28)nition of Morita morphism of Lie
2-groupoids.
φ
Γ −→ ∆
De(cid:28)nition 2.5. A morphism of Lie 2-groupoids is a Morita morphism
φ
if is the
omposition of two morphisms
Γ2 //∆2[Γ0] //∆2
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
Γ1 //∆1[Γ0] //∆1
Γ(cid:15)(cid:15) (cid:15)(cid:15) id // Γ(cid:15)(cid:15) (cid:15)(cid:15) φ0 //∆(cid:15)(cid:15) (cid:15)(cid:15)
0 0 0
Γ → ∆ Γ → ∆ [Γ ]
0 0 1 1 0
su
h that and are surje
tive submersions and
Γ //∆ [Γ ]
2 2 0
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
Γ // ∆ [Γ ]
1 1 0
is a Morita morphism of 1-groupoids.
6
The (weakest) equivalen
e relation generated by the Morita morphism is
alled
Γ ∆
Morita equivalen
e. More pre
isely, two Lie 2-groupoids and are Morita
E ,E ,...,E
0 1 n
equivalent if there exists a (cid:28)nite
olle
tion of Lie 2-groupoids with
E = Γ E = ∆ i ∈ {1,...,n}
0 n
and , and, for ea
h , either a Morita morphism
E → E E → E
i−1 i i i−1
or a Morita morphism . In fa
t, by Lemma 2.12(d), one
has the following well-known lemma.
Γ ∆
Lemma 2.6. If and are Morita equivalent, there exits a
hain of Morita
Γ ← E → ∆ Γ ∆
morphisms of length 2 in between and .
φ :Γ → ∆
Remark 2.7. In the
ategori
al point of view, a Morita morphism is
in parti
ular a 2-equivalen
e of 2-
ategories preserving the smooth stru
tures.
Remark 2.8. Similar to [7℄, one
an de(cid:28)ne di(cid:27)erentiable 2-sta
ks. Two Lie 2-
groupoids de(cid:28)ne the same di(cid:27)erentiable 2-sta
k if, and only if, they are Morita
equivalent. Infa
taLie2-groupoid
anbethoughtofasa
hoi
eofadi(cid:27)erentiable
atlas on a di(cid:27)erentiable 2-sta
k.
2.2 Generalized morphisms of Lie 2-groupoids
Generalized morphisms of Lie 2-groupoids are a straightforward generalization of
generalized morphisms of Lie (1-)groupoids [24,34℄. They also have been
onsid-
2Gpd
ered in [42℄. Let denote the
ategory of Lie 2-groupoids and morphisms of
F
Lie 2-groupoids. A generalized morphism is a zigzag
∼ ∼
Γ ←− E → ... ←− E → ∆,
1 n
F :
where all leftward arrows are Morita morphisms. We use a squig arrow
Γ ∆
to denote a generalized morphism. The
omposition of two generalized
morphismsisde(cid:28)nedbythe
on
atenationoftwozigzags. Infa
tweareinterested
in equivalen
e
lasses of generalized morphisms:
f : Γ → ∆
In the sequel, we will
onsider two morphisms of 2-groupoids and
g : Γ → ∆ ϕ : Γ → ∆
0 1
to be equivalent if there exists two smooth appli
ations
ψ : Γ → ∆ x ∈ Γ
1 2 2
and su
h that, for any and any pair of
omposable arrows
i,j ∈ Γ
1
, the following relations are satis(cid:28)ed:
g (x)∗1 ⋆ψ(l(x)) = ψ(u(x))⋆ 1 ∗f (x) ,
2 ϕ(s(x)) ϕ(t(x)) 2
(cid:0) (cid:1) (cid:0) (cid:1) (4)
ψ(j ∗i) = 1 ∗ψ(i) ⋆ ψ(j)∗1 .
(cid:0) g1(j) (cid:1) (cid:0) f1(i)(cid:1) (5)
f g ψ
In other words, and are (cid:16)
onjugate(cid:17) by a (invertible) map
ompatible with
the horizontal multipli
ation.
It is easy to
he
k that the
onditions (4) and (5) are equivalent to the data
f g
of a natural 2-transformation from to [8,30℄. Re
all that a natural 2-
ϕ(m) ∈ ∆
1
transformation is given by the following data: an arrow for ea
h
m ∈ Γ ψ(γ) ∈ ∆ γ ∈ Γ
0 2 1
obje
t , and a 2-arrow for ea
h arrow as in the diagram
ϕ(s(j))
f(s(j)) //g(s(j))
tttttt5=
f(j) ψ(j) g(j)
(cid:15)(cid:15) tttttt (cid:15)(cid:15)
f(t(j)) // g(t(j))
ϕ(t(j))
and satisfying obvious
ompatibility
onditions with respe
t to the
ompositions
of arrows and 2-arrows.
7
We now introdu
e the notion of equivalen
e of generalized morphisms; it is the
natural equivalen
e relation on generalized morphism extending the equivalen
e
of groupoids morphisms. Namely, we
onsider the weakest equivalen
e relation
satisfying the following three properties:
f,g
(a) If there exists a natural transformation between a pair of homomor-
f g
phisms of 2-groupoids, and are equivalent as generalized morphisms.
φ
Γ −→ ∆
(b) If is a Morita morphism of 2-groupoids, the generalized morphisms
φ φ φ φ id id
∆ ←− Γ −→ ∆ Γ −→ ∆ ←− Γ ∆ −→ ∆ Γ −→ Γ
and are equivalent to and ,
respe
tively.
(
) Pre- and post-
omposition with a third generalized morphism preserves the
equivalen
e.
F1 : Γ ←φ−1 E1 −f→1 ∆ F2 : Γ ←φ−2 E2 −f→2 ∆
Example 2.9. Let and ε be two
E1 −→ E2
generalized morphisms. Suppose that there exists a morphism su
h
that the diagram
E1
xxqqφq1qqq MMMfM1MM&&
Γ ε ∆
ffMMφM2MMM E(cid:15)(cid:15)2 qqqfq2qq88
F F
1 2

ommutes up to 2-transformations. Then and are equivalent generalized
morphisms.
∼
Γ ←−
EExa−→m∼ p.l.e.2←.∼−10E. By−→∼its∆very de(cid:28)nition, a Morita equivalen
e oFf g:roΓup oid∆s
1 n
de(cid:28)nes two generalized morphisms and
G : ∆ Γ F ◦ G G ◦ F
. The
ompositions and are both equivalent to the
F G
identity. Furthermore, the equivalen
e
lasses of and are independent of the

hoi
e of the Morita equivalen
e.
Remark 2.11. Roughly speaking, generalized morphisms are obtained by for-
mally inverting the Morita morphisms. In fa
t, the following Lemma is easy to

he
k.
M
Lemma 2.12. The
olle
tion of all Morita morphisms of 2-groupoids is a left
2Gpd
multipli
ative system [28,De(cid:28)nition 7.1.5;41,De(cid:28)nition 10.3.4℄ in . Indeed,
the following properties hold:
id
(Γ −→ Γ) ∈ M ∀Γ∈ 2Gpd
(a) , ;
M
(b) is
losed under
omposition;
f φ ψ g
Γ −→ ∆ ←− E 2Gpd φ ∈ M Γ ←− Z −→ E
(
) given ∼ in with , there exists ∼ in
2Gpd ψ ∈ M
with su
h that
Z
wwppψpp∼pp NNNNgNN''
Γ E
NNfNNNN'' wwpp∼ppφpp
∆

ommutes;
φ f
(d) given Γ ∼ //∆ g //// E in 2Gpd with φ ∈ M, f◦φ= g◦φ implies f = g.
8
M 2Gpd
Sin
e isaleftmultipli
ative systeminthe
ategory ,we
an
onsiderthe
2Gpd 2Gpd M
M
lo
alization of withrespe
tto [28,Chapter7;41,Se
tion10.3℄.
2Gpd 2Gpd
M
This new
ategory has the same obje
ts as but its arrows are
2Gpd
M
equivalen
e
lasses of generalized morphisms. An isomorphism in
orre-
2Gpd
sponds to (the equivalen
e
lass of) a Morita equivalen
e in .
Lemma 2.12(C) implies that any generalized morphism
an be represented by a

hain of length 2:
Γ ∆
Lemma 2.13. Any generalized morphism between two Lie 2-groupoids and
is equivalent to a diagram
φ f
Γ ←− E −→ ∆
2Gpd φ φ∈ M
in the
ategory su
h that is a Morita morphism (i.e. ).
Remark 2.14. There is a bije
tion between maps of di(cid:27)erentiable 2-sta
ks and
equivalen
e
lasses of generalized morphisms of Lie 2-groupoids up to Morita
equivalen
es.
2.3 2-group bundles
In this se
tion, we give a de(cid:28)nition of 2-group bundles of a global nature and
formulated in terms of generalized morphisms of Lie 2-groupoids.
[G → H]
De(cid:28)nition 2.15. A prin
ipal (2-group) -bundle over a Lie groupoid
Γ ⇉ Γ B Γ ⇉ Γ
1 0 1 0
is a generalized morphism from (seen as a Lie 2-groupoid)
[G → H] (G → H)
to the 2-group asso
iated to the
rossed module .
[G → Aut(G)] Γ ⇉ Γ
1 0
In parti
ular, a prin
ipal -bundle over a groupoid is
Γ ⇉ Γ
1 0
a generalized morphism from (seen as a 2-groupoid) to the 2-group
[G→ Aut(G)]
.
[G → H] B B′ Γ ⇉ Γ
1 0
Two prin
ipal -bundles and over the groupoid are said to
be isomorphi
if, and only if, these two generalized morphisms are equivalent.
[G → H] M [G → H]
A -bundle over a manifold is a (2-group) -bundle over the
M ⇉ M
groupoid .
B [G → H] Γ ⇉ Γ Γ′ ⇉ Γ′
Let be a -bundle over a Lie groupoid 1 0. If 1 0 and
[G′ → H′] Γ ⇉ Γ [G → H]
1 0
are Morita equivalent to and respe
tively, then the

omposition
Γ′ ⇉ Γ′ ! Γ ⇉ Γ B [G → H]! [G′ → H′].
(cid:0) 1 0(cid:1) (cid:0) 1 0(cid:1)
[G′ → H′] Γ′ ⇉ Γ′ B
de(cid:28)nes a prin
ipal -bundle over 1 0 denoted by abuse of no-
tation. Here the left and right squig arrows are the Morita equivalen
es seens as
invertible generalized morphisms as in Example 2.10.
[G → H] B
De(cid:28)nition 2.16. A prin
ipal (2-group) -bundle over a Lie groupoid
Γ ⇉ Γ [G′ → H′] B′
1 0
and a prin
ipal (2-group) -bundle over a Lie groupoid
Γ′ ⇉ Γ′ B
1 0
aresaidtobe Moritaequivalent if, and onlyif, (viewed as ageneralized
Γ′ ⇉ Γ′ [G′ → H′] B′
morphism(cid:0) 1 0(cid:1) ) and are equivalent generalized morphisms.
B B′ Γ ⇉ Γ [G → H]
1 0
In parti
ular, if and are Morita equivalent, and are
Γ′ ⇉ Γ′ [G′ → H′]
1 0
Morita equivalent to and respe
tively.
9
Remark 2.17. When thegroupoid isjustamanifold, our de(cid:28)nition isequivalent
totheusualde(cid:28)nitionof2-groupbundlesin[4,6,36℄assuggestedbyExamples2.18
and 2.19 below.
π
P −→ M H
Example 2.18. Let be a prin
ipal -bundle. Then the diagram
M oo P × P //H
M
(cid:15)(cid:15) (cid:15)(cid:15) φ (cid:15)(cid:15) (cid:15)(cid:15) f (cid:15)(cid:15) (cid:15)(cid:15)
M oo P × P //H
M
t s
(cid:15)(cid:15) (cid:15)(cid:15) π (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
M oo P //∗
s(x,y) = x t(x,y) = y π(x) = φ(x,y) = π(y) x·f(x,y) = y
where , , and , de(cid:28)nes a
M 2 [1 → H]
generalized morphism from the manifold and to the -group . Hen
e,
2 M H P M
it is a -group bundle over . Note that a prin
ipal -bundle over is
H′ P′ M′
Morita equivalent (as a 2-group bundle) to a prin
ipal -bundle over if,
H M H′ M′ P P′
and only if, and are isomorphi
to and respe
tively and and
are isomorphi
prin
ipal bundles.
M G
Example 2.19. Let be a smooth manifold and be a (non abelian) Lie
M G
group. A non abelian 2-
o
y
le [15,16,21,33℄ on with values in relative to
{U } M
an open
overing i i∈I of is a
olle
tion of smooth maps
λ : U → Aut(G) g :U → G
ij ij ijk ijk
and
satisfying the following relations:
λ ◦λ = Ad ◦λ
ij jk gijk ik
g g = g λ−1(g ).
ijl jkl ikl kl ijk
[G → Aut(G)]
Su
h a non abelian 2-
o
y
le de(cid:28)nes a -bundle over the manifold
M
; for it
an be seen as the generalized morphism
M oo `i,jUij ×G×G f //G⋉Aut(G)
u l
(cid:15)(cid:15) (cid:15)(cid:15) φ (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
M oo `i,jUij ×G //Aut(G)
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
M oo `iUi //∗
M [G → Aut(G)]
between the manifold and the 2-group . Here
l(x ,g ,g ) = (x ,g ) φ(x ,g) = x
ij 1 2 ij 1 ij
u(x ,g ,g ) = (x ,g ) f(x ,g ,g )= g g−1,Ad ◦λ (x)
ij 1 2 ij 2 ij 1 2 (cid:0) 2 1 g1 ij (cid:1)
x x ∈ M U = U ∩U
ij ij i j
where denotes a point seen asa point of the open subset ,
x x ∈ M U g,g ,g
i i 1 2
the point seen as a point of the open subset , and arbitrary
G
elements of . The horizontal and verti
al multipli
ation are given by
(x ,α)∗(x ,β) = x ,g λ−1(α)β ,
ij jk (cid:0) ik ijk jk (cid:1)
(x ,g ,g )⋆(x ,g ,g )= (x ,g ,g ).
ij 1 2 ij 2 3 ij 1 3
10