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ATIYAH SEQUENCES AND CONNECTIONS ON PRINCIPAL

BUNDLES OVER LIE GROUPOIDS AND DIFFERENTIABLE STACKS

INDRANIL BISWAS, SAIKAT CHATTERJEE, PRAPHULLA KOUSHIK, AND FRANK NEUMANN

Abstract. We construct and study general and integrable connections on Lie groupoidsand differentiable stacks, as well as on principal bundles over them using Atiyah sequencesof vector bundles associated to transversal tangential distributions.

Contents

1. Introduction 1

2. Smooth spaces, groupoids, and stacks 3

3. Principal bundles over differentiable stacks and Lie groupoids 9

4. Connections on Lie groupoids and differentiable stacks 13

5. Connections on principal bundles over Lie groupoids and differentiable stacks 18

Acknowledgements 24

References 25

1. Introduction

We develop a general theory of connections for principal bundles over Lie groupoidsand differentiable stacks using Atiyah sequences associated to transversal tangential dis-tributions. The constructions presented here are inspired by the classical work of Atiyah[At] on connections for fiber bundles in complex geometry.

Given a Lie groupoid X := [X1 ⇒ X0] with both X0 and X1 smooth manifolds, suchthat the source map s is a submersion, we introduce connections on X as a distributionH ⊂ TX1 transversal to the fibers of the source map s. Such a connection is said to beintegrable or flat if, in addition, the corresponding distribution H is integrable. Theseconnections and flat connections also give rise to connections and flat connections ondifferentiable stacks under certain compatibility conditions. For the particular case ofDeligne–Mumford stacks, which are presented by etale Lie groupoids and include descrip-tions for orbifolds and foliations, such a connection always exists. For Deligne–Mumfordstacks a natural tangential distribution is given by the tangent bundle itself. Given now aLie group G and a principal G-bundle over a Lie groupoid X := [X1 ⇒ X0] equipped withsuch a connection H,⊂ TX1, we can define the notion of a connection on the principal

2010 Mathematics Subject Classification. Primary 53C08, Secondary 22A22, 58H05, 53D50.Key words and phrases. Atiyah sequence, principal bundles, Lie groupoids, connections, differentiable

stacks.1

http://arxiv.org/abs/2012.08442v2

2 I. BISWAS, S. CHATTERJEE, P. KOUSHIK, AND F. NEUMANN

G-bundle. A principal G-bundle over the Lie groupoid X is basically given by a principalG-bundle α : EG −→ X0 and some extra compatibility data reflecting the groupoidstructure. More precisely, a connection for a principal G-bundle over X corresponds to asplitting of the associated Atiyah sequence of vector bundles [At]

0 −→ ad(EG) −→ At(EG) −→ TX0 −→ 0

which satisfies the condition of being compatible with the various data of the structuresinvolved. This then allows us to define the associated curvature and characteristic differ-ential forms for connections on principal bundles over Lie groupoids.

Furthermore, using adequate groupoid presentations, these constructions extend to aframework that enables us to define and study connections and characteristic forms forprincipal G-bundles over differentiable stacks. In the particular case of Deligne-Mumfordstacks, which are represented by etale Lie groupoids, we develop the theory of connectionsintrinsically and construct the Atiyah sequence out of the stack data. This relies on thefact that in the case of Deligne-Mumford stacks, the associated Atiyah sequence is againa sequence of vector bundles in a natural way. These constructions also corresponds torelated ones in [BMW] for the algebraic geometrical context (compare also with [LM]).Though we will work throughout this article mainly in the differentiable setting, we remarkthat most of the concepts and constructions presented here also work equally well in theholomorphic and algebraic geometrical setting for the differential geometry of complexanalytic and algebraic stacks. An associated Chern-Weil theory of characteristic classesfor our setting is developed in a related article by the authors [BCKN]. Some of theconstructions and results presented here were also announced in [BN].

Versions of connections and flat connections on differentiable, complex analytic, andalgebraic stacks were also introduced using cofoliations on stacks by Behrend [Be]. In-dependently, Tang in [Ta] defined in a similar fashion flat connections for Lie groupoids,which he called etalizations. More recently, Arias Abad and Crainic in [AC] introducedgeneral Ehresmann connections for Lie groupoids and relate them to their framework ofhomotopy representations of Lie groupoids. Connections for principal bundles over Liegroupoids using pseudo-connection forms were studied earlier also by Laurent-Gengoux,Tu, and Xu [LGTX]. They also describe the associated Chern-Weil theory. Herrera andOrtiz, [HO], have informed us that they are currently also developing a similar theory forprincipal 2-bundles over Lie groupoids involving Atiyah LA-groupoids. We refer the readeralso to other related work on the differential geometry of Lie groupoids and differentiablestacks [FN, TXLG, CLW, Pi, Tr, DE].

Outline and organization of the article. In the first section (Section 2), we introducethe category of smooth spaces over which our stacks and groupoids are constructed andpresent the basic notions of fibered categories, stacks, and groupoids. We also discussthe relation between differentiable stacks on the one side and Lie groupoids on the other.In the following section (Section 3), we study principal bundles over differentiable stacksand Lie groupoids and their categorical interplay. In Section 4 we introduce notions andbasic properties of connections on Lie groupoids and differentiable stacks using verticaltangential distributions and compare our constructions with other existing frameworks inthe literature. We also construct characteristic differential forms for our general connec-tions. In the final section (Section 5), we define and study connections on Lie groupoids

ATIYAH SEQUENCES AND CONNECTIONS FOR BUNDLES ON LIE GROUPOIDS 3

as splittings of associated Atiyah sequences. Finally, we apply the theory to study con-nections on principal bundles over differentiable stacks and give general characterizationsfor connections on principal bundles over Deligne-Mumford stacks, which corresponds toetale Lie groupoids.

2. Smooth spaces, groupoids, and stacks

In this section, we will recall the main notions and constructions needed for our set-up.For a background on the theory of stacks and its main properties we refer to [Be, BX, Fa,He, Ne].

2.1. Smooth spaces and fibered categories. We shall refer to any of the category ofC∞-manifolds, the category of complex analytic manifolds, and the category of smoothschemes of finite type over the field of complex numbers, as the category S of smoothspaces and smooth maps. The tangent bundle of any smooth space X will be denotedby TX . A smooth space will mean an object in S, and a smooth map will refer to amorphism in S which is a submersion, and by a submersion, we mean a smooth mapwhose differential restricted to TxX is surjective for every point x of the domain X . Anetale map is a smooth immersion in S, which is also a submersion.

For a smooth space X , the structure sheaf of it will be denoted by OX . A vector bundleon X will be identified with its sheaf of sections, which is a finitely generated locally freesheaf of OX-modules. The cotangent bundle of X will be denoted by Ω1

X = T ∗X ,and the p-th exterior power of T ∗X will be denoted by ΩpX :=

∧p T ∗X . An integrabledistribution on a smooth space X is a subbundle H ⊂ TX of the tangent bundle ofX such that its annihilator H⊥ generates an ideal in

⊕k≥0Ω

kX which is preserved by

the exterior differential of the de Rham complex; equivalently, H is closed under the Liebracket operation on vector fields.

The big etale siteSet on the category S is given by the following Grothendieck topologyon S. We call a family Ui −→ X of morphisms in S with target X a covering familyof X , if all smooth maps Ui −→ X are etale and the total map from the disjoint union

∐

i

Ui −→ X

is surjective. This defines a pretopology on S generating a Grothendieck topology, whichis known as the big etale topology on S (compare [SGA4, Expose II] and [Vi]). If eitheror both of two morphisms U −→ X and V −→ X in S is a submersion, then their fiberproduct U ×X V −→ X exists.

Definition 2.1 (Groupoid fibration). A groupoid fibration over S is a category X , to-gether with a functor πX : X −→ S, satisfying the following axioms:

(i) For every morphism V −→ U in S, and every object x of X lying over U , thereexists an arrow y −→ x in X lying over V −→ U .

(ii) For every commutative triangleW −→ V −→ U inS and all morphisms z −→ xand y −→ x in X lying over W −→ U and V −→ U respectively, there existsa unique arrow z −→ y in X lying over W −→ V such that the compositionz −→ y −→ x is the morphism z −→ x.


The condition (ii) in Definition 2.1 ensures that the object y over V , which exists byDefinition 2.1(i), is unique up to a unique isomorphism. Any choice of such an object yis called a pullback of x via the morphism f : V −→ U . We will write as usual y = x|Vor y = f ∗x.

Let X be a groupoid fibration over S. The subcategory of X consisting of all objectslying over a fixed object U of S with the morphisms being those lying over the identitymorphism idU is called the fiber or category of sections of X over U . The fiber of X overU will be denoted by X (U).

Groupoid fibrations overS form a 2-category in which fiber products exist (see [Gr, Vi]).For two groupoid fibrations

πX : X −→ S and πY : Y −→ S ,

the 1-morphisms from X to Y are given by functors φ : X −→ Y such that

πY φ = πX .

The 2-morphisms are given by natural transformations between these projection functorpreserving functors.

Example 2.2. Let F : S −→ (Sets) be a presheaf, meaning a contravariant functor.We get a groupoid fibration X , where the objects are pairs of the form (U, x), with Ua smooth space and x ∈ F (U), while a morphism (U, x) −→ (V, y) is a smooth mapf : U −→ V such that x = y|F (U), equivalently, x = F (f)(y). The projection functoris given by

π : X −→ S , (U, x) 7−→ U .

Therefore, any sheaf F : S −→ (Sets) gives a groupoid fibration overS. In particular,every smooth space X gives a groupoid fibration X over S as the sheaf represented byX , in other words,

X(U) = HomS(U, X) .

To simplify notation, we will identify X with the smooth space X without further clari-fication.

A category X fibered in groupoids over S is representable if there exists a smoothspace X isomorphic to X as groupoid fibrations over S. We call a morphism of groupoidfibrations X −→ Y a representable submersion if for every smooth space U and ev-ery morphism U −→ Y , the fiber product X ×Y U is representable, and the inducedmorphism of smooth spaces X ×Y U −→ U is a submersion.

2.2. Stacks over smooth spaces. Now let us recall the definition of a stack [Be, BX].We clarify that a stack X here will always mean a stack over the big etale site Set ofsmooth spaces.

Definition 2.3 (Stack). A groupoid fibration X over S is a stack if the following gluingaxioms hold with respect to the site Set:

(i) Take any smooth space X in S, any two objects x, y in X lying over X and anytwo isomorphisms φ , ψ : x −→ y over X . If the condition φ|Ui = ψ|Ui holds forall Ui in a covering Ui −→ X, then φ = ψ.


(ii) Take any smooth space X in S, any two objects x , y ∈ X lying over X andany covering Ui −→ X with isomorphisms φi : x|Ui −→ y|Ui for every i. Ifthe condition φi|Uij = φj |Uij holds for all i, j, then there exists an isomorphismφ : x −→ y with φ|Ui = φi for all i.

(iii) For any smooth space X in S, any covering Ui −→ X, any family xi of objectsxi in the fiber XUi and any family of morphisms φij, where φij : xi|Uij −→ xj|Uijsatisfies the cocycle condition φjk φij = φik in X (Uijk), there exists an objectx lying over X with isomorphisms φi : x|Ui −→ xi such that φij φi = φj inX (Uij).

The isomorphism φ in Definition 2.3(ii) is unique by Definition 2.3(i). Similarly, fromDefinition 2.3(i) and Definition 2.3(ii), it follows that the object x whose existence is as-serted in Definition 2.3(iii) is unique up to a unique isomorphism. All pullbacks mentionedin Definition 2.3 are only unique up to isomorphism, but the properties do not depend onparticular choices.

In order to be able to do geometry on stacks, we need to compare them with smoothspaces and extend the geometry to stacks. From now on, we will restrict ourselves tostacks over the category S of C∞-manifolds, but we remark that we have incarnations ofthe analogue concepts and constructions for the category S of complex analytic manifoldsas well as smooth schemes of finite type over the complex numbers.

Definition 2.4 (Differentiable stack). A stack X over the site Set is called differentiableif there exists a smooth space X in S and a surjective representable submersion x :X −→ X , i.e., there exists a smooth space X together with a morphism of stacksx : X −→ X , such that for every smooth space U and every morphism of stacksU −→ X , the following two hold:

(1) the fiber product X ×X U is representable, and(2) the induced morphism of smooth spaces X×X U −→ U is a surjective submersion.

If X is a differentiable stack, such a surjective representable submersion x : X −→ X

is called a presentation or atlas for the stack X . It need not be unique, in other words,a differentiable stack can have different presentations.

Definition 2.5. A differentiable stack X is a (proper) Deligne–Mumford stack if it hasa (proper) etale presentation.

Orbifolds correspond to proper Deligne–Mumford stacks (see [Be, LM]).

The incarnations of these concepts over the site Set of complex analytic manifoldsrespectively, smooth schemes of finite type over C will be referred to as complex analyticstacks, respectively, algebraic stacks.

2.3. Lie groupoids and differentiable stacks. Differentiable stacks are also incarna-tions of Lie groupoids, as we will recall now in detail (see also [BX, dH]).

Definition 2.6 (Lie groupoid). A Lie groupoid X = [X1 ⇒ X0] is a groupoid internal tothe category S of smooth spaces, meaning the space X1 of arrows and the space X0 ofobjects are objects of S and all structure morphisms

s, t : X1 −→ X0 , m : X1 ×s,X0,t X1 −→ X1 ,


i : X1 −→ X1 , e : X0 −→ X1

are morphisms in S (so they are smooth maps). Here s is the source map, t is the targetmap, m is the multiplication map, e is the identity section, and i is the inversion map ofthe groupoid. The source map s is a submersion. Using i, this implies that the targetmap t is also a submersion.

If s and t are etale, the groupoid X = [X1 ⇒ X0] is called etale. If the anchor map

(s, t) : X1 −→ X0 ×X0

is proper, the groupoid is called a proper groupoid. If the

A Lie group G is a Lie groupoid [G ⇒ ∗] with one object, meaning the space X0 isjust a point in S.

Every Lie groupoid X = [X1 ⇒ X0] gives rise to the associated tangent groupoidTX := [TX1 ⇒ TX0].

Example 2.7. Let X be a smooth space. The groupoid fibration X is, in fact, a differ-entiable stack over S. A presentation is given by the identity morphism idX .

Example 2.8 (Classifying stack). For a Lie group G, let BG be the category which hasas objects all pairs (P, S), where S is a smooth space of S and P is a principal G-bundleover S; a morphism (P, S) −→ (Q, T ) is a commutative diagram

Pϕ

//

Q

S // T

where ϕ : P −→ Q is a G-equivariant map. Note that the above diagram is Cartesian.Then BG together with the projection functor

π : BG −→ S , (P, S) 7−→ S

is a groupoid fibration over S. We note that BG is in fact a differentiable stack, and itis known as the classifying stack of G (see also Example 2.11 below). A presentation isgiven by the representable surjective submersion ∗ −→ BG where ∗ is a “point” in S.

Definition 2.9. Let X = [X1 ⇒ X0] be a Lie groupoid. A (left) X-space is given by anobject P of S together with a smooth map π : P −→ X0 and a map

σ : Q := X1 ×s,X0,π P −→ P , σ(γ, x) := γ · x

such that,

(i) π(γ · x) = t(γ) for all (γ, x) ∈ X1 ×s,X0,π P ,(ii) e(π(x)) · x = x for all x ∈ X1, and(iii) (δ · γ) · x = δ · (γ · x) for all (γ, δ, x) ∈ X1 ×s,X0,t X1 ×s,X0,π P .

Similarly, we can also define a (right) X-space.

Let X be a differentiable stack with a given presentation x : X −→ X . We canassociate to X a Lie groupoid X = [X1 ⇒ X0] as follows: Let X0 := X and X1 :=X ×X X and the source and target morphisms

s, t : X ×X X ⇒ X


of X being the first and second canonical projection morphisms. The composition ofmorphisms m in X is given as projection to the first and third factor

X ×X X ×X X ∼= (X ×X X)×X (X ×X X) −→ X ×X X .

The morphism X×X X −→ X×X X that interchanges the two factors gives the inversemorphism i for the groupoid, while the unit morphism e is given by the diagonal morphismX −→ X ×X X . As a presentation x : X −→ X of a differentiable stack is asubmersion, it follows that the source and target morphisms s, t : X×X X ⇒ X , beinginduced maps from the fiber product, are also submersions.

In the opposite direction, given a Lie groupoid X we can associate a differentiable stackX to it. Basically this is a generalization of associating to a Lie group G its classifyingstack BG (see Example 2.8). For this we now define (compare also [BX]):

Definition 2.10 (Torsors). Let X = [X1 ⇒ X0] be a Lie groupoid, and let S be a smoothspace. A (right) X-torsor over S is a smooth space P together with a surjective submersionπ : P −→ S and a right action of X on P (see the second part of Definition 2.9) satisfyingthe condition that π is X-invariant, i.e., π(γ ·p) = π(p) for all (γ, x) ∈ X1×s,X0,aP and theextra condition that for all p, p′ ∈ P with π(p) = π(p′), there exists a unique x ∈ X1

such that p · x is defined and p · x = p′.

Let π : P −→ S and ρ : Q −→ T be X-torsors. A morphism of X-torsors from thefirst one to the second one is given by a Cartesian diagram of smooth maps

Pϕ

//

Q

S // T

(2.1)

such that ϕ is a X1-equivariant map. Again, there is also a similar notion of a (left)X-torsor over a smooth space S.

Given a X-torsor P , the map a : P −→ X0 is called the anchor map or momentummap. The surjective submersion π : P −→ S is called the structure map.

For a Lie groupoid X = [X1 ⇒ X0], let BX denote the category of X-torsors, meaningthe category whose objects are pairs (P, S), where S is a smooth space and P a X-torsorover S. The morphisms (P, S) −→ (Q, T ) are given by Cartesian diagrams as in (2.1).Then BX is a groupoid fibration over S with a canonical projection functor

π : BX −→ S , (P, S) 7−→ S .

It turns out that BX is, in fact, a differentiable stack, the classifying stack of X-torsors.The proof of [BX, Proposition 2.3] works in any of the three categories S.

Example 2.11. In the case where the Lie groupoid X = [G ⇒ ∗] is a Lie group, aX-torsor over a smooth space S is simply a G-torsor or principal G-bundle over S. Theassociated classifying stack is the classifying stack BG of the Lie group G.

As presentations for a given differentiable stack are not unique, the associated Liegroupoids might be different. In order to define algebraic and geometric invariants fordifferentiable stacks, like cohomology or differential forms, they should, however, be inde-pendent from the choice of a presentation for the given stack. Therefore, it is important


to know under which conditions two different Lie groupoids give rise to isomorphic differ-entiable stacks.

Definition 2.12. Let X = [X1 ⇒ X0] and Y = [Y1 ⇒ Y0] be Lie groupoids. Amorphismof Lie groupoids is a functor φ : X −→ Y given by two smooth maps φ = (φ1, φ0) with

φ0 : X0 −→ Y0 , φ1 : X1 −→ Y1

which commute with all structure morphisms of the groupoids.

A morphism φ : X −→ Y of Lie groupoids is an etale morphism, if both maps φ0 :X0 −→ Y0 and φ1 : X1 −→ Y1 are etale.

A morphism φ : X −→ Y of Lie groupoids is a Morita morphism or essential equiva-lence if

(i) φ0 : X0 −→ Y0 is a surjective submersion, and(ii) the diagram

X1

(s,t)//

φ1

X0 ×X0

φ0×φ0

Y1(s,t)

// Y0 × Y0

is Cartesian, or in other words, X1∼= Y1 ×Y0×Y0 (X0 ×X0).

Two Lie groupoids X and Y are Morita equivalent, if there exists a third Lie groupoid W

and Morita morphisms

Xφ←− W

ψ−→ Y .

Theorem 2.13 (Morita equivalence). Let X = [X1 ⇒ X0] and Y = [Y1 ⇒ Y0] be Liegroupoids. Let X and Y be the associated differentiable stacks, i.e., X is the classifyingstack BX of X-torsors and Y is the classifying stack BY of Y-torsors. Then the followingthree statements are equivalent:

(i) The differentiable stacks X and Y are isomorphic.(ii) The Lie groupoids X and Y are Morita equivalent.(iii) there exists a smooth space Q together with two smooth maps f : Q −→ X0 and

g : Q −→ Y0 and (commuting actions) of X1 and Y1 in such a way that Q is atthe same time a left X-torsor over Y0 via g and a right Y-torsor over X0 via f , inother words, Q is a X-Y-bitorsor.

Proof. For the differentiable category of C∞-manifolds, this is [BX, Theorem 2.2]. It isimmediate to see that the proof works verbatim for any of the other categories S ofsmooth spaces, meaning for complex analytic as well as algebraic stacks.

Therefore different presentations of the same differentiable stack are given by Moritaequivalent Lie groupoids. Conversely, Morita equivalent Lie groupoids present isomorphicdifferentiable stacks. The case of etale groupoids presenting Deligne-Mumford stacks isof particular importance as these can be used naturally also to describe orbifolds andfoliations [Co, CM, L].


3. Principal bundles over differentiable stacks and Lie groupoids

In this section, we will define the notion of a principal G-bundle over a differentiablestack and over a Lie groupoid.

3.1. Principal bundles over differentiable stacks. Let us start with the definition ofprincipal bundles over differentiable stacks.

Definition 3.1 (Principal bundle over differentiable stack). Let G be a Lie group andX a differentiable stack. A principal G-bundle or G-torsor EG over X is given by adifferentiable stack EG, a morphism of differentiable stacks π : EG −→ X , and a 2-Cartesian diagram

EG ×Gσ

//

p1

EG

π

v~

EGπ

// X .

such that for any submersion f : U −→ X from a smooth space U , the pullback byf in the above diagram defines a principal G-bundle on U . Morphisms ρ : EG −→ FG

between two principal G-bundles EG and FG are defined in the obvious way.

Principal G-bundles over a differentiable stack X can also be defined directly by usinga presentation or atlas. In other words, a principal G-bundle EG over a differentiablestack X is given by a principal G-bundle EG −→ X0 for an atlas X0 −→ X togetherwith an isomorphism of the pullbacks p∗1EG

∼−→ p∗2EG on the fiber product X0 ×X X0

satisfying the cocycle condition on X0×X X0×X X0. It turns out that for any submersionf : U −→ X , this datum defines a principal G-bundle EG −→ U over U , becauseX0 ×X U −→ U has local sections. Therefore we get a differentiable stack EG and theG-multiplication map glues and comes with a natural morphism of stacks EG×G −→ EG

(see [BMW, He, Ne]). We can reformulate all this more explicit also as follows: For eachsmooth atlas u : U −→ X we are given a principal G-bundle EG,u over U and for each2-commutative diagram of the form

U

u

ϕ// V

v~~⑤⑤⑤⑤⑤⑤⑤⑤

X

(3.1)

with a 2-isomorphism α : u ⇒ v ϕ, where u, v are smooth atlases, we are given anisomorphism

θϕ,α : EG,u∼=−→ ϕ∗

EG,v

satisfying the cocycle condition which says that for each 2-commutative diagram of theform

U

u

ϕ// V

ψ//

v

W

w④④④④④④④④

X


together with 2-isomorphisms α : u ⇒ v ϕ and β : v ⇒ w ψ we have a commutativediagram

EG,uθψϕ,βα

∼=//

θϕ,α ∼=

(ψ ϕ)∗EG,w

ϕ∗EG,v∼=

ϕ∗θψ,β

// (ϕ∗ ψ∗)EG,w

If FG is another principal G-bundle over X , then a morphism f : EG −→ FG is givenby a morphism fu : EG,u −→ FG,u for each smooth atlas u : U −→ X such that forany 2-commutative diagram of the form (3.1) we have a commutative diagram

EG,ufu

//

θEGϕ,α

∼=

FG,u

θFGϕ,α

∼=

ϕ∗EG,vϕ∗fv

// ϕ∗FG,v

The category of principal G-bundles over a differentiable stack X forms in a natural waya groupoid fibration BunG(X ) over S. In fact, we get the following characterizationfrom the above (compare also [BMW, He])

Proposition 3.2. Let X be a differentiable stack and G a Lie group with classifyingstack BG. Giving a principal G-bundle over X is equivalent to giving a morphism ofstacks X −→ BG and two principal G-bundles over X are isomorphic if and only ifthe corresponding morphisms of stacks X −→ BG are 2-isomorphic.

We consider the groupoid fibration BunG(X ) over S whose objects over a smoothspace U are principal G-bundles EG over X × U and whose morphisms are given bypullback diagrams of principal G-bundles. Given two differentiable stacks X and Y ,we also have the groupoid fibration H om(X , Y ) over S, whose groupoid of sectionsover a smooth space U is the groupoid of 1-morphisms or functors H om(X , Y )(U) =HomU(X × U, Y × U). Let us remark that if in addition X is proper and Y of finitepresentation, then H om(X , Y ) is again a differentiable stack and in particular if Y isa Deligne–Mumford stack, then the Hom stack H om(X , Y ) is also a Deligne–Mumfordstack (compare [Ao], [Ol]). We can now make the following straightforward observationusing Proposition 3.2 (compare with [BMW]).

Proposition 3.3. There is an equivalence of groupoid fibrations over S

BunG(X ) ∼= H om(X , BG) .

3.2. Principal bundles over Lie groupoids. Let us now recall the general notion ofa principal G-bundle over a Lie groupoid (see [LGTX, TXLG]).

Given a X-space π : P −→ X0 for a Lie groupoid X = [X1 ⇒ X0], we have for anyγ ∈ X1 a smooth isomorphism in S

lγ : π−1(u) −→ π−1(v) , x 7−→ γ · x ,

where u = s(γ) and v = t(γ). Associated to this X-space is a transformation groupoidP = [Q = X1×s,X0,πP ⇒ P ], where the source and target maps are given by s(γ, x) = x


and t(γ, x) = γ · x. The multiplication map is given by

(γ, y) · (δ, x) = (γ · δ, x) ,

where y = δ · x. The first projection defines a strict homomorphism of Lie groupoidsfrom P = [Q⇒ P ] to X = [X1 ⇒ X0].

Definition 3.4 (Principal bundle over Lie groupoid). Let G be a Lie group and X =[X1 ⇒ X0] a Lie groupoid. A principal G-bundle or G-torsor over X, denoted by EG :=[s∗EG ⇒ EG], is given by a principal (right) G-bundle π : EG −→ X0, which is also aX-space such that for all x ∈ EG and all γ ∈ X1 with s(γ) = π(x), we have

(γ · x) · g = γ · (x · g) for all g ∈ G .

Let s∗EG = X1×s,X0,πEG be the pullback along the source map s. Then EG = [s∗EG ⇒

EG] is in a natural way a Lie groupoid, the transformation groupoid with respect to theX1-action.

Example 3.5. Let G and H be Lie groups, and let X = [H × X0 ⇒ X0] be thetransformation groupoid over a smooth space X0. Then a principal G-bundle over X isan H-equivariant principal G-bundle over X0. In particular, if X = [X0 ⇒ X0] with bothstructure maps s, t being the identity map idX0

, then a principal G-bundle over X is justa principal G-bundle over X0.

Similarly, we can define vector bundles of rank n over a Lie groupoid X and over adifferentiable stack X . They can be identified with the principal GLn-bundles over X

and X respectively.

The following theorem generalizes [BX, Proposition 4.1] for arbitrary principal bundlesand the proof given below is a variation of the argument for S1-bundles.

Theorem 3.6. Let G be a Lie group, X be a differentiable stack and X = [X1 ⇒ X0] bea Lie groupoid presenting X . Then there is a canonical equivalence of categories:

BunG(X ) ∼= BunG(X) ,

where BunG(X ) is the category of principal G-bundles over X and BunG(X) is thecategory of principal G-bundles over X.

Proof. Let us first assume we have given a principal G-bundle π : EG −→ X over thedifferentiable stack X . Let EG be the pullback of X0 −→ X via π, in other words, wehave a 2-Cartesian diagram

EG //

X0

EGπ

// X

This means that EG −→ X0 is a principal G-bundle over X0 and EG −→ EG is arepresentable surjective submersion. Let EG = [s∗EG ⇒ EG] be the associated Liegroupoid, which comes with an induced morphism of Lie groupoids

EG = [s∗EG ⇒ EG] −→ X = [X1 ⇒ X0]


and gives rise to the Cartesian diagram

s∗EG //

X1

EG // X0

In other words, we get a pullback diagram of smooth spaces in which the vertical maps aresource maps. This implies that s∗EG −→ EG is a principal G-bundle, and the verticalmaps are morphisms of G-bundle. Therefore, X1 acts on EG and EG becomes an X-spacewhich actually turns EG = [s∗EG ⇒ EG] into a principal G-bundle over the groupoidX = [X1 ⇒ X0].

In fact, we get a functor. Ψ : BunG(X ) −→ BunG(X), which associates the principalG-bundle EG = [s∗EG ⇒ EG] over the Lie groupoid X to the given principal G-bundleEG −→ X .

Now let us assume conversely, that we have given a principal G-bundle EG = [s∗EG ⇒

EG] over the Lie groupoid X = [X1 ⇒ X0]. This means that we have a principal G-bundleπ : EG −→ X0, which is also a X-space satisfying the conditions given in Definition 3.4.Recall that

s∗EG = X1 ×s,X0,π EG

is the pullback along the source map s, and EG = [s∗EG ⇒ EG] is in a natural way a Liegroupoid. It follows also that s∗EG becomes in a natural way a G-bundle over X1, andwe have a morphism of Lie groupoids

EG = [s∗EG ⇒ EG] −→ X

which respects the G-bundle structures of τ : s∗EG −→ X1 and π : EG −→ X0. LetEG := BEG be the associated differentiable stack of EG-torsors. The morphism of Liegroupoids EG −→ X induces a morphism between the associated differentiable stacksEG −→ X , which is representable, as its pullback to X0 is the map π : EG −→ X0

between smooth spaces:

EG //

EG

X0// X

Furthermore, it follows that the pullback of the morphism of stacks EG −→ X along anymorphism U −→ X from a smooth space U is a principal G-bundle. Therefore EG is aprincipal G-bundle over the differentiable stack X .

Finally, we get a functor in the opposite direction Φ : BunG(X) −→ BunG(X ),which associates the principal G-bundle EG −→ X to the given principal G-bundleEG = [s∗EG ⇒ EG] over the Lie groupoid X. It is easy to see that the functors Ψ and Φare mutually inverse and give the desired equivalence of categories.

The following definition allows for yet another alternative description of principal G-bundles over Lie groupoids.


Definition 3.7. Let X = [X1 ⇒ X0] be a Lie groupoid and G a Lie group. A principalG-groupoid over X is a Lie groupoid P = [Q⇒ P ] together with a groupoid morphism

G // Qτ

//

t

s

X1

t

s

G // Pπ

// X0

such that both G −→ Qτ−→ X1 and G −→ P

π−→ X0 are principal G-bundles, and,

furthermore, the following conditions hold:

(1) the source and target maps s and t on Q are G-equivariant,(2) the identity section e : P −→ Q is G-equivariant,

(3) the inversion map i : Q −→ Q satisfies the identity i(zg) = i(z)g−1 for all z ∈ Qand g ∈ G, and

(4) the multiplication map Q×s,X0,tQ −→ Q is G-equivariant for the diagonal actionof G on Q×s,X0,t Q.

There is an obvious notion of a morphism between principal G-groupoids, and wecan speak of the category BunG,X of principal G-groupoids over X, which yields thefollowing equivalent characterization of the category of principal G-bundles over a givenLie groupoid X.

Proposition 3.8. Let G be a Lie group and X = [X1 ⇒ X0] a Lie groupoid. Then thereis a canonical equivalence of categories

BunG(X) ∼= BunG,X ,

where BunG(X) is the category of principal G-bundles over X, and BunG,X is the categoryof principal G-groupoids over X.

Proof. The proof for the differentiable category given in [LGTX, Lemma 2.5] and worksequally well in the complex analytic and algebraic context.

Remark 3.9. It can be shown that principal G-bundles over a Lie groupoid X = [X1 ⇒

X0] are also equivalent to generalized homomorphisms from X to the Lie groupoid BG =[G ⇒ ∗], which is Morita equivalent to the gauge groupoid PGauge := [P × P/G ⇒ X0].In [LGTX, Prop. 2.13, Prop. 2.14 and Thm 2.15] this is discussed in the differentiablesetting, but this makes sense again in any of the categories S of smooth spaces.

4. Connections on Lie groupoids and differentiable stacks

In this section, we define and study the notion of connections on Lie groupoids anddifferentiable stacks.

4.1. Connections on Lie groupoids and differentiable stacks. Let us start withthe general definition of a connection on a Lie groupoid.

Let (X = [X1 ⇒ X0] , s , t ,m , e , i) be a Lie groupoid. Let

K := kernel(ds) ⊂ TX1 (4.1)


be the vertical tangent bundle for s. Since the source map s : X1 −→ X0 is a submersion,K is a subbundle of TX1. Fix a distribution on X1 given by a subbundle H ⊂ TX1 suchthat the natural homomorphism

K ⊕H −→ TX1 (4.2)

is an isomorphism, so H is a complement of K. Let

dHs := (ds)|H : H −→ s∗TX0

be the restriction of ds toH. Note that dHs is an isomorphism because the homomorphismin (4.2) is an isomorphism.

Let dt : TX1 −→ t∗TX0 be the differential of the map t. Consider the homomorphism

θ := dt (dHs)−1 : s∗TX0 −→ t∗TX0 . (4.3)

For any y ∈ X1, let

θy := θ|(s∗TX0)y : Ts(y)X0 −→ Tt(y)X0 (4.4)

be the restriction of θ to the fiber (s∗TX0)y = Ts(y)X0.

We can then define generally

Definition 4.1 (Connection on a Lie groupoid). A connection on a Lie groupoid X =[X1 ⇒ X0] is a distribution H ⊂ TX1 such that

(i) for every x ∈ X0, the image of the differential

de(x) : TxX0 −→ Te(x)X1

coincides with the subspace He(x) ⊂ Te(x)X1, and(ii) for every y , z ∈ X1 with t(y) = s(z), the homomorphism

θm(z,y) : Ts(y)X0 −→ Tt(z)X0

coincides with the composition θz θy (see (4.4)).

A connection on X is said to be flat (or integrable) if the distribution H ⊂ TX1 definingthe connection is integrable.

Henceforth we will denote m(y, z) = y z. Similarly for the pairs (y, v), (z, w) ∈ TX1

such that s(y) = t(z), dty(v) = dsz(w), the composition dm((y, v), (z, w)

)in the induced

tangent Lie groupoid TX = [TX1 ⇒ TX0] will be denoted as (y z, v w).

We will now discuss some functoriality properties for our notion of connection on Liegroupoids.

Proposition 4.2. Let X = [X1 ⇒ X0] and Y = [Y1 ⇒ Y0] be Lie groupoids, and let

φ = (φ1, φ0) : X −→ Y

be a Morita morphism. Let

H ⊂ TY1

be a distribution that defines a connection on the groupoid Y. ThenH induces a connectionon the groupoid X.


Proof. Recall from Definition 2.12(i) that the map φ0 : X0 −→ Y0 is a surjective sub-mersion. Hence from Definition 2.12(ii) it follows that

φ1 : X1 −→ Y1

is also a surjective submersion. Let

dφ1 : TX1 −→ TY1

be the differential of the map φ1. Let H ⊂ TY1 be the distribution inducing a connectionon the groupoid Y. Now define

H := (dφ1)−1(φ∗

1H) ⊂ TX1 .

Since φ1 is a surjective submersion, it follows that H ⊂ TX1 is a distribution and thereforedefines a connection on the groupoid X.

Similarly, as in [Be], we can study functorial behaviour with respect to horizontalmorphisms of Lie groupoids.

Definition 4.3. Let X = [X1 ⇒ X0] and Y = [Y1 ⇒ Y0] be Lie groupoids and φ =(φ1, φ0) : X −→ Y be a morphism. Suppose that H ⊂ TX1 defines a connection on X

and K ⊂ TY1 a connection on Y. The morphism φ is called horizontal if the differentialdφ1 : TX1 −→ TY1 maps H into K.

Proposition 4.4. Let X and Y be Lie groupoids together with an etale morphism φ =(φ1, φ0) : X −→ Y. Then any connection K on Y induces a unique connection H onX such that φ is a horizontal morphism. If the connection K is integrable, then also theinduced connection H is integrable.

Proof. Given a connection K on the groupoid Y, the unique connection H on the groupoidX is given by the fiber product H := K ×TY1 TX1.

In particular, a connection (respectively, flat connection) on an etale groupoid inducesthen a connection (respectively, flat connection) on the associated Deligne-Mumford stackBX, the classifying stack of X-torsors.

Remark 4.5. Related constructions of connections and flat connections on groupoidsand on stacks in the algebro-geometric, differentiable and holomorphic context were alsostudied by Behrend in [Be]. Flat connections in the differentiable setting for Lie groupoidswere also independently introduced as etalifications by Tang [Ta]. These constructionsgive all rise to subgroupoids of the associated tangent groupoid TX of a groupoidX , whichin the differentiable category is equivalent to the horizontal paths forming a subgroupoidof the path groupoid of X. This is used by Laurent-Gengoux, Stienon and Xu in [LGSX]to define connections in the general framework of non-abelian differentiable gerbes viaEhresmann connections on Lie groupoid extensions. They also appear as multiplicativedistributions on Lie groupoids in recent work by Drummond, and Egea [DE] and Trenti-naglia [Tr]. Another definition of a general Ehresmann connection for Lie groupoids wasmore recently also given and discussed by Arias Abad and Crainic [AC].

We will now point out some additional properties of our constructions of connections forLie groupoids to highlight the relation with other constructions existing in the literatureand here in particular with those in [Be, AC].


Let H ⊂ TX1 be a connection on the Lie groupoid X = [X0 ⇒ X1]. Henceforthwe will denote m(y, z) = y z. Similarly for the pairs (y, v), (z, w) ∈ TX1 such thats(y) = t(z), dty(v) = dsz(w), the composition dm

((y, v), (z, w)

)in the induced tangent

Lie groupoid TX = [TX1 ⇒ TX0] will be denoted as (y z, v w). The connectiondefines a groupoid [s∗TX0 ⇒ TX0] as follows. Without loss of any information we denotean element (γ, s(γ), v) of s∗TX0 by (γ, v), where v ∈ Ts(γ)X0. The source, target andcomposition maps are then respectively given by

s : (γ, v) 7−→ (s(γ), v), γ ∈ X1, v ∈ Ts(γ)X0,

t : (γ, v) 7−→ (t(γ), θγ(v)),

(γ2, v2) (γ1, v1) = (γ2 γ1, v1).

(4.5)

The inversion and unit maps are obvious. Note that the composition is well definedbecause of condition (ii) in Definition 4.1. Similarly [t∗TX0 ⇒ TX0] is a groupoid withthe corresponding structure maps as given below.

s : (γ, u) 7−→ (s(γ), θγ−1(u)), γ ∈ X1, u ∈ Tt(γ)X0,

t : (γ, u) 7−→ (t(γ), u),

(γ2, u2) (γ1, u1) = (γ2 γ1, u2).

(4.6)

In fact

θ : (γ, v) 7−→ (γ, dHs−1γ (v)) 7−→ (γ, dtγ dHs

−1γ︸︷︷︸

θγ

(v))

in (4.3) defines an isomorphism of Lie groupoids

s∗TX0θ

//

t∗TX0

TX0Id

// TX0

(4.7)

Moreover, the condition (ii) in Definition 4.1 implies that

dHs−1γ2γ1

(v)− dHs−1γ2(θγ1(v)) dHs

−1γ1(v) ∈ Ker(dtγ2γ1). (4.8)

for any pair of composable γ1, γ2 ∈ X1 and v ∈ Ts(γ1)X0. We denote the correspondingelement in Ker(dtγ2γ1) by

K(γ2, γ1, v) := dHs−1γ2γ1

(v)− dHs−1γ2(θγ1(v)) dHs

−1γ1(v).

The outcome of the above observation is the following lemma.

Lemma 4.6. (dHs−1, Id) : [s∗TX0 ⇒ TX0] −→ [TX1 ⇒ TX0] defines an essentially

surjective, faithful functor if and only if K(γ2, γ1, v) vanishes for all pair of composableγ1, γ2 ∈ X1 and v ∈ Ts(γ1)X0.

Proof. That it is essentially surjective is evident. Now

dHs−1((γ2, v2) (γ1, v1)

)

= dHs−1((γ2 γ1, v1)

)

= (γ2 γ1, dHs−1γ2γ1

(v1)).


On the other hand,

dHs−1((γ2, v2)

) dHs

−1((γ1, v1)

)

=(γ2, dHs

−1γ2(v2)

)(γ1, dHs

−1γ1(v1)

)

=(γ2 γ1, dHs

−1γ2(v2) dHs

−1γ1(v1)

).

(4.9)

Then functoriality is immediate from vanishing of the left hand side of (4.8). Since dHs−1

is injective, the functor is faithful.

Hence, K(γ2, γ1, v) gives an obstruction for the groupoid [H ⇒ TX0] to be a sub-groupoid of the tangent groupoid TX = [TX1 ⇒ TX0]. Indeed vanishing of the elementin (4.8) is equivalent to the necessary and sufficient condition for a splitting as in (4.2) toyield a subgroupoid, mentioned in Lemma 2.13 of [AC]. Moreover (4.7) implies that boththe diagrams

H ////

TX0

X1 //// X0

are Cartesian and thus we arrive at the definition of a connection on a Lie groupoid in[Be, Def. 2.1]. In conclusion, we see that our definition of a connection on a Lie groupoidis more general than the one in [Be], but more strict than the definition in [AC].

4.2. Making TX0 a vector bundle over X. Let H be a connection on a given Liegroupoid X = [X1 ⇒ X0], and let

τ : TX0 −→ X0

be the natural projection. Consider the fiber product

Y1 := X1 ×s,X0,τ TX0 .

Let s′ : Y1 −→ TX0 be the projection to the second factor. Define the morphism

t′ : Y1 −→ TX0 , (x, v) 7−→ θx(v) ∈ Tt(x)X0 ,

where θx is constructed as in (4.4). Furthermore, let

p : Y1 = X1 ×s,X0,τ TX0 −→ X1

be the projection to the first factor. For any z, y ∈ Y1 with t′(y) = s′(z), define

m′(z, y) := (m(p(z), p(y)), s′(y)) .

Note that s(p(z)) = t(p(y)), so m(p(z), p(y)) is defined. Let e′ be the morphism

e′ : TX0 −→ Y1 , v 7−→ (e(τ(v)), v).

Finally, let

i′ : Y1 −→ Y1, (z, v) 7−→ (i(z), θz(v))

be the involution. It is straightforward to check that ([Y1 ⇒ TX0], s′, t′, m′, e′, i′) is a

Lie groupoid. In other words, TX0 is a vector bundle over the groupoid X.


4.3. Characteristic differential forms for connections on X. Let H be a connectionon the Lie groupoid X = [X1 ⇒ X0]. Using the canonical decomposition of the tangentspace

TX1 = H⊕ kernel(ds) ,

we get a projection∧jTX1 −→ ∧

jH → ∧jTX1 .

The composition ∧jTX1 −→ ∧jTX1 gives by duality an endomorphism of the space of

j-forms on X1.

For a differential form ω on X1, the differential form on X1 induced by the aboveendomorphism for the given connection H will be denoted by H(ω).

Definition 4.7. A differential j-form on the Lie groupoid [X1 ⇒ X0] is a differentialj-form ω on X0 such that H(s∗ω) = H(t∗ω).

Induced differential forms for an integrable distribution observe the following basicproperty.

Proposition 4.8. Let H be a connection on a Lie groupoid X = [X1 ⇒ X0] and assumethat the distribution H ⊂ TX1 is integrable. Let ω be a differential form on X0 such thatH(s∗ω) = H(t∗ω). Then H(s∗dω) = H(t∗dω).

Proof. Take a point x ∈ X1. Let L be the locally defined leaf of H passing through x.Let

ι : L → X1

be the inclusion map. SinceH(s∗ω) = H(t∗ω), it follows immediately that ι∗s∗ω = ι∗t∗ω.Therefore we have,

ι∗s∗(dω) = dι∗s∗ω = dι∗t∗ω = ι∗t∗(dω) .

But this implies thatH(s∗(dω))(x) = H(t∗(dω))(x) and so we conclude thatH(s∗(dω)) =H(t∗(dω)), which finishes the proof.

5. Connections on principal bundles over Lie groupoids and

differentiable stacks

We will now study in detail the notion and interplay of connections on principal G-bundles over Lie groupoids and differentiable stacks.

5.1. Connections on principal G-bundles over Lie groupoids. Let us start withthe groupoid picture. Let (X = [X1 ⇒ X0], s, t, m, e, i) be a Lie groupoid and G a Liegroup. The Lie algebra of G will be denoted by g. We shall consider g as a G-moduleusing the adjoint action. Let

α : EG −→ X0 (5.1)

be a principal G-bundle over X0. The adjoint vector bundle for EG

ad(EG) := EG ×G g −→ X0

is the bundle associated to EG for the adjoint action of G on g. The action of G on EGinduces an action of G on the direct image α∗TEG, where α is the projection as given in(5.1).


Definition 5.1. The Atiyah bundle for a principal G-bundle EG over X0 is the invariantdirect image

At(EG) := (α∗TEG)G ⊂ α∗TEG . (5.2)

Therefore, we have At(EG) = (TEG)/G (see also [At] for the classical and analoguenotion of the Atiyah bundle in the complex analytic context).

Let

qE : TEG −→ (TEG)/G = At(EG) (5.3)

be the quotient map.

Let Trel ⊂ TEG be the relative tangent bundle for the projection α in (5.1). It fits intothe short exact sequence of vector bundles

0 −→ Trelι0−→ TEG

dα−→ α∗TX0 −→ 0 , (5.4)

where dα is the differential of α. Using the action of G on EG, the vector bundle Trel −→EG gets identified with the trivial vector bundle EG × g −→ EG. Therefore,

ad(EG) = (α∗Trel)G ,

so ad(EG) = (α∗Trel)/G. The map of quotients Trel/G −→ (TEG)/G induced by theinclusion Trel → TEG makes ad(EG) a subbundle of At(EG). The differential of dα in(5.4) being G-equivariant therefore produces a homomorphism

(dα)′ : At(EG) −→ TX0 . (5.5)

Combining these we obtain the Atiyah exact sequence (see [At]):

0 −→ ad(EG) −→ At(EG)(dα)′

−→ TX0 −→ 0 . (5.6)

This exact sequence is the quotient of the exact sequence in (5.4) by the action of G. Nowwe can define the general notion of a connection on a principal G-bundle (compare [At])

Definition 5.2. A connection on a principal G-bundle EG over X0 is a splitting of theAtiyah exact sequence

0 −→ ad(EG) −→ At(EG)(dα)′

−→ TX0 −→ 0 . (5.7)

Giving a splitting of the exact sequence in (5.6) is evidently equivalent to giving aG-equivariant splitting of the exact sequence in (5.4). Therefore, a connection on EG is ahomomorphism

DEG : TEG −→ Trel

such that

• DEG is G-equivariant, and• DEG ι0 = IdTrel , where ι0 is the homomorphism in (5.4).

Recall that Trel = EG×g. Therefore, any homomorphism DEG : TEG −→ Trel satisfying

the above two conditions is a g-valued 1-form DEG on EG such that the correspondinghomomorphism

DEG : TEG −→ g (5.8)

is G-equivariant for the adjoint action of G on g.


Now we equip EG with the structure of a principal G-bundle on the fixed Lie groupoidX = [X1 ⇒ X0]. First we shall use the description of a principal G-bundle over X givenin Definition 3.7. So we have a Lie groupoid EG = [Q ⇒ P ] together with a groupoidmorphism

G // Qτ

//

t

s

X1

t

s

G // EGπ

// X0

(5.9)

such that both Qτ−→ X1 and EG

π−→ X0 are principal G-bundles, and the four condi-

tions in Definition 3.7 are satisfied.

Let ∇ be a connection on the principal G-bundle EG −→ X0. Let

∇ : TEG −→ g (5.10)

be the G-equivariant g-valued 1-form on EG as in (5.8) corresponding to ∇.

Definition 5.3 (Connection on a principal bundles over a Lie groupoid). A connectionon a principal G-bundle EG = [Q ⇒ EG] over the Lie groupoid X = [X1 ⇒ X0] is aconnection ∇ on the principal G-bundle EG −→ X0 such that the two g-valued 1-forms

s∗∇ and t∗∇ on Q coincide, where ∇ is the 1-form ∇ : TEG −→ g associated to theconnection ∇.

Now we adopt Definition 3.4. Let

EG := [s∗EG ⇒ EG]

be a principal G-bundle over X as in Definition 3.4. Let s : s∗EG −→ EG and t :s∗EG −→ EG respectively be the source map and the target map.

The following definition is then evidently equivalent to Definition 5.3.

Definition 5.4. A connection on the principal G-bundle EG = [s∗EG ⇒ EG] over theLie groupoid X = [X1 ⇒ X0] is a connection ∇ on the principal G-bundle EG −→ X0

such that the two g-valued 1-forms s∗∇ and t∗∇ on s∗EG coincide, where ∇ is the 1-form

∇ : TEG −→ g associated to the connection ∇.

Definition 5.5. Let ∇ be a connection on a principal G-bundle EG = [s∗EG ⇒ EG] over

the Lie groupoid X = [X1 ⇒ X0] given by a connection ∇ on EG. The curvature of ∇

is defined to be the curvature of ∇. In particular, the connection ∇ is called flat if ∇ isintegrable.

Proposition 5.6. Let X = [X1 ⇒ X0] and Y = [Y1 ⇒ Y0] be Lie groupoids, and letφ : X −→ Y be a morphism given by maps

φ0 : X0 −→ Y0 , φ1 : X1 −→ Y1

(see Definition 2.12). LetEG := [s∗EG ⇒ EG]

be a principal G-bundle on Y equipped with a connection ∇ given by a connection ∇ on theprincipal G-bundle EG −→ Y0. Then the pulled back connection φ∗

0∇ on φ∗EG −→ X0

is a connection on the principal G-bundle φ∗EG over the Lie groupoid X.


Proof. Let s and s′ (respectively, t and t′) be the source maps (respectively, target maps)of X and Y respectively. The two maps s′φ1 and φ0s from X1 to Y0 coincide. Similarly,the two maps t′ φ1 and φ0 t from X1 to Y0 also coincide.

Using this, it is straightforward to check that the connection φ∗0∇ on φ∗EG −→ X0 is

a connection on the principal G-bundle φ∗EG over the groupoid X.

The space of connections on a principal G-bundle EG over the Lie groupoid X is anaffine space for the space of all ad(EG)-valued 1-forms on the groupoid.

Henceforth, given a Lie groupoid X = [X1 ⇒ X0] we will assume that there exist agiven integrable distribution H ⊂ TX1, in other words, we have given a flat connectionon the Lie groupoid X.

Consider now again the Atiyah sequence as constructed in (5.6). The Lie bracket ofvector fields defines a Lie algebra structure on the sheaves of sections of all three vectorbundles. The Lie algebra structure on the sheaf of sections of ad(EG) is linear with respectto the multiplication by functions on X0, or in other words, the fibers of ad(EG) are Liealgebras.

Let g be again the Lie algebra for the Lie group G. Recall that ad(EG) = (EG× g)/Gis the vector bundle on X0 associated to principal G-bundle EG for the adjoint action ofG on g. Since this adjoint action of G preserves the Lie algebra structure of g, it followsthat the fibers of ad(EG) are Lie algebras identified with g up to conjugations.

Given any splitting of the Atiyah sequence (5.6)

D : TX0 −→ At(EG) ,

the obstruction for D to be compatible with the Lie algebra structure is given by a section

K(D) ∈ H0(X0, ad(EG)⊗ Ω2X0) ,

which is an ad(EG)-valued 2-form on the groupoid X.

Definition 5.7. For a principal G-bundle EG = [s∗EG ⇒ EG] over a Lie groupoid X =[X1 ⇒ X0] with a connection D, the section K(D) is called the curvature of the connectionD.

The above constructions readily imply now the following result, which allows for thedevelopment of a general Chern-Weil theory and the construction of characteristic classes,which will be discussed systematically in [BCKN].

Theorem 5.8. For any invariant form ν ∈ (Symk(g∗))G, the evaluation ν(K(D)) on thecurvature K(D) is a closed 2k-form on the Lie groupoid X.

5.2. Connections on principal G-bundles over differentiable stacks. Finally, weshall turn to the stacky picture and study connections for principal G-bundles on a dif-ferentiable stack (compare also [BMW]).

Definition 5.9 (Connection on principal bundle over differentiable stack). Let EG be aprincipal G-bundle on a differentiable stack X . A connection∇ on EG consists of the dataof a connection ∇u on each principal G-bundle EG,u, where u : U −→ X is a smooth


atlas for X , which pulls back naturally with respect to each 2-commutative diagram ofthe form

U

u!!

ϕ// V

v④④④④④④④④

X .

A connection ∇ on EG is flat or integrable if it is in addition integrable on each principalG-bundle EG,u.

Let us unravel this definition with more details. We can realize the connections foreach atlas u in terms of g-valued 1-forms such that ωEG,u

∈ H0(EG,u, Ω1EG,u⊗ g) is the

corresponding 1-form for the connection ∇u on EG,u. From the Cartesian diagram

ϕ∗EG,vϕ

//

EG,v

Uϕ

// V

we get a connection ϕ∗ωEG,von ϕ∗EG,v. The condition for the existence of a connection is

then given as follows

ωEG,u= θ∗ϕ,αϕ

∗ωEG,v.

From now on, we will assume that X is a Deligne–Mumford stack, which in particularmeans that the tangent stack TX gives rise to a vector bundle TX −→ X over X .For a general differentiable stack, this is not always the case (see [Be], [LGTX], [Hp]).

Given a principal G-bundle EG over X , we define the associated Atiyah bundle At(EG)over X by setting

At(EG)u := At(EG,u) .

for any etale morphism u : U −→ X . In addition, for each 2-commutative diagram ofthe form

U

u

ϕ// V

v~~⑤⑤⑤⑤⑤⑤⑤⑤

X

(5.11)

where u, v.ϕ are etale morphisms, we can compose the isomorphism At(EG,u)∼=−→ At(ϕ∗EG,v)

with the isomorphism At(ϕ∗EG,v)∼=−→ ϕ∗At(EG,u) to get an isomorphism

At(EG)u∼=−→ ϕ∗At(EG)v.

Similarly, given a principal G-bundle EG over X , we can define the associated adjointbundle ad(EG) over X arising from the adjoint representation of G by setting (compare[BMW, Sect. 1.4])

ad(EG) := EG ×G g ,

where ad(EG)u = ad(EG,u). We then obtain a commutative diagram of vector bundles


0 // ad(EG)u //

At(EG)u //

TU //

0

0 // ϕ∗ad(EG)v // ϕ∗At(EG)v // ϕ∗TV // 0.

which gives a well-defined short exact sequence of vector bundles over the Deligne–Mumford stack X , called the Atiyah sequence associated to the principal G-bundle EG

0 // ad(EG) // At(EG) // TX // 0.

Remark 5.10. In the situation of a general differentiable stack X , similarly as for thetangent stack, At(EG) and ad(EG) will generally not give vector bundles over X .

From the constructions and considerations above, it follows now

Proposition 5.11. A principal G-bundle EG over a Deligne–Mumford stack X admitsa connection if and only if its associated Atiyah sequence has a splitting.

Finally, we have the following comparison theorem for the existence of a connection ofa principal G-bundle over a Deligne–Mumford stack:

Theorem 5.12. Giving a connection (respectively, flat connection) on a principal G-bundle EG over a Deligne–Mumford stack X with etale atlas x : X0 −→ X is equivalentto giving a connection (respectively, flat connection) on the associated principal G-bundleEG over the groupoid X = [X1 ⇒ X0]. Giving a connection (respectively, flat connection)on a principal G-bundle EG = [s∗EG ⇒ EG] over an etale Lie groupoid X = [X1 ⇒ X0] isequivalent to giving a connection (respectively, flat connection) on the associated principalG-bundle EG of EG-torsors over the classifying stack BX of X-torsors.

Proof. This follows from the categorical equivalence of Theorem 3.6 between the categoryBunG(X ) of principal G-bundles over X and the category BunG(X) of principal G-bundles over the associated Lie groupoid X = [X1 ⇒ X0] and the fact that a splitting ofthe Atiyah sequence of associated vector bundles over the stack X

0 // ad(EG) // At(EG) // TX // 0

corresponds to a splitting of the Atiyah sequence of associated vector bundles over thesmooth space X0

0 −→ ad(EG) −→ At(EG)−→TX0 −→ 0

and vice versa involving the 2-Cartesian diagram

EG //

X0

EGπ

// X

and unraveling the explicit constructions of associated bundles as given in the proof ofTheorem 3.6.


Let us now consider the classifying stack B∇G of principal G-bundles with connections.The objects are triples (P, ω, S) where S is a smooth space of S and P is a principalG-bundle over S and ω ∈ Ω1(S, g)G a connection 1-form. A morphisms (P, ω, S) −→(P ′, ω′, S ′) is given by a commutative diagram

Pϕ

//

P ′

S // S ′

where ϕ : P −→ P ′ is a G-equivariant map and ϕ∗ω′ = ω. Then B∇G together withthe projection functor

π : B∇G −→ S , (P, ω, S) 7−→ S

is a groupoid fibration over S. We note that B∇G is, in fact, a stack as principal G-bundles glue and connection 1-forms on the principal bundles glue as well (compare also[CLW]).

Remark 5.13. It is not clear in general if and under which conditions B∇G is actuallya differentiable or Deligne–Mumford stack and not just a stack over S.

The above constructions now allow us to characterize principal G-bundles over differ-entiable stacks equivalently as follows.

Proposition 5.14. Let X be a differentiable stack. Giving a principal G-bundle withconnection over X is equivalent to giving a morphism of stacks X −→ B∇G and twoprincipal G-bundles with connections over X are isomorphic if and only if the corre-sponding morphisms of stacks X −→ B∇G are 2-isomorphic.

Similarly, as before, we can consider the groupoid fibration Bun∇G(X ) over S whose

objects over a smooth space U are principal G-bundles EG over X × U with connec-tions and whose morphisms are given by pullback diagrams of principal G-bundles withconnections as above. So from Proposition 5.14 we get the following.

Proposition 5.15. There is an equivalence of groupoid fibrations over S

Bun∇G(X ) ∼= H om(X , B

∇G) .

Collier–Lerman–Wolbert [CLW, Theorem 6.4] proved a similar result involving holo-nomy and parallel transport for principal G-bundles over stacks.

Acknowledgements

The first author acknowledges the support of a J. C. Bose Fellowship. The secondauthor acknowledges research support from SERB, DST, Government of India grantMTR/2018/000528. The fourth author would like to thank the Tata Institute of Funda-mental Research (TIFR) in Mumbai and the University of Leicester for financial support.


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School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road,

Mumbai 400005, India

Email address : [email protected]

School of Mathematics, Indian Institute of Science Education and Research Thiru-

vananthapuram, Maruthamala P.O., Vihtura, Kerala 695551, India


School of Mathematics, Indian Institute of Science Education and Research Thiru-

vananthapuram, Maruthamala P.O., Vihtura, Kerala 695551, India


Pure Mathematics Group, School of Mathematics and Actuarial Science, University

of Leicester, University Road, Leicester LE1 7RH, England, UK


http://arxiv.org/abs/1306.1164

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