# Atiyah sequences and connections on principal bundles over Lie groupoids and
differentiable stacks

Indranil Biswas  School of Mathematics, Tata Institute of Fundamental
Research, Homi Bhabha Road, Mumbai 400005, India<EMAIL_ADDRESS>,
Saikat Chatterjee  School of Mathematics, Indian Institute of Science
Education and Research Thiruvananthapuram, Maruthamala P.O., Vihtura, Kerala
695551, India<EMAIL_ADDRESS>,  Praphulla Koushik  School of
Mathematics, Indian Institute of Science Education and Research
Thiruvananthapuram, Maruthamala P.O., Vihtura, Kerala 695551, India
<EMAIL_ADDRESS>and  Frank Neumann  Pure Mathematics Group, School of
Computing and Mathematical Sciences, University of Leicester, University Road,
Leicester LE1 7RH, England, UK<EMAIL_ADDRESS>
###### Abstract.

We construct and study both general and integrable connections on Lie
groupoids and differentiable stacks, as well as on principal bundles over them
using an Atiyah exact sequence of vector bundles associated to transversal
tangential distributions.

###### Key words and phrases:

Atiyah exact sequence, principal bundles, Lie groupoids, connections,
differentiable stacks

###### 2010 Mathematics Subject Classification:

Primary 53C08, Secondary 22A22, 58H05, 53D50

###### Contents

  1. 1 Introduction
  2. 2 Smooth spaces, groupoids, and stacks
    1. 2.1 Smooth spaces and fibered categories
    2. 2.2 Stacks over smooth spaces
    3. 2.3 Lie groupoids and differentiable stacks
  3. 3 Principal bundles over differentiable stacks and Lie groupoids
    1. 3.1 Principal bundles over differentiable stacks
    2. 3.2 Principal bundles over Lie groupoids
  4. 4 Connections on Lie groupoids and differentiable stacks
    1. 4.1 Connections on Lie groupoids and differentiable stacks
    2. 4.2 Making $TX_{0}$ a vector bundle over ${\mathbb{X}}$
    3. 4.3 Characteristic differential forms for connections on ${\mathbb{X}}$
  5. 5 Connections on principal bundles over Lie groupoids and differentiable stacks
    1. 5.1 Connections on principal $G$-bundles over Lie groupoids
    2. 5.2 Connections on principal $G$-bundles over differentiable stacks

##  1\. Introduction

We develop a general theory of connections for principal bundles over Lie
groupoids and differentiable stacks using Atiyah exact sequences associated to
transversal tangential distributions. 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 ${\mathbb{X}}\,:=\,[X_{1}\rightrightarrows X_{0}]$ with
both $X_{0}$ and $X_{1}$ smooth manifolds, such that the source map $s$ is a
submersion, we introduce connections on ${\mathbb{X}}$ as a distribution
${\mathcal{H}}\,\subset\,TX_{1}$ transversal to the fibers of the source map
$s$. Such a connection is said to be integrable or flat if, in addition, the
corresponding distribution $\mathcal{H}$ is integrable. These connections
(respectively, flat connections) also give rise to connections (respectively,
flat connections) on differentiable stacks under certain compatibility
conditions. For the particular case of Deligne–Mumford stacks, which are
presented by étale Lie groupoids and include descriptions for orbifolds and
foliations, such a connection always exists. Indeed, for a Deligne–Mumford
stack a natural tangential distribution is given by the tangent bundle itself.

Now given a Lie group $G$ and a principal $G$-bundle over a Lie groupoid
${\mathbb{X}}\,:=\,[X_{1}\rightrightarrows X_{0}]$ equipped with such a
connection ${\mathcal{H}}\,\subset\,TX_{1}$, we can define the notion of a
connection on the principal $G$-bundle. A principal $G$-bundle over the Lie
groupoid ${\mathbb{X}}$ is basically given by a principal $G$-bundle
$\alpha\,:\,E_{G}\,\longrightarrow\,X_{0}$ and some extra compatibility data
reflecting the groupoid structure. More precisely, a connection for a
principal $G$-bundle over ${\mathbb{X}}$ corresponds to a splitting of the
associated Atiyah exact sequence of vector bundles [At]

$0\,\longrightarrow\,\text{ad}(E_{G})\,\longrightarrow\,\text{At}(E_{G})\,\longrightarrow\,TX_{0}\,\longrightarrow\,0$

which satisfies the condition of being compatible with the various data of the
structures involved; the details are in Section 5. This then allows us to
define the associated curvature and characteristic differential forms for
connections on principal bundles over Lie groupoids.

Furthermore, using adequate groupoid presentations, these constructions extend
to a framework that enables us to define and study connections and
characteristic forms for principal $G$-bundles over differentiable stacks. In
the particular case of Deligne-Mumford stacks, which are represented by étale
Lie groupoids, we develop the theory of connections intrinsically and
construct the Atiyah exact sequence out of the stack data. This relies on the
fact that in the case of Deligne-Mumford stacks, the associated Atiyah exact
sequence is again a sequence of vector bundles in a natural way. These
constructions also corresponds to related 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 remark that most of the
concepts and constructions presented here also work equally well in the
holomorphic and algebraic geometrical setting for the differential geometry of
complex analytic and algebraic stacks. An associated Chern-Weil theory of
characteristic classes for our setting is developed in a related article by
the authors [BCKN]. Some of the constructions and results presented here were
announced earlier in [BN].

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

Outline and organization of the article. In the first section (Section 2), we
introduce the category of smooth spaces over which our stacks and groupoids
are constructed and present the basic notions of fibered categories, stacks,
and groupoids. We also discuss the 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 stacks and Lie groupoids
and their categorical interplay. In Section 4 we introduce notions and basic
properties of connections on Lie groupoids and differentiable stacks using
vertical tangential distributions and compare our constructions with other
existing frameworks in the literature. We also construct characteristic
differential forms for our general connections. In the final section (Section
5), we define and study connections on Lie groupoids as splittings of
associated Atiyah exact sequences. Finally, we apply the theory to study
connections on principal bundles over differentiable stacks and give general
characterizations for connections on principal bundles over Deligne-Mumford
stacks, which corresponds to étale 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 of ${\mathcal{C}}^{\infty}$-manifolds,

  * •

the category of complex analytic manifolds, and

  * •

the category of smooth schemes of finite type over the field of complex
numbers,

as the category $\mathfrak{S}$ of smooth spaces and smooth maps. The tangent
bundle of any smooth space $X$ will be denoted by $TX$. A smooth space will
mean an object in $\mathfrak{S}$, and a smooth map will refer to a morphism in
$\mathfrak{S}$ which is a submersion, and by a submersion, we mean a smooth
map whose differential restricted to $T_{x}X$ is surjective for every point
$x$ of the domain $X$. An étale map is a smooth immersion in $\mathfrak{S}$
(so it is also a submersion).

For a smooth space $X$, the structure sheaf of it will be denoted by
${\mathcal{O}}_{X}$. A vector bundle on $X$ will be identified with its sheaf
of sections, which is a finitely generated locally free sheaf of
${\mathcal{O}}_{X}$-modules. The cotangent bundle of $X$ will be denoted by
$\Omega^{1}_{X}\,=\,T^{*}X$, and the $p$-th exterior power of $T^{*}X$ will be
denoted by $\Omega^{p}_{X}\,:=\,\bigwedge^{p}T^{*}X$. An integrable
distribution on a smooth space $X$ is a subbundle ${\mathcal{H}}\,\subset\,TX$
of the tangent bundle of $X$ such that its annihilator ${\mathcal{H}}^{\bot}$
generates an ideal in $\bigoplus_{k\geq 0}\Omega^{k}_{X}$ which is preserved
by the exterior differential of the de Rham complex; equivalently,
${\mathcal{H}}$ is closed under the Lie bracket operation on vector fields.

The big étale site ${\mathfrak{S}}_{et}$ on the category $\mathfrak{S}$ is
given by the following Grothendieck topology on $\mathfrak{S}$. We call a
family $\\{U_{i}\longrightarrow X\\}$ of morphisms in $\mathfrak{S}$ with
target $X$ a covering family of $X$, if all smooth maps
$U_{i}\,\longrightarrow\,X$ are étale and the total map from the disjoint
union

$\coprod_{i}U_{i}\longrightarrow X$

is surjective. This defines a pretopology on $\mathfrak{S}$ generating a
Grothendieck topology, which is known as the big étale topology on
$\mathfrak{S}$ (compare [SGA4, Exposé II] and [Vi]). If either or both of two
morphisms $U\,\longrightarrow\,X$ and $V\,\longrightarrow\,X$ in
$\mathfrak{S}$ is a submersion, then their fiber product
$U\times_{X}V\,\longrightarrow\,X$ exists.

######  Definition 2.1 (Groupoid fibration).

A groupoid fibration over $\mathfrak{S}$ is a category $\mathscr{X}$, together
with a functor

$\pi_{\mathscr{X}}\,:\,\mathscr{X}\,\longrightarrow\,\mathfrak{S}$

satisfying the following axioms:

  * (i)

For every morphism $V\longrightarrow U$ in $\mathfrak{S}$, and every object
$x$ of $\mathscr{X}$ lying over $U$, there exists an arrow $y\longrightarrow
x$ in $\mathscr{X}$ lying over $V\longrightarrow U$.

  * (ii)

For every commutative triangle

$W\,\longrightarrow\,V\,\longrightarrow\,U$

in $\mathfrak{S}$ and all morphisms $z\,\longrightarrow\,x$ and
$y\,\longrightarrow\,x$ in $\mathscr{X}$ lying over $W\,\longrightarrow\,U$
and $V\,\longrightarrow\,U$ respectively, there exists a unique arrow
$z\,\longrightarrow\,y$ in $\mathscr{X}$ lying over $W\,\longrightarrow\,V$
such that the composition

$z\,\longrightarrow\,y\,\longrightarrow\,x$

is the morphism $z\,\longrightarrow\,x$.

The condition (ii) in Definition 2.1 ensures that the object $y$ over $V$,
which exists by Definition 2.1(i), is unique up to a unique isomorphism. Any
choice of such an object $y$ is called a pullback of $x$ via the morphism
$f\,:\,V\,\longrightarrow\,U$. We will write as usual $y\,=\,x|_{V}$ or
$y\,=\,f^{*}x$.

Let $\mathscr{X}$ be a groupoid fibration over $\mathfrak{S}$. The subcategory
of $\mathscr{X}$ consisting of all objects lying over a fixed object $U$ of
$\mathfrak{S}$ with the morphisms being those lying over the identity morphism
$id_{U}$ is called the fiber or category of sections of $\mathscr{X}$ over
$U$. The fiber of $\mathscr{X}$ over $U$ will be denoted by
${\mathscr{X}}(U)$.

Groupoid fibrations over $\mathfrak{S}$ form a 2-category in which fiber
products exist (see [Gr, Vi]). For two groupoid fibrations

$\pi_{\mathscr{X}}\,:\,{\mathscr{X}}\,\longrightarrow\,\mathfrak{S}\ ~{}\
\text{ and }\ ~{}\
\pi_{\mathscr{Y}}\,:\,{\mathscr{Y}}\,\longrightarrow\,\mathfrak{S}\,,$

the $1$-morphisms from ${\mathscr{X}}$ to ${\mathscr{Y}}$ are given by
functors $\phi\,:\,{\mathscr{X}}\,\longrightarrow\,{\mathscr{Y}}$ such that

$\pi_{\mathscr{Y}}\circ\phi\,=\,\pi_{\mathscr{X}}\,.$

The $2$-morphisms are given by natural transformations between these
projection functor preserving functors.

######  Example 2.2.

Let $F\,:\,{\mathfrak{S}}\,\longrightarrow\,(Sets)$ be a presheaf, meaning a
contravariant functor. We get a groupoid fibration ${\mathscr{X}}$, where the
objects are pairs of the form $(U,\,x)$, with $U$ a smooth space and
$x\,\in\,F(U)$, while a morphism $(U,\,x)\,\longrightarrow\,(V,\,y)$ is a
smooth map $f\,:\,U\,\longrightarrow\,V$ such that $x\,=\,y|_{F(U)}$,
equivalently, $x\,=\,F(f)(y)$. The projection functor is given by

$\pi\,:\,{\mathscr{X}}\,\longrightarrow\,{\mathfrak{S}}\,,\,\,\
(U,\,x)\,\longmapsto\,U\,.$

Therefore, any sheaf $F\,:\,{\mathfrak{S}}\,\longrightarrow\,(Sets)$ gives a
groupoid fibration over $\mathfrak{S}$. In particular, every smooth space $X$
gives a groupoid fibration $\underline{X}$ over $\mathfrak{S}$ as the sheaf
represented by $X$, in other words,

${\underline{X}}(U)\,=\,Hom_{\mathfrak{S}}(U,\,X)\,.$

To simplify notation, we will identify $\underline{X}$ with the smooth space
$X$ without further clarification.

A category $\mathscr{X}$ fibered in groupoids over $\mathfrak{S}$ is
representable if there exists a smooth space $X$ isomorphic to $\mathscr{X}$
as groupoid fibrations over $\mathfrak{S}$. We call a morphism of groupoid
fibrations $\mathscr{X}\,\longrightarrow\,\mathscr{Y}$ a representable
submersion if for every smooth space $U$ and every morphism
$U\,\longrightarrow\,{\mathscr{Y}}$, the fiber product
${\mathscr{X}}\times_{\mathscr{Y}}U$ is representable, and the induced
morphism of smooth spaces
${\mathscr{X}}\times_{\mathscr{Y}}U\,\longrightarrow\,U$ 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
$\mathscr{X}$ here will always mean a stack over the big étale site
${\mathfrak{S}}_{et}$ of smooth spaces.

######  Definition 2.3 (Stack).

A groupoid fibration $\mathscr{X}$ over $\mathfrak{S}$ is a stack if the
following gluing axioms hold with respect to the site ${\mathfrak{S}}_{et}$:

  * (i)

Take any smooth space $X$ in $\mathfrak{S}$, any two objects $x,\,y$ in
${\mathscr{X}}$ lying over $X$ and any two isomorphisms

$\phi\,,\,\psi\,:\,x\,\longrightarrow\,y$

over $X$. If the condition $\phi|_{U_{i}}\,=\,\psi|_{U_{i}}$ holds for all
$U_{i}$ in a covering $\\{U_{i}\longrightarrow X\\}$, then $\phi\,=\,\psi$.

  * (ii)

Take any smooth space $X$ in $\mathfrak{S}$, any two objects
$x\,,y\,\in\,{\mathscr{X}}$ lying over $X$ and any covering
$\\{U_{i}\longrightarrow X\\}$ with isomorphisms

$\phi_{i}\,:\,x|_{U_{i}}\,\longrightarrow\,y|_{U_{i}}$

for every $i$. If the condition $\phi_{i}|_{U_{ij}}\,=\,\phi_{j}|_{U_{ij}}$
holds for all $i,\,j$, then there exists an isomorphism
$\phi\,:\,x\,\longrightarrow\,y$ with $\phi|_{U_{i}}\,=\,\phi_{i}$ for all
$i$.

  * (iii)

For any smooth space $X$ in $\mathfrak{S}$, any covering
$\\{U_{i}\longrightarrow X\\}$, any family $\\{x_{i}\\}$ of objects $x_{i}$ in
the fiber ${\mathscr{X}}_{U_{i}}$ and any family of morphisms
$\\{\phi_{ij}\\}$, where

$\phi_{ij}\,:\,x_{i}|_{U_{ij}}\,\longrightarrow\,x_{j}|_{U_{ij}}$

satisfies the cocycle condition $\phi_{jk}\circ\phi_{ij}\,=\,\phi_{ik}$ in
${\mathscr{X}}(U_{ijk})$, there exists an object $x$ lying over $X$ with
isomorphisms

$\phi_{i}\,:\,x|_{U_{i}}\,\longrightarrow\,x_{i}$

such that $\phi_{ij}\circ\phi_{i}\,=\,\phi_{j}$ in ${\mathscr{X}}(U_{ij})$.

The isomorphism $\phi$ in Definition 2.3(ii) is unique by Definition 2.3(i).
Similarly, from Definition 2.3(i) and Definition 2.3(ii), it follows that the
object $x$ whose existence is asserted in Definition 2.3(iii) is unique up to
a unique isomorphism. All pullbacks mentioned in Definition 2.3 are only
unique up to isomorphism, but the properties do not depend on particular
choices.

In order to be able to do geometry on stacks, we need to compare them with
smooth spaces and extend the geometry to stacks. From now on, we will restrict
ourselves to stacks over the category $\mathfrak{S}$ of
${\mathcal{C}}^{\infty}$-manifolds, but we remark that we have incarnations of
the analogue concepts and constructions for the category $\mathfrak{S}$ of
complex analytic manifolds as well as smooth schemes of finite type over the
complex numbers.

######  Definition 2.4 (Differentiable stack).

A stack $\mathscr{X}$ over the site $\mathfrak{S}_{et}$ is called
differentiable if there exists a smooth space $X$ in $\mathfrak{S}$ and a
surjective representable submersion

$x\,:\,X\,\longrightarrow\,{\mathscr{X}},$

i.e., there exists a smooth space $X$ together with a morphism of stacks

$x\,:\,X\,\longrightarrow\,{\mathscr{X}},$

such that for every smooth space $U$ and every morphism of stacks
$U\,\longrightarrow\,{\mathscr{X}}$, the following two hold:

  1. (1)

the fiber product $X\times_{\mathscr{X}}U$ is representable, and

  2. (2)

the induced morphism of smooth spaces
$X\times_{\mathscr{X}}U\,\longrightarrow\,U$ is a surjective submersion.

If $\mathscr{X}$ is a differentiable stack, such a surjective representable
submersion $x\,:\,X\,\longrightarrow\,{\mathscr{X}}$ is called a presentation
or atlas for the stack $\mathscr{X}$. It need not be unique, in other words, a
differentiable stack can have different presentations.

######  Definition 2.5.

A differentiable stack $\mathscr{X}$ is a (proper) Deligne–Mumford stack if it
has a (proper) étale presentation.

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

The incarnations of these concepts over the site $\mathfrak{S}_{et}$ of
complex analytic manifolds respectively, smooth schemes of finite type over
${\mathbb{C}}$ will be referred to as complex analytic stacks, respectively,
algebraic stacks.

###  2.3. Lie groupoids and differentiable stacks

Differentiable stacks are also incarnations of Lie groupoids, as we will
recall now in detail (see also [BX, dH]).

######  Definition 2.6 (Lie groupoid).

A Lie groupoid $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ is a groupoid
internal to the category $\mathfrak{S}$ of smooth spaces, meaning the space
$X_{1}$ of arrows and the space $X_{0}$ of objects are objects of
$\mathfrak{S}$ and all structure morphisms

$s,\,t\,:\,X_{1}\,\longrightarrow\,X_{0}\,,\ \
m\,:\,X_{1}\times_{s,X_{0},t}X_{1}\longrightarrow X_{1}\,,$
$i\,:\,X_{1}\,\longrightarrow\,X_{1}\,,\ \
e\,:\,X_{0}\,\longrightarrow\,X_{1}$

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

If $s$ and $t$ are étale, the groupoid
${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ is called étale. If the
anchor map

$(s,\,t)\,:\,X_{1}\,\longrightarrow\,X_{0}\times X_{0}$

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

A Lie group $G$ is a Lie groupoid $[G\,\rightrightarrows\,*]$ with one object,
meaning the space $X_{0}$ is just a point in $\mathfrak{S}$.

Every Lie groupoid ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ gives
rise to the associated tangent groupoid
$T{\mathbb{X}}\,:=\,[TX_{1}\rightrightarrows TX_{0}]$.

######  Example 2.7.

Let $X$ be a smooth space. The groupoid fibration $\underline{X}$ is, in fact,
a differentiable stack over $\mathfrak{S}$. A presentation is given by the
identity morphism $id_{X}$.

######  Example 2.8 (Classifying stack).

For a Lie group $G$, let ${\mathscr{B}}G$ be the category which has as objects
all pairs $(P,\,S)$, where $S$ is a smooth space of $\mathfrak{S}$ and $P$ is
a principal $G$-bundle over $S$; a morphism
$(P,\,S)\,\longrightarrow\,(Q,\,T)$ is a commutative diagram

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T}$

where $\varphi\,:\,P\,\longrightarrow\,Q$ is a $G$-equivariant map. Note that
the above diagram is Cartesian. Then ${\mathscr{B}}G$ together with the
projection functor

$\pi:{\mathscr{B}}G\,\longrightarrow\,{\mathfrak{S}}\,,\
~{}(P,\,S)\,\longmapsto\,S$

is a groupoid fibration over $\mathfrak{S}$. We note that ${\mathscr{B}}G$ is
in fact a differentiable stack, and it is known as the classifying stack of
$G$ (see also Example 2.11 below). A presentation is given by the
representable surjective submersion $*\,\longrightarrow\,{\mathscr{B}}G$ where
$*$ is a “point” in $\mathfrak{S}$.

######  Definition 2.9.

Let ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ be a Lie groupoid. A
(left) $\mathbb{X}$-space is given by an object $P$ of $\mathfrak{S}$ together
with a smooth map $\pi\,:\,P\,\longrightarrow\,X_{0}$ and a map

$\sigma\,:\,Q\,:=\,X_{1}\times_{s,X_{0},\pi}P\,\longrightarrow\,P\,,\ \
~{}\sigma(\gamma,\,x)\,:=\,\gamma\cdot x$

such that,

  * (i)

$\pi(\gamma\cdot x)\,=\,t(\gamma)$ for all
$(\gamma,\,x)\,\in\,X_{1}\times_{s,X_{0},\pi}P$,

  * (ii)

$e(\pi(x))\cdot x\,=\,x$ for all $x\,\in\,X_{1}$, and

  * (iii)

$(\delta\cdot\gamma)\cdot x\,=\,\delta\cdot(\gamma\cdot x)$ for all
$(\gamma,\,\delta,\,x)\,\in\,X_{1}\times_{s,X_{0},t}X_{1}\times_{s,X_{0},\pi}P$.

Similarly, we can also define a (right) $\mathbb{X}$-space.

Let $\mathscr{X}$ be a differentiable stack with a given presentation
$x\,:\,X\,\longrightarrow\,{\mathscr{X}}$. We can associate to $\mathscr{X}$ a
Lie groupoid ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ as follows: Let
$X_{0}\,:=\,X$ and $X_{1}\,:=\,X\times_{\mathscr{X}}X$ and the source and
target morphisms

$s,\,\,t\,:\,X\times_{\mathscr{X}}X\,\rightrightarrows\,X$

of $\mathbb{X}$ being the first and second canonical projection morphisms. The
composition of morphisms $m$ in $\mathbb{X}$ is given as projection to the
first and third factor

$X\times_{\mathscr{X}}X\times_{\mathscr{X}}X\,\cong\,(X\times_{\mathscr{X}}X)\times_{X}(X\times_{\mathscr{X}}X)\,\longrightarrow\,X\times_{\mathscr{X}}X\,.$

The morphism $X\times_{\mathscr{X}}X\,\longrightarrow\,X\times_{\mathscr{X}}X$
that interchanges the two factors gives the inverse morphism $i$ for the
groupoid, while the unit morphism $e$ is given by the diagonal morphism
$X\,\longrightarrow\,X\times_{\mathscr{X}}X$. As a presentation
$x\,:\,X\,\longrightarrow\,{\mathscr{X}}$ of a differentiable stack is a
submersion, it follows that the source and target morphisms

$s,\,\,t\,:\,X\times_{\mathscr{X}}X\,\rightrightarrows\,X,$

being induced maps from the fiber product, are also submersions.

In the opposite direction, given a Lie groupoid $\mathbb{X}$ we can associate
a differentiable stack $\mathscr{X}$ to it. Basically this is a generalization
of associating to a Lie group $G$ its classifying stack ${\mathscr{B}}G$ (see
Example 2.8). For this we now define (compare also [BX]):

######  Definition 2.10 (Torsors).

Let ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ be a Lie groupoid, and
let $S$ be a smooth space. A (right) $\mathbb{X}$-torsor over $S$ is a smooth
space $P$ together with a surjective submersion
$\pi\,:\,P\,\longrightarrow\,S$ and a right action of $\mathbb{X}$ on $P$ (see
the second part of Definition 2.9) satisfying the condition that $\pi$ is
$\mathbb{X}$-invariant, i.e., $\pi(\gamma\cdot p)\,=\,\pi(p)$ for all
$(\gamma,\,x)\,\in\,X_{1}\times_{s,X_{0},a}P$ and the extra condition that for
all $p,\,\,p^{\prime}\,\in\,P$ with $\pi(p)\,=\,\pi(p^{\prime})$, there exists
a unique $x\,\in\,X_{1}$ such that $p\cdot x$ is defined and $p\cdot
x\,=\,p^{\prime}$.

Let $\pi\,:\,P\,\longrightarrow\,S$ and $\rho\,:\,Q\,\longrightarrow\,T$ be
$\mathbb{X}$-torsors. A morphism of $\mathbb{X}$-torsors from the first one to
the second one is given by a Cartesian diagram of smooth maps

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T}$
(2.1)

such that $\varphi$ is a $X_{1}$-equivariant map. Again, there is also a
similar notion of a (left) $\mathbb{X}$-torsor over a smooth space $S$.

Given a $\mathbb{X}$-torsor $P$, the map $a:P\,\longrightarrow\,X_{0}$ is
called the anchor map or momentum map. The surjective submersion
$\pi\,:\,P\,\longrightarrow\,S$ is called the structure map.

For a Lie groupoid ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$, let
${\mathscr{B}}{\mathbb{X}}$ denote the category of $\mathbb{X}$-torsors,
meaning the category whose objects are pairs $(P,\,S)$, where $S$ is a smooth
space and $P$ a $\mathbb{X}$-torsor over $S$. The morphisms
$(P,\,S)\,\longrightarrow\,(Q,\,T)$ are given by Cartesian diagrams as in
(2.1). Then ${\mathscr{B}}{\mathbb{X}}$ is a groupoid fibration over
$\mathfrak{S}$ with a canonical projection functor

$\pi\,:\,{\mathscr{B}}{\mathbb{X}}\,\longrightarrow\,{\mathfrak{S}}\,,\
~{}(P,\,S)\,\longmapsto\,S\,.$

It turns out that ${\mathscr{B}}{\mathbb{X}}$ is, in fact, a differentiable
stack, the classifying stack of $\mathbb{X}$-torsors. The proof of [BX,
Proposition 2.3] works in any of the three categories $\mathfrak{S}$.

######  Example 2.11.

In the case where the Lie groupoid ${\mathbb{X}}\,=\,[G\rightrightarrows*]$ is
a Lie group, a $\mathbb{X}$-torsor over a smooth space $S$ is simply a
$G$-torsor or principal $G$-bundle over $S$. The associated classifying stack
is the classifying stack ${\mathscr{B}}G$ of the Lie group $G$.

As presentations for a given differentiable stack are not unique, the
associated Lie groupoids might be different. In order to define algebraic and
geometric invariants for differentiable stacks, like cohomology or
differential forms, they should, however, be independent 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
differentiable stacks.

######  Definition 2.12.

Let ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ and
${\mathbb{Y}}\,=\,[Y_{1}\rightrightarrows Y_{0}]$ be Lie groupoids. A morphism
of Lie groupoids is a functor
$\phi\,:\,{\mathbb{X}}\,\longrightarrow\,{\mathbb{Y}}$ given by two smooth
maps $\phi\,=\,(\phi_{1},\,\phi_{0})$ with

$\phi_{0}\,:\,X_{0}\,\longrightarrow\,Y_{0}\,,\ \
\phi_{1}\,:\,X_{1}\,\longrightarrow\,Y_{1}$

which commute with all structure morphisms of the groupoids.

A morphism $\phi\,:\,{\mathbb{X}}\,\longrightarrow\,{\mathbb{Y}}$ of Lie
groupoids is an étale morphism, if both maps
$\phi_{0}\,:\,X_{0}\,\longrightarrow\,Y_{0}$ and
$\phi_{1}\,:\,X_{1}\,\longrightarrow\,Y_{1}$ are étale.

A morphism $\phi\,:\,{\mathbb{X}}\,\longrightarrow\,{\mathbb{Y}}$ of Lie
groupoids is a Morita morphism or essential equivalence if

  * (i)

$\phi_{0}\,:\,X_{0}\,\longrightarrow\,Y_{0}$ is a surjective submersion, and

  * (ii)

the diagram

$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(s,t)\,\,\,}$$\scriptstyle{\phi_{1}}$$\textstyle{X_{0}\times
X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{0}\times\phi_{0}}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(s,t)\,\,\,}$$\textstyle{Y_{0}\times
Y_{0}}$

is Cartesian, or in other words, $X_{1}\,\cong\,Y_{1}\times_{Y_{0}\times
Y_{0}}(X_{0}\times X_{0})$.

Two Lie groupoids $\mathbb{X}$ and $\mathbb{Y}$ are Morita equivalent, if
there exists a third Lie groupoid $\mathbb{W}$ and Morita morphisms

${\mathbb{X}}\,\stackrel{{\scriptstyle\phi}}{{\longleftarrow}}\,{\mathbb{W}}\,\stackrel{{\scriptstyle\psi}}{{\longrightarrow}}\,{\mathbb{Y}}\,.$

######  Theorem 2.13 (Morita equivalence).

Let ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ and
${\mathbb{Y}}\,=\,[Y_{1}\rightrightarrows Y_{0}]$ be Lie groupoids. Let
$\mathscr{X}$ and $\mathscr{Y}$ be the associated differentiable stacks, i.e.,
$\mathscr{X}$ is the classifying stack ${\mathscr{B}}{\mathbb{X}}$ of
$\mathbb{X}$-torsors and $\mathscr{Y}$ is the classifying stack
${\mathscr{B}}{\mathbb{Y}}$ of $\mathbb{Y}$-torsors. Then the following three
statements are equivalent:

  * (i)

The differentiable stacks $\mathscr{X}$ and $\mathscr{Y}$ are isomorphic.

  * (ii)

The Lie groupoids $\mathbb{X}$ and $\mathbb{Y}$ are Morita equivalent.

  * (iii)

there exists a smooth space $Q$ together with two smooth maps
$f\,:\,Q\,\longrightarrow\,X_{0}$ and $g\,:\,Q\,\longrightarrow\,Y_{0}$ and
(commuting actions) of $X_{1}$ and $Y_{1}$ in such a way that $Q$ is at the
same time a left ${\mathbb{X}}$-torsor over $Y_{0}$ via $g$ and a right
$\mathbb{Y}$-torsor over $X_{0}$ via $f$, in other words, $Q$ is a
${\mathbb{X}}$-$\mathbb{Y}$-bitorsor.

###### Proof.

For the differentiable category of ${\mathcal{C}}^{\infty}$-manifolds, this is
[BX, Theorem 2.2]. It is immediate to see that the proof works verbatim for
any of the other categories $\mathfrak{S}$ of smooth spaces, meaning for
complex analytic as well as algebraic stacks. ∎

Therefore different presentations of the same differentiable stack are given
by Morita equivalent Lie groupoids. Conversely, Morita equivalent Lie
groupoids present isomorphic differentiable stacks. The case of étale
groupoids presenting Deligne-Mumford stacks is of particular importance as
these can be used naturally also to describe orbifolds and foliations [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
differentiable stack and over a Lie groupoid.

###  3.1. Principal bundles over differentiable stacks

Let us start with the definition of principal bundles over differentiable
stacks.

######  Definition 3.1 (Principal bundle over differentiable stack).

Let $G$ be a Lie group and $\mathscr{X}$ a differentiable stack. A principal
$G$-bundle or $G$-torsor ${\mathscr{E}}_{G}$ over $\mathscr{X}$ is given by a
differentiable stack $\mathscr{E}_{G}$, a morphism of differentiable stacks
$\pi\,:\,\mathscr{E}_{G}\,\longrightarrow\,\mathscr{X}$, and a $2$-Cartesian
diagram

$\textstyle{{\mathscr{E}}_{G}\times
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\scriptstyle{p_{1}}$$\textstyle{{\mathscr{E}}_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{{\mathscr{E}}_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathscr{X}.}$

such that for any submersion

$f\,:\,U\,\longrightarrow\,\mathscr{X}$

from a smooth space $U$, the pullback by $f$ in the above diagram defines a
principal $G$-bundle on $U$. Morphisms
$\rho\,:\,{\mathscr{E}}_{G}\,\longrightarrow\,{\mathscr{F}}_{G}$ between two
principal $G$-bundles ${\mathscr{E}}_{G}$ and ${\mathscr{F}}_{G}$ are defined
in the obvious way.

Principal $G$-bundles over a differentiable stack $\mathscr{X}$ can also be
defined directly by using a presentation or atlas. In other words, a principal
$G$-bundle ${\mathscr{E}}_{G}$ over a differentiable stack $\mathscr{X}$ is
given by a principal $G$-bundle $E_{G}\,\longrightarrow\,X_{0}$ for an atlas
$X_{0}\,\longrightarrow\,{\mathscr{X}}$ together with an isomorphism of the
pullbacks
$p_{1}^{*}E_{G}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,p_{2}^{*}E_{G}$
on the fiber product $X_{0}\times_{\mathscr{X}}X_{0}$ satisfying the cocycle
condition on $X_{0}\times_{\mathscr{X}}X_{0}\times_{\mathscr{X}}X_{0}$. It
turns out that for any submersion $f\,:\,U\,\longrightarrow\,{\mathscr{X}}$,
this datum defines a principal $G$-bundle $E_{G}\,\longrightarrow\,U$ over
$U$, because $X_{0}\times_{\mathscr{X}}U\,\longrightarrow\,U$ has local
sections. Therefore we get a differentiable stack ${\mathscr{E}}_{G}$ and the
$G$-multiplication map glues and comes with a natural morphism of stacks
${{\mathscr{E}}_{G}}\times G\,\longrightarrow\,{{\mathscr{E}}_{G}}$ (see [BMW,
He, Ne]). We can reformulate all this more explicit also as follows: For each
smooth atlas $u\,:\,U\,\longrightarrow\,{\mathscr{X}}$ we are given a
principal $G$-bundle ${\mathscr{E}}_{G,u}$ over $U$ and for each
$2$-commutative diagram of the form

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{\varphi}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v}$$\textstyle{\mathscr{X}}$
(3.1)

with a $2$-isomorphism $\alpha\,:\,u\,\Rightarrow\,v\circ\varphi$, where
$u,\,v$ are smooth atlases, we are given an isomorphism

$\theta_{\varphi,\alpha}\,\,\,:\,\,{\mathscr{E}}_{G,u}\,\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\,\varphi^{*}{\mathscr{E}}_{G,v}$

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

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{\varphi}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\scriptstyle{v}$$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\textstyle{\mathscr{X}}$

together with $2$-isomorphisms $\alpha\,:\,u\,\Rightarrow\,v\circ\varphi$ and
$\beta\,:\,v\,\Rightarrow\,w\circ\psi$ we have a commutative diagram

$\textstyle{{\mathscr{E}}_{G,u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta_{\psi\circ\varphi,\beta\circ\alpha}}$$\scriptstyle{\cong}$$\scriptstyle{\theta_{\varphi,\alpha}}$$\scriptstyle{\cong}$$\textstyle{(\psi\circ\varphi)^{*}{\mathscr{E}}_{G,w}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varphi^{*}{\mathscr{E}}_{G,v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\varphi^{*}\theta_{\psi,\beta}}$$\textstyle{(\varphi^{*}\circ\psi^{*}){\mathscr{E}}_{G,w}}$

If ${\mathscr{F}}_{G}$ is another principal $G$-bundle over $\mathscr{X}$,
then a morphism $f\,:\,{\mathscr{E}}_{G}\,\longrightarrow\,{\mathscr{F}}_{G}$
is given by a morphism
$f_{u}\,:\,{\mathscr{E}}_{G,u}\,\longrightarrow\,{\mathscr{F}}_{G,u}$ for each
smooth atlas $u\,:\,U\,\longrightarrow\,{\mathscr{X}}$ such that for any
$2$-commutative diagram of the form (3.1) we have a commutative diagram

$\textstyle{{\mathscr{E}}_{G,u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{u}}$$\scriptstyle{\theta^{{\mathscr{E}}_{G}}_{\varphi,\alpha}}$$\scriptstyle{\cong}$$\textstyle{{\mathscr{F}}_{G,u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta^{{\mathscr{F}}_{G}}_{\varphi,\alpha}}$$\scriptstyle{\cong}$$\textstyle{\varphi^{*}{\mathscr{E}}_{G,v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{*}f_{v}}$$\textstyle{\varphi^{*}{\mathscr{F}}_{G,v}}$

The category of principal $G$-bundles over a differentiable stack
$\mathscr{X}$ forms in a natural way a groupoid fibration
${\mathscr{B}}un_{G}({\mathscr{X}})$ over $\mathfrak{S}$. In fact, we get the
following characterization from the above (compare also [BMW, He])

######  Proposition 3.2.

Let $\mathscr{X}$ be a differentiable stack and $G$ a Lie group with
classifying stack ${\mathscr{B}}G$. Giving a principal $G$-bundle over
$\mathscr{X}$ is equivalent to giving a morphism of stacks
${\mathscr{X}}\,\longrightarrow\,{\mathscr{B}}G$ and two principal $G$-bundles
over $\mathscr{X}$ are isomorphic if and only if the corresponding morphisms
of stacks ${\mathscr{X}}\,\longrightarrow\,{\mathscr{B}}G$ are $2$-isomorphic.

We consider the groupoid fibration ${\mathscr{B}}un_{G}({\mathscr{X}})$ over
$\mathfrak{S}$ whose objects over a smooth space $U$ are principal $G$-bundles
${\mathscr{E}}_{G}$ over ${\mathscr{X}}\times U$ and whose morphisms are given
by pullback diagrams of principal $G$-bundles. Given two differentiable stacks
$\mathscr{X}$ and $\mathscr{Y}$, we also have the groupoid fibration
${\mathscr{H}}om({\mathscr{X}},\,{\mathscr{Y}})$ over $\mathfrak{S}$, whose
groupoid of sections over a smooth space $U$ is the groupoid of $1$-morphisms
or functors
${\mathscr{H}}om({\mathscr{X}},\,{\mathscr{Y}})(U)\,=\,\mathrm{Hom}_{U}({\mathscr{X}}\times
U,\,{\mathscr{Y}}\times U)$. Let us remark that if in addition ${\mathscr{X}}$
is proper and ${\mathcal{Y}}$ of finite presentation, then
${\mathscr{H}}om({\mathscr{X}},\,{\mathscr{Y}})$ is again a differentiable
stack and in particular if ${\mathscr{Y}}$ is a Deligne–Mumford stack, then
the Hom stack ${\mathscr{H}}om({\mathscr{X}},\,{\mathscr{Y}})$ is also a
Deligne–Mumford stack (compare [Ao], [Ol]). We can now make the following
straightforward observation using Proposition 3.2 (compare with [BMW]).

######  Proposition 3.3.

There is an equivalence of groupoid fibrations over $\mathfrak{S}$

${\mathscr{B}}un_{G}({\mathscr{X}})\,\cong\,{\mathscr{H}}om({\mathscr{X}},\,{\mathscr{B}}G)\,.$

###  3.2. Principal bundles over Lie groupoids

Let us now recall the general notion of a principal $G$-bundle over a Lie
groupoid (see [LGTX, TXLG]).

Given a $\mathbb{X}$-space $\pi\,:\,P\,\longrightarrow\,X_{0}$ for a Lie
groupoid ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$, we have for any
$\gamma\,\in\,X_{1}$ a smooth isomorphism in $\mathfrak{S}$

$l_{\gamma}\,:\,\pi^{-1}(u)\,\longrightarrow\,\pi^{-1}(v)\,,\
~{}x\,\longmapsto\,\gamma\cdot x\,,$

where $u\,=\,s(\gamma)$ and $v\,=\,t(\gamma)$. Associated to this
$\mathbb{X}$-space is a transformation groupoid
${\mathbb{P}}\,=\,[Q\,=\,X_{1}\times_{s,X_{0},\pi}P\rightrightarrows P]$,
where the source and target maps are given by $s(\gamma,\,x)\,=\,x$ and
$t(\gamma,x)\,=\,\gamma\cdot x$. The multiplication map is given by

$(\gamma,\,y)\cdot(\delta,\,x)\,=\,(\gamma\cdot\delta,\,x)\,,$

where $y\,=\,\delta\cdot x$. The first projection defines a strict
homomorphism of Lie groupoids from ${\mathbb{P}}\,=\,[Q\rightrightarrows P]$
to ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$.

######  Definition 3.4 (Principal bundle over Lie groupoid).

Let $G$ be a Lie group and ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ a
Lie groupoid. A principal $G$-bundle or $G$-torsor over $\mathbb{X}$, denoted
by ${\mathbb{E}}_{G}\,:=\,[s^{*}E_{G}\rightrightarrows E_{G}]$, is given by a
principal (right) $G$-bundle $\pi\,:\,E_{G}\,\longrightarrow\,X_{0}$, which is
also a $\mathbb{X}$-space such that for all $x\,\in\,E_{G}$ and all
$\gamma\,\in\,X_{1}$ with $s(\gamma)\,=\,\pi(x)$, we have

$(\gamma\cdot x)\cdot g\,=\,\gamma\cdot(x\cdot g)~{}\ \mathrm{for\,\,all}~{}\
g\,\in\,G\,.$

Let $s^{*}E_{G}\,=\,X_{1}\times_{s,X_{0},\pi}E_{G}$ be the pullback along the
source map $s$. Then ${\mathbb{E}}_{G}\,=\,[s^{*}E_{G}\rightrightarrows
E_{G}]$ is in a natural way a Lie groupoid, the transformation groupoid with
respect to the $X_{1}$-action.

######  Example 3.5.

Let $G$ and $H$ be Lie groups, and let ${\mathbb{X}}\,=\,[H\times
X_{0}\rightrightarrows X_{0}]$ be the transformation groupoid over a smooth
space $X_{0}$. Then a principal $G$-bundle over $\mathbb{X}$ is an
$H$-equivariant principal $G$-bundle over $X_{0}$. In particular, if
${\mathbb{X}}\,=\,[X_{0}\rightrightarrows X_{0}]$ with both structure maps
$s,t$ being the identity map $id_{X_{0}}$, then a principal $G$-bundle over
$\mathbb{X}$ is just a principal $G$-bundle over $X_{0}$.

######  Example 3.6.

Let $G,\,H$ be a pair of Lie groups, and
${\mathbb{X}}\,=\,[H\rightrightarrows*]$ the single object Lie groupoid
associated to the Lie group $H$. Then a left-action of $H$ on $G$ satisfying
$h\cdot(gg^{\prime})\,=\,(h\cdot g)g^{\prime}$ for all $h\,\in\,H$ and
$g,\,g^{\prime}\,\in\,G$ defines a principal $G$-bundle over ${\mathbb{X}}$,
and vice versa.

Similarly, we can define vector bundles of rank $n$ over a Lie groupoid
${\mathbb{X}}$ and over a differentiable stack $\mathscr{X}$. They can be
identified with the principal $\mathrm{GL}_{n}$-bundles over ${\mathbb{X}}$
and $\mathscr{X}$ respectively.

The following theorem generalizes [BX, Proposition 4.1] for arbitrary
principal bundles and the proof given below is a variation of the argument for
$S^{1}$-bundles.

######  Theorem 3.7.

Let $G$ be a Lie group, $\mathscr{X}$ be a differentiable stack and
${\mathbb{X}}=[X_{1}\rightrightarrows X_{0}]$ be a Lie groupoid presenting
$\mathscr{X}$. Then there is a canonical equivalence of categories:

${\mathscr{B}}un_{G}({\mathscr{X}})\,\cong\,{\mathbb{B}}un_{G}({\mathbb{X}})\,,$

where ${\mathscr{B}}un_{G}({\mathscr{X}})$ is the category of principal
$G$-bundles over $\mathscr{X}$ and ${\mathbb{B}}un_{G}({\mathbb{X}})$ is the
category of principal $G$-bundles over $\mathbb{X}$.

###### Proof.

Let us first assume we have given a principal $G$-bundle
$\pi\,:\,{\mathscr{E}}_{G}\,\longrightarrow\,{\mathscr{X}}$ over the
differentiable stack ${\mathscr{X}}$. Let $E_{G}$ be the pullback of
$X_{0}\,\longrightarrow\,{\mathscr{X}}$ via $\pi$, in other words, we have a
$2$-Cartesian diagram

$\textstyle{E_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{E}}_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathscr{X}}$

This means that $E_{G}\,\longrightarrow\,X_{0}$ is a principal $G$-bundle over
$X_{0}$ and $E_{G}\,\longrightarrow\,{\mathscr{E}}_{G}$ is a representable
surjective submersion. Let ${\mathbb{E}}_{G}\,=\,[s^{*}E_{G}\rightrightarrows
E_{G}]$ be the associated Lie groupoid, which comes with an induced morphism
of Lie groupoids

${\mathbb{E}}_{G}\,=\,[s^{*}E_{G}\rightrightarrows
E_{G}]\,\longrightarrow\,{\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$

and gives rise to the Cartesian diagram

$\textstyle{s^{*}E_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}}$

In other words, we get a pullback diagram of smooth spaces in which the
vertical maps are source maps. This implies that
$s^{*}E_{G}\,\longrightarrow\,E_{G}$ is a principal $G$-bundle, and the
vertical maps are morphisms of $G$-bundle. Therefore, $X_{1}$ acts on $E_{G}$
and $E_{G}$ becomes an $\mathbb{X}$-space which actually turns
${\mathbb{E}}_{G}\,=\,[s^{*}E_{G}\rightrightarrows E_{G}]$ into a principal
$G$-bundle over the groupoid ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows
X_{0}]$.

In fact, we get a functor.
$\Psi\,:\,{\mathscr{B}}un_{G}({\mathscr{X}})\,\longrightarrow\,{\mathbb{B}}un_{G}({\mathbb{X}})$,
which associates the principal $G$-bundle
${\mathbb{E}}_{G}\,=\,[s^{*}E_{G}\rightrightarrows E_{G}]$ over the Lie
groupoid ${\mathbb{X}}$ to the given principal $G$-bundle
${\mathscr{E}}_{G}\,\longrightarrow\,{\mathscr{X}}$.

Now let us assume conversely, that we have given a principal $G$-bundle
${\mathbb{E}}_{G}\,=\,[s^{*}E_{G}\rightrightarrows E_{G}]$ over the Lie
groupoid ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$. This means that we
have a principal $G$-bundle $\pi\,:\,E_{G}\,\longrightarrow\,X_{0}$, which is
also a ${\mathbb{X}}$-space satisfying the conditions given in Definition 3.4.
Recall that

$s^{*}E_{G}\,=\,X_{1}\times_{s,X_{0},\pi}E_{G}$

is the pullback along the source map $s$, and
${\mathbb{E}}_{G}\,=\,[s^{*}E_{G}\rightrightarrows E_{G}]$ is in a natural way
a Lie groupoid. It follows also that $s^{*}E_{G}$ becomes in a natural way a
$G$-bundle over $X_{1}$, and we have a morphism of Lie groupoids

${\mathbb{E}}_{G}\,=\,[s^{*}E_{G}\rightrightarrows
E_{G}]\,\longrightarrow\,{\mathbb{X}}$

which respects the $G$-bundle structures of
$\tau\,:\,s^{*}E_{G}\,\longrightarrow\,X_{1}$ and
$\pi\,:\,E_{G}\,\longrightarrow\,X_{0}$. Let
${\mathscr{E}}_{G}:={\mathscr{B}}{\mathbb{E}}_{G}$ be the associated
differentiable stack of ${\mathbb{E}}_{G}$-torsors. The morphism of Lie
groupoids ${\mathbb{E}}_{G}\,\longrightarrow\,{\mathbb{X}}$ induces a morphism
between the associated differentiable stacks
${\mathscr{E}}_{G}\,\longrightarrow\,{\mathscr{X}}$, which is representable,
as its pullback to $X_{0}$ is the map $\pi\,:\,E_{G}\,\longrightarrow\,X_{0}$
between smooth spaces:

$\textstyle{E_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{E}}_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{X}}$

Furthermore, it follows that the pullback of the morphism of stacks
${\mathscr{E}}_{G}\,\longrightarrow\,{\mathscr{X}}$ along any morphism
$U\,\longrightarrow\,\mathscr{X}$ from a smooth space $U$ is a principal
$G$-bundle. Therefore ${\mathscr{E}}_{G}$ is a principal $G$-bundle over the
differentiable stack ${\mathscr{X}}$.

Finally, we get a functor in the opposite direction
$\Phi\,:\,{\mathbb{B}}un_{G}({\mathbb{X}})\,\longrightarrow\,{\mathscr{B}}un_{G}({\mathscr{X}})$,
which associates the principal $G$-bundle
${\mathscr{E}}_{G}\,\longrightarrow\,{\mathscr{X}}$ to the given principal
$G$-bundle ${\mathbb{E}}_{G}\,=\,[s^{*}E_{G}\rightrightarrows E_{G}]$ over the
Lie groupoid ${\mathbb{X}}$. It is easy to see that the functors $\Psi$ and
$\Phi$ are 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 3.8.

Let ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ be a Lie groupoid and
$G$ a Lie group. A principal $G$-groupoid over $\mathbb{X}$ is a Lie groupoid
${\mathbb{P}}\,=\,[Q\rightrightarrows P]$ together with a groupoid morphism

$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{\widetilde{t}}$$\scriptstyle{\widetilde{s}}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\scriptstyle{s}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X_{0}}$

such that both
$G\,\longrightarrow\,Q\,\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}\,X_{1}$
and
$G\,\longrightarrow\,P\,\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\,X_{0}$
are principal $G$-bundles, and, furthermore, the following conditions hold:

  1. (1)

the source and target maps $\widetilde{s}$ and $\widetilde{t}$ on $Q$ are
$G$-equivariant,

  2. (2)

the identity section $\widetilde{e}\,:\,P\,\longrightarrow\,Q$ is
$G$-equivariant,

  3. (3)

the inversion map $\widetilde{i}\,:\,Q\,\longrightarrow\,Q$ satisfies the
identity $\widetilde{i}(zg)\,=\,\widetilde{i}(z)g^{-1}$ for all $z\,\in\,Q$
and $g\,\in\,G$, and

  4. (4)

the multiplication map $Q\times_{s,X_{0},t}Q\,\longrightarrow\,Q$ is
$G$-equivariant for the diagonal action of $G$ on $Q\times_{s,X_{0},t}Q$.

There is an obvious notion of a morphism between principal $G$-groupoids, and
we can speak of the category ${\mathbb{B}}un_{G,{\mathbb{X}}}$ of principal
$G$-groupoids over $\mathbb{X}$, which yields the following equivalent
characterization of the category of principal $G$-bundles over a given Lie
groupoid $\mathbb{X}$.

######  Proposition 3.9.

Let $G$ be a Lie group and ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ a
Lie groupoid. Then there is a canonical equivalence of categories

${\mathbb{B}}un_{G}({\mathbb{X}})\,\cong\,{\mathbb{B}}un_{G,{\mathbb{X}}}\,,$

where ${\mathbb{B}}un_{G}({\mathbb{X}})$ is the category of principal
$G$-bundles over $\mathbb{X}$, and ${\mathbb{B}}un_{G,{\mathbb{X}}}$ is the
category of principal $G$-groupoids over $\mathbb{X}$.

###### Proof.

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

######  Remark 3.10.

It can be shown that principal $G$-bundles over a Lie groupoid
${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ are also equivalent to
generalized homomorphisms from $\mathbb{X}$ to the Lie groupoid
${\mathbb{B}}G\,=\,[G\rightrightarrows*]$, which is Morita equivalent to the
gauge groupoid ${\mathbb{P}}_{\text{Gauge}}:=[P\times P/G\rightrightarrows
X_{0}]$. In [LGTX, Prop. 2.13, Prop. 2.14 and Thm 2.15] this is discussed in
the differentiable setting, but this makes sense again in any of the
categories $\mathfrak{S}$ 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 and differentiable stacks.

###  4.1. Connections on Lie groupoids and differentiable stacks

Let us start with the general definition of a connection on a Lie groupoid.

Let $({\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]\,,s\,,t\,,m\,,e\,,i)$ be
a Lie groupoid. Let

${\mathcal{K}}\,:=\,\text{kernel}(ds)\,\subset\,TX_{1}$ (4.1)

be the vertical tangent bundle for $s$. Since the source map
$s\,:\,X_{1}\,\longrightarrow\,X_{0}$ is a submersion, ${\mathcal{K}}$ is a
subbundle of $TX_{1}$. Fix a distribution on $X_{1}$ given by a subbundle
${\mathcal{H}}\,\subset\,TX_{1}$ such that the natural homomorphism

${\mathcal{K}}\oplus{\mathcal{H}}\,\longrightarrow\,TX_{1}$ (4.2)

is an isomorphism, so ${\mathcal{H}}$ is a complement of ${\mathcal{K}}$. Let

$d_{\mathcal{H}}s\,:=\,(ds)|_{\mathcal{H}}\,:\,{\mathcal{H}}\,\longrightarrow\,s^{*}TX_{0}$

be the restriction of $ds$ to ${\mathcal{H}}$. Note that $d_{\mathcal{H}}s$ is
an isomorphism because the homomorphism in (4.2) is an isomorphism.

Let $dt\,:\,TX_{1}\,\longrightarrow\,t^{*}TX_{0}$ be the differential of the
map $t$. Consider the homomorphism

$\theta\,:=\,dt\circ(d_{\mathcal{H}}s)^{-1}\,:\,s^{*}TX_{0}\,\longrightarrow\,t^{*}TX_{0}\,.$
(4.3)

For any $y\,\in\,X_{1}$, let

$\theta_{y}\,:=\,\theta|_{(s^{*}TX_{0})_{y}}\,:\,T_{s(y)}X_{0}\,\longrightarrow\,T_{t(y)}X_{0}$
(4.4)

be the restriction of $\theta$ to the fiber
$(s^{*}TX_{0})_{y}\,=\,T_{s(y)}X_{0}$.

We can then define generally

######  Definition 4.1 (Connection on a Lie groupoid).

A connection on a Lie groupoid ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows
X_{0}]$ is a distribution ${\mathcal{H}}\,\subset\,TX_{1}$ given by a
subbundle complementing $\mathcal{K}$ such that

  * (i)

for every $x\,\in\,X_{0}$, the image of the differential

$de(x)\,:\,T_{x}X_{0}\,\longrightarrow\,T_{e(x)}X_{1}$

coincides with the subspace ${\mathcal{H}}_{e(x)}\,\subset\,T_{e(x)}X_{1}$,
and

  * (ii)

for every $y\,,z\,\in\,X_{1}$ with $t(y)\,=\,s(z)$, the homomorphism

$\theta_{m(z,y)}\,:\,T_{s(y)}X_{0}\,\longrightarrow\,T_{t(z)}X_{0}$

coincides with the composition $\theta_{z}\circ\theta_{y}$ (see (4.4)).

A connection on ${\mathbb{X}}$ is said to be flat (or integrable) if the
distribution ${\mathcal{H}}\,\subset\,TX_{1}$ defining the connection is
integrable.

Henceforth we will denote $m(y,z)=y\circ z.$ Similarly for the pairs
$(y,v),(z,w)\in TX_{1}$ such that $s(y)=t(z),dt_{y}(v)=ds_{z}(w),$ the
composition $dm\bigl{(}(y,v),(z,w)\bigr{)}$ in the induced tangent Lie
groupoid $T{\mathbb{X}}=[TX_{1}\rightrightarrows TX_{0}]$ will be denoted as
$(y\circ z,v\circ w).$

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

######  Proposition 4.2.

Let ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ and
${\mathbb{Y}}\,=\,[Y_{1}\rightrightarrows Y_{0}]$ be Lie groupoids, and let

$\phi\,=\,(\phi_{1},\,\phi_{0})\,:\,{\mathbb{X}}\,\longrightarrow\,{\mathbb{Y}}$

be a Morita morphism. Let

${\mathcal{H}}\,\subset\,TY_{1}$

be a distribution that defines a connection on the groupoid ${\mathbb{Y}}$.
Then $\mathcal{H}$ induces a connection on the groupoid ${\mathbb{X}}$.

###### Proof.

Recall from Definition 2.12(i) that the map
$\phi_{0}\,:\,X_{0}\,\longrightarrow\,Y_{0}$ is a surjective submersion. Hence
from Definition 2.12(ii) it follows that

$\phi_{1}\,:\,X_{1}\,\longrightarrow\,Y_{1}$

is also a surjective submersion. Let

$d\phi_{1}\,:\,TX_{1}\,\longrightarrow\,TY_{1}$

be the differential of the map $\phi_{1}$. Let
${\mathcal{H}}\,\subset\,TY_{1}$ be the distribution inducing a connection on
the groupoid ${\mathbb{Y}}$. Now define

$\widetilde{\mathcal{H}}\,:=\,(d\phi_{1})^{-1}(\phi^{*}_{1}{\mathcal{H}})\,\subset\,TX_{1}\,.$

Since $\phi_{1}$ is a surjective submersion, it follows that
$\widetilde{\mathcal{H}}\,\subset\,TX_{1}$ is a distribution and therefore
defines a connection on the groupoid ${\mathbb{X}}$. ∎

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

######  Definition 4.3.

Let ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ and
${\mathbb{Y}}\,=\,[Y_{1}\rightrightarrows Y_{0}]$ be Lie groupoids and
$\phi\,=\,(\phi_{1},\,\phi_{0})\,:\,{\mathbb{X}}\,\longrightarrow\,{\mathbb{Y}}$
be a morphism. Suppose that ${\mathcal{H}}\subset TX_{1}$ defines a connection
on ${\mathbb{X}}$ and ${\mathcal{L}}\subset TY_{1}$ a connection on
${\mathbb{Y}}$. The morphism $\phi$ is called horizontal if the differential
$d\phi_{1}\,:\,TX_{1}\,\longrightarrow\,TY_{1}$ maps ${\mathcal{H}}$ into
${\mathcal{L}}$.

######  Proposition 4.4.

Let ${\mathbb{X}}$ and ${\mathbb{Y}}$ be Lie groupoids together with an étale
morphism
$\phi\,=\,(\phi_{1},\,\phi_{0})\,:\,{\mathbb{X}}\,\longrightarrow\,{\mathbb{Y}}$.
Then any connection ${\mathcal{L}}$ on ${\mathbb{Y}}$ induces a unique
connection ${\mathcal{H}}$ on ${\mathbb{X}}$ such that $\phi$ is a horizontal
morphism. If the connection ${\mathcal{L}}$ is integrable, then also the
induced connection ${\mathcal{H}}$ is integrable.

###### Proof.

Given a connection ${\mathcal{L}}$ on the groupoid ${\mathbb{Y}}$, the unique
connection ${\mathcal{H}}$ on the groupoid ${\mathbb{X}}$ is given by the
fiber product ${\mathcal{H}}:={\mathcal{L}}\times_{TY_{1}}TX_{1}.$ ∎

In particular, a connection (respectively, flat connection) on an étale
groupoid induces then a connection (respectively, flat connection) on the
associated Deligne-Mumford stack ${\mathscr{B}}{\mathbb{X}}$, the classifying
stack of $\mathbb{X}$-torsors.

######  Example 4.5.

Let $\mathbb{X}=[X\rightrightarrows X]$ be the Lie groupoid associated to a
smooth space $X$. Then $\text{kernel}(ds)\,=\,\\{0\\}$ and the map
$x\,\longmapsto\,\mathcal{H}_{x}X\,:=\,T_{x}M$, $x\,\in\,X$, defines an
integrable connection $\mathcal{H}_{\mathbb{X}}\subset TX$ on $\mathbb{X}$.

######  Example 4.6.

More generally, for an étale Lie groupoid
$\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}],$ since the differential
$ds\,\colon\,TX_{1}\,\longrightarrow\,s^{*}TX_{0}$ is an isomorphism, the
distribution $\mathcal{H}_{\mathbb{X}}=\,TX_{1}$ is an integrable connection
on the Lie groupoid $\mathbb{X}$.

######  Example 4.7.

Let $\pi\,\colon\,E\,\longrightarrow\,X$ be a finite rank vector bundle over a
smooth space $X.$ We get a Lie groupoid $\mathbb{X}=[E\rightrightarrows X]$
with the source and target maps both being $\pi.$ Then any two composable
morphisms belong to the same fibre, and the composition can be defined as
addition of vectors in that fibre. Any connection on the vector bundle
$\pi\,\colon\,E\,\longrightarrow\,X$ smoothly splits $E$ into the horizontal
component and $\text{kernel}(ds).$ This gives a connection on the associated
Lie groupoid $\mathbb{X}=[E\rightrightarrows X]$. Integrability of one
connection implies integrability of the other.

######  Example 4.8.

Let $G$ be a Lie group. Given a principal $G$-bundle
$\pi\,\colon\,P\,\longrightarrow\,X$ over a smooth space $X$, one defines the
Atiyah or gauge groupoid $\mathbb{P}_{\rm Gauge}:=\,[\frac{P\times
P}{G}\rightrightarrows X\,]$ by building the quotient of the groupoid
$[P\times P\rightrightarrows P]$ with respect to the diagonal action of $G$ on
$P\times P.$ A connection $\omega$ on the principal bundle
$\pi\,\colon\,P\,\longrightarrow\,X$ gives a $G$-invariant horizontal
distribution $\mathcal{H}\subset TP$ complementing the $\text{kernel}.$ We
define a connection $\mathcal{H}_{\mathbb{P}_{\rm Gauge}}$ on the Lie groupoid
$\mathbb{P}_{\rm Gauge}$ by

${\mathcal{H}}_{[p,q]}\,\,:=\,\,\frac{{\mathcal{H}}_{p}\oplus
T_{q}P}{T_{p,q}\,{(p,q)\cdot G}}\,\,\subset\,\,T_{[p,q]}\frac{P\times
P}{G}\,,$

where $(p,\,q)\cdot G\,\subset\,P\times P$ is the orbit of the element
$(p,\,q)\,\in\,P\times P.$ Integrability of the connection $\omega$ implies
integrability of $\mathcal{H}_{\mathbb{P}_{\rm Gauge}}.$

######  Remark 4.9.

Related constructions of connections and flat connections on groupoids and on
stacks in the algebro-geometric, differentiable and holomorphic context were
also studied by Behrend in [Be]. Flat connections in the differentiable
setting for Lie groupoids were also independently introduced as étalifications
by Tang [Ta]. These constructions give all rise to subgroupoids of the
associated tangent groupoid $T{\mathbb{X}}$ of a groupoid $X$, which in the
differentiable category is equivalent to the horizontal paths forming a
subgroupoid of the path groupoid of ${\mathbb{X}}$. This is used by Laurent-
Gengoux, Stiénon and Xu in [LGSX] to define connections in the general
framework of non-abelian differentiable gerbes via Ehresmann connections on
Lie groupoid extensions. They also appear as multiplicative distributions on
Lie groupoids in recent work by Drummond, and Egea [DE] and Trentinaglia [Tr].
Another definition of a general Ehresmann connection for Lie groupoids was
more recently also given and discussed by Arias Abad and Crainic [AC].

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

Let ${\mathcal{H}}\subset TX_{1}$ be a connection on a Lie groupoid
${\mathbb{X}}=[X_{0}\rightrightarrows X_{1}]$. The connection defines a
groupoid $[s^{*}TX_{0}\rightrightarrows TX_{0}]$ as follows. Without loss of
any information we denote an element $(\gamma,s(\gamma),v)$ of $s^{*}TX_{0}$
by $(\gamma,v)$, where $v\in T_{s(\gamma)}X_{0}$. The source, target and
composition maps are then respectively given by

$\begin{split}&s\,\colon\,(\gamma,\,v)\,\longmapsto\,(s(\gamma),\,v),\qquad\gamma\,\in\,X_{1},\,v\,\in\,T_{s(\gamma)}X_{0},\\\
&t\,\colon\,(\gamma,\,v)\,\longmapsto\,(t(\gamma),\,\theta_{\gamma}(v)),\\\
&(\gamma_{2},\,v_{2})\circ(\gamma_{1},\,v_{1})\,=\,(\gamma_{2}\circ\gamma_{1},\,v_{1}).\\\
\end{split}$ (4.5)

The inversion and unit maps are obvious. Note that the composition is well
defined because of condition (ii) in Definition 4.1. Similarly
$[t^{*}TX_{0}\rightrightarrows TX_{0}]$ is a groupoid with the corresponding
structure maps given as:

$\begin{split}&s\,\colon\,(\gamma,\,u)\,\longmapsto\,(s(\gamma),\,\theta_{{\gamma}^{-1}}(u)),\qquad\gamma\,\in\,X_{1},\,u\,\in\,T_{t(\gamma)}X_{0},\\\
&t\,\colon\,(\gamma,\,u)\,\longmapsto\,(t(\gamma),\,u),\\\
&(\gamma_{2},\,u_{2})\circ(\gamma_{1},\,u_{1})\,=\,(\gamma_{2}\circ\gamma_{1},\,u_{2}).\\\
\end{split}$ (4.6)

In fact

$\theta\,\colon\,(\gamma,\,v)\,\longmapsto\,(\gamma,\,{d_{\mathcal{H}}s}_{{\gamma}}^{-1}(v))\,\longmapsto\,(\gamma,\,\underbrace{dt_{\gamma}\circ{d_{\mathcal{H}}s}_{{\gamma}}^{-1}}_{\theta_{\gamma}}(v))$

in (4.3) defines an isomorphism of Lie groupoids

$\textstyle{s^{*}TX_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\theta}$$\textstyle{t^{*}TX_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TX_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
Id}}$$\textstyle{TX_{0}}$ (4.7)

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

${d_{\mathcal{H}}s}_{{\gamma_{2}\circ\gamma_{1}}}^{-1}(v)-{d_{\mathcal{H}}s}_{{\gamma_{2}}}^{-1}(\theta_{\gamma_{1}}(v))\circ{d_{\mathcal{H}}s}_{{\gamma_{1}}}^{-1}(v)\,\in\,{\rm
ker}(dt_{\gamma_{2}\circ\gamma_{1}})$ (4.8)

for any pair of composable $\gamma_{1},\,\gamma_{2}\,\in\,X_{1}$ and
$v\,\in\,T_{s(\gamma_{1})}X_{0}.$ We denote the corresponding element in ${\rm
ker}(dt_{\gamma_{2}\circ\gamma_{1}})$ by

${\mathfrak{K}}(\gamma_{2},\,\gamma_{1},\,v)\,:=\,{d_{\mathcal{H}}s}_{{\gamma_{2}\circ\gamma_{1}}}^{-1}(v)-{d_{\mathcal{H}}s}_{{\gamma_{2}}}^{-1}(\theta_{\gamma_{1}}(v))\circ{d_{\mathcal{H}}s}_{{\gamma_{1}}}^{-1}(v).$

The outcome of the above observation is the following lemma.

######  Lemma 4.10.

$({d_{\mathcal{H}}s}^{-1},\,{\rm{Id}})\,\colon\,[s^{*}{TX_{0}}\rightrightarrows
TX_{0}]\,\longrightarrow\,[TX_{1}\rightrightarrows TX_{0}]$ defines an
essentially surjective, faithful functor if and only if
${\mathfrak{K}}(\gamma_{2},\,\gamma_{1},\,v)$ vanishes for all pairs of
composable arrows $\gamma_{1},\,\gamma_{2}\,\in\,X_{1}$ and
$v\,\in\,T_{s(\gamma_{1})}X_{0}.$

###### Proof.

That it is essentially surjective is evident. Now we have:

${d_{\mathcal{H}}s}^{-1}\big{(}(\gamma_{2},v_{2})\circ(\gamma_{1},v_{1})\big{)}={d_{\mathcal{H}}s}^{-1}\big{(}(\gamma_{2}\circ\gamma_{1},v_{1})\big{)}=(\gamma_{2}\circ\gamma_{1},{d_{\mathcal{H}}s}_{\gamma_{2}\circ\gamma_{1}}^{-1}(v_{1})).$

On the other hand, we get:

$\displaystyle{d_{\mathcal{H}}s}^{-1}\big{(}(\gamma_{2},v_{2})\big{)}\circ{d_{\mathcal{H}}s}^{-1}\big{(}(\gamma_{1},v_{1})\big{)}=\big{(}\gamma_{2},{d_{\mathcal{H}}s}_{\gamma_{2}}^{-1}(v_{2})\big{)}\circ\big{(}\gamma_{1},{d_{\mathcal{H}}s}_{\gamma_{1}}^{-1}(v_{1})\big{)}$
$\displaystyle=\big{(}\gamma_{2}\circ\gamma_{1},{d_{\mathcal{H}}s}_{\gamma_{2}}^{-1}(v_{2})\circ{d_{\mathcal{H}}s}_{\gamma_{1}}^{-1}(v_{1})\big{)}.$

Then functoriality is now immediate from the vanishing of the left hand side
in (4.8). Since ${d_{\mathcal{H}}s}^{-1}$ is injective, the functor is also
faithful. ∎

######  Remark 4.11.

From the above we see that ${\mathfrak{K}}(\gamma_{2},\,\gamma_{1},\,v)$ gives
an obstruction for the groupoid $[{\mathcal{H}}\,\rightrightarrows\,TX_{0}]$
to be a subgroupoid of the tangent groupoid
$T{\mathbb{X}}\,=\,[TX_{1}\rightrightarrows TX_{0}]$. Indeed vanishing of the
element in (4.8) is equivalent to the necessary and sufficient condition for a
splitting as in (4.2) to yield a subgroupoid, mentioned in Lemma 2.13 of [AC].
Moreover (4.7) implies that both diagrams

$\textstyle{{\mathcal{H}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TX_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}}$

are Cartesian and thus we arrive at the definition of a connection on a Lie
groupoid as given in [Be, Def. 2.1]. In conclusion, we see that our definition
of a connection on a Lie groupoid is more general than the one in [Be], but
more strict than the definition in [AC]. In fact, a partition of unity
argument shows that any Lie groupoid admits a connection in the sense of [AC].
If such a connection satisfies in addition the conditions (i) and (ii) of
Definition 4.1, then we obtain a connection as defined in this article.

###  4.2. Making $TX_{0}$ a vector bundle over ${\mathbb{X}}$

We will now state another useful geometric interpretation. Let $\mathcal{H}$
be a connection on a given Lie groupoid
${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$, and let

$\tau\,:\,TX_{0}\,\longrightarrow\,X_{0}$

be the natural projection. Consider the fiber product

$Y_{1}\,:=\,X_{1}\times_{s,X_{0},\tau}TX_{0}\,.$

Let $s^{\prime}\,:\,Y_{1}\,\longrightarrow\,TX_{0}$ be the projection to the
second factor. Define the morphism

$t^{\prime}\,:\,Y_{1}\,\longrightarrow\,TX_{0}\,,\
~{}(x,\,v)\,\longmapsto\,\theta_{x}(v)\,\in\,T_{t(x)}X_{0}\,,$

where $\theta_{x}$ is constructed as in (4.4). Furthermore, let

$p\,:\,Y_{1}\,=\,X_{1}\times_{s,X_{0},\tau}TX_{0}\,\longrightarrow\,X_{1}$

be the projection to the first factor. For any $z,\,y\,\in\,Y_{1}$ with
$t^{\prime}(y)\,=\,s^{\prime}(z)$, define

$m^{\prime}(z,\,y)\,:=\,(m(p(z),\,p(y)),\,s^{\prime}(y))\,.$

Note that $s(p(z))\,=\,t(p(y))$, so $m(p(z),\,p(y))$ is defined. Let
$e^{\prime}$ be the morphism

$e^{\prime}\,:\,TX_{0}\,\longrightarrow\,Y_{1}\,,\
~{}v\,\longmapsto\,(e(\tau(v)),\,v).$

Finally, let

$i^{\prime}\,:\,Y_{1}\,\longrightarrow\,Y_{1},\,\
~{}(z,\,v)\,\longmapsto\,(i(z),\,\theta_{z}(v))$

be the involution. It is straightforward to check that
$([Y_{1}\rightrightarrows
TX_{0}],\,s^{\prime},\,t^{\prime},\,m^{\prime},\,e^{\prime},\,i^{\prime})$ is
a Lie groupoid. In other words, $TX_{0}$ is a vector bundle over the groupoid
${\mathbb{X}}$.

###  4.3. Characteristic differential forms for connections on ${\mathbb{X}}$

We will now study the behaviour and interpretation of connections on Lie
groupoids in terms of differential forms. Let $\mathcal{H}$ be a connection on
the Lie groupoid ${\mathbb{X}}=[X_{1}\rightrightarrows X_{0}]$. Using the
canonical decomposition of the tangent space

$TX_{1}\,=\,{\mathcal{H}}\oplus\text{kernel}(ds)\,,$

we get a projection

$\wedge^{j}TX_{1}\,\longrightarrow\,\wedge^{j}{\mathcal{H}}\,\hookrightarrow\,\wedge^{j}TX_{1}\,.$

The composition $\wedge^{j}TX_{1}\,\longrightarrow\,\wedge^{j}TX_{1}$ gives by
duality an endomorphism of the space of $j$-forms on $X_{1}$.

For a differential form $\omega$ on $X_{1}$, the differential form on $X_{1}$
induced by the above endomorphism for the given connection $\mathcal{H}$ will
be denoted by $H(\omega)$.

######  Definition 4.12.

A differential $j$-form on the Lie groupoid $[X_{1}\rightrightarrows X_{0}]$
is a differential $j$-form $\omega$ on $X_{0}$ such that
$\mathcal{H}(s^{*}\omega)\,=\,\mathcal{H}(t^{*}\omega)$.

Induced differential forms for an integrable distribution observe the
following basic property.

######  Proposition 4.13.

Let $\mathcal{H}$ be a connection on a Lie groupoid
${\mathbb{X}}=[X_{1}\rightrightarrows X_{0}]$ and assume that the distribution
${\mathcal{H}}\,\subset\,TX_{1}$ is integrable. Let $\omega$ be a differential
form on $X_{0}$ such that
$\mathcal{H}(s^{*}\omega)\,=\,{\mathcal{H}}(t^{*}\omega)$. Then
$\mathcal{H}(s^{*}d\omega)\,=\,\mathcal{H}(t^{*}d\omega)$.

###### Proof.

Take a point $x\,\in\,X_{1}$. Let ${\mathcal{L}}$ be the locally defined leaf
of ${\mathcal{H}}$ passing through $x$. Let

$\iota\,:\,{\mathcal{L}}\,\hookrightarrow\,X_{1}$

be the inclusion map. Since
$\mathcal{H}(s^{*}\omega)\,=\,\mathcal{H}(t^{*}\omega)$, it follows
immediately that $\iota^{*}s^{*}\omega\,=\,\iota^{*}t^{*}\omega$. Therefore we
have,

$\iota^{*}s^{*}(d\omega)\,=\,d\iota^{*}s^{*}\omega\,=\,d\iota^{*}t^{*}\omega\,=\,\iota^{*}t^{*}(d\omega)\,.$

But this implies that
$\mathcal{H}(s^{*}(d\omega))(x)\,=\,\mathcal{H}(t^{*}(d\omega))(x)$ and so we
conclude that $\mathcal{H}(s^{*}(d\omega))\,=\,\mathcal{H}(t^{*}(d\omega))$,
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 with the groupoid picture. Let
$({\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}],\,s,\,t,\,m,\,e,\,i)$ be a
Lie groupoid and $G$ a Lie group. The Lie algebra of $G$ will be denoted by
$\mathfrak{g}$. We shall consider $\mathfrak{g}$ as a $G$-module using the
adjoint action. Let

$\alpha\,:\,E_{G}\,\longrightarrow\,X_{0}$ (5.1)

be a principal $G$-bundle over $X_{0}$. The adjoint vector bundle for $E_{G}$

$\text{ad}(E_{G})\,:=\,E_{G}\times^{G}{\mathfrak{g}}\,\longrightarrow\,X_{0}$

is the bundle associated to $E_{G}$ for the adjoint action of $G$ on
$\mathfrak{g}$. The action of $G$ on $E_{G}$ induces an action of $G$ on the
direct image $\alpha_{*}TE_{G}$, where $\alpha$ is the projection as given in
(5.1).

######  Definition 5.1.

The Atiyah bundle for a principal $G$-bundle $E_{G}$ over $X_{0}$ is the
invariant direct image

$\text{At}(E_{G})\,:=\,(\alpha_{*}TE_{G})^{G}\,\subset\,\alpha_{*}TE_{G}\,.$
(5.2)

Therefore, we have $\text{At}(E_{G})\,=\,(TE_{G})/G$ (see also [At] for the
classical and analogue notion of the Atiyah bundle in the complex analytic
context).

Let

$q_{E}\,:\,TE_{G}\,\longrightarrow\,(TE_{G})/G\,=\,\text{At}(E_{G})$ (5.3)

be the quotient map.

Let $T_{\rm rel}\,\subset\,TE_{G}$ be the relative tangent bundle for the
projection $\alpha$ in (5.1). It fits into the short exact sequence of vector
bundles

$0\,\longrightarrow\,T_{\rm
rel}\,\stackrel{{\scriptstyle\iota_{0}}}{{\longrightarrow}}\,TE_{G}\,\stackrel{{\scriptstyle
d\alpha}}{{\longrightarrow}}\,\alpha^{*}TX_{0}\,\longrightarrow\,0\,,$ (5.4)

where $d\alpha$ is the differential of $\alpha$. Using the action of $G$ on
$E_{G}$, the vector bundle $T_{\rm rel}\,\longrightarrow\,E_{G}$ gets
identified with the trivial vector bundle
$E_{G}\times\mathfrak{g}\,\longrightarrow\,E_{G}$. Therefore, we see that

$\text{ad}(E_{G})\,=\,(\alpha_{*}T_{\rm rel})^{G}\,,$

and so $\text{ad}(E_{G})\,=\,(\alpha_{*}T_{\rm rel})/G$. The map of quotients
$T_{\rm rel}/G\,\longrightarrow\,(TE_{G})/G$ induced by the inclusion $T_{\rm
rel}\,\hookrightarrow\,TE_{G}$ makes $\text{ad}(E_{G})$ a subbundle of
$\text{At}(E_{G})$. The differential of $d\alpha$ in (5.4) being
$G$-equivariant therefore produces a homomorphism

$(d\alpha)^{\prime}\,:\,\text{At}(E_{G})\,\longrightarrow\,TX_{0}\,.$ (5.5)

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

$0\,\longrightarrow\,\text{ad}(E_{G})\,\longrightarrow\,\text{At}(E_{G})\,\stackrel{{\scriptstyle(d\alpha)^{\prime}}}{{\longrightarrow}}\,TX_{0}\,\longrightarrow\,0\,.$
(5.6)

This exact sequence is the quotient of the exact sequence in (5.4) by the
action of $G$. Now we 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 $E_{G}$ over $X_{0}$ is a splitting of
the Atiyah exact sequence

$0\,\longrightarrow\,\text{ad}(E_{G})\,\longrightarrow\,\text{At}(E_{G})\,\stackrel{{\scriptstyle(d\alpha)^{\prime}}}{{\longrightarrow}}\,TX_{0}\,\longrightarrow\,0\,.$
(5.7)

Giving a splitting of the exact sequence in (5.6) is evidently equivalent to
giving a $G$-equivariant splitting of the exact sequence in (5.4). Therefore,
a connection on $E_{G}$ is a homomorphism

$D_{E_{G}}\,:\,TE_{G}\,\longrightarrow\,T_{\rm rel}$

such that

  * •

$D_{E_{G}}$ is $G$-equivariant, and

  * •

$D_{E_{G}}\circ\iota_{0}\,=\,\text{Id}_{T_{\rm rel}}$, where $\iota_{0}$ is
the homomorphism in (5.4).

Recall that $T_{\rm rel}\,=\,E_{G}\times\mathfrak{g}$. Therefore, any
homomorphism $D_{E_{G}}\,:\,TE_{G}\,\longrightarrow\,T_{\rm rel}$ satisfying
the above two conditions is a $\mathfrak{g}$-valued $1$-form
$\widehat{D}_{E_{G}}$ on $E_{G}$ such that the corresponding homomorphism

$\widehat{D}_{E_{G}}\,:\,TE_{G}\,\longrightarrow\,\mathfrak{g}$ (5.8)

is $G$-equivariant for the adjoint action of $G$ on $\mathfrak{g}$.

Now we equip $E_{G}$ with the structure of a principal $G$-bundle on the fixed
Lie groupoid ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$. First we shall
use the description of a principal $G$-bundle over ${\mathbb{X}}$ given in
Definition 3.8. So we have a Lie groupoid
${\mathbb{E}}_{G}\,=\,[Q\rightrightarrows P]$ together with a groupoid
morphism

$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Q\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\scriptstyle{\widetilde{t}}$$\scriptstyle{\widetilde{s}}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t}$$\scriptstyle{s}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{X_{0}}$
(5.9)

such that both $Q\,\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}\,X_{1}$
and $E_{G}\,\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}\,X_{0}$ are
principal $G$-bundles, and the four conditions in Definition 3.8 are
satisfied.

Let $\nabla$ be a connection on the principal $G$-bundle
$E_{G}\,\longrightarrow\,X_{0}$. Let

$\widehat{\nabla}\,:\,TE_{G}\,\longrightarrow\,\mathfrak{g}$ (5.10)

be the $G$-equivariant $\mathfrak{g}$-valued $1$-form on $E_{G}$ as in (5.8)
corresponding to $\nabla$.

######  Definition 5.3 (Connection on a principal bundles over a Lie
groupoid).

A connection on a principal $G$-bundle
${\mathbb{E}}_{G}\,=\,[Q\rightrightarrows E_{G}]$ over the Lie groupoid
${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ is a connection $\nabla$ on
the principal $G$-bundle $E_{G}\,\longrightarrow\,X_{0}$ such that the two
$\mathfrak{g}$-valued $1$-forms $\widetilde{s}^{*}\widehat{\nabla}$ and
$\widetilde{t}^{*}\widehat{\nabla}$ on $Q$ coincide, where $\widehat{\nabla}$
is the $1$-form $\widehat{\nabla}\,:\,TE_{G}\,\longrightarrow\,\mathfrak{g}$
associated to the connection $\nabla$.

Now we adopt Definition 3.4. Let

${\mathbb{E}}_{G}\,:=\,[s^{*}E_{G}\rightrightarrows E_{G}]$

be a principal $G$-bundle over $\mathbb{X}$ as in Definition 3.4. Let
$\widehat{s}\,:\,s^{*}E_{G}\,\longrightarrow\,E_{G}$ and
$\widehat{t}\,:\,s^{*}E_{G}\,\longrightarrow\,E_{G}$ 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
${\mathbb{E}}_{G}\,=\,[s^{*}E_{G}\rightrightarrows E_{G}]$ over the Lie
groupoid ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ is a connection
$\nabla$ on the principal $G$-bundle $E_{G}\,\longrightarrow\,X_{0}$ such that
the two $\mathfrak{g}$-valued $1$-forms $\widehat{s}^{*}\widehat{\nabla}$ and
$\widehat{t}^{*}\widehat{\nabla}$ on $s^{*}E_{G}$ coincide, where
$\widehat{\nabla}$ is the $1$-form
$\widehat{\nabla}\,:\,TE_{G}\,\longrightarrow\,\mathfrak{g}$ associated to the
connection $\nabla$.

######  Definition 5.5.

Let $\widehat{\nabla}$ be a connection on a principal $G$-bundle
${\mathbb{E}}_{G}\,=\,[s^{*}E_{G}\rightrightarrows E_{G}]$ over the Lie
groupoid ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ given by a
connection $\nabla$ on $E_{G}$. The curvature of $\widehat{\nabla}$ is defined
to be the curvature of $\nabla$. In particular, the connection
$\widehat{\nabla}$ is called flat if $\nabla$ is integrable.

######  Proposition 5.6.

Let ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ and
${\mathbb{Y}}\,=\,[Y_{1}\rightrightarrows Y_{0}]$ be Lie groupoids, and let
$\phi\,:\,{\mathbb{X}}\,\longrightarrow\,{\mathbb{Y}}$ be a morphism given by
maps

$\phi_{0}\,:\,X_{0}\,\longrightarrow\,Y_{0}\,,\ \
\phi_{1}\,:\,X_{1}\,\longrightarrow\,Y_{1}$

(see Definition 2.12). Let

${\mathbb{E}}_{G}\,:=\,[s^{*}E_{G}\rightrightarrows E_{G}]$

be a principal $G$-bundle on $\mathbb{Y}$ equipped with a connection
$\widehat{\nabla}$ given by a connection $\nabla$ on the principal $G$-bundle
$E_{G}\,\longrightarrow\,Y_{0}$. Then the pulled back connection
$\phi^{*}_{0}\nabla$ on $\phi^{*}E_{G}\,\longrightarrow\,X_{0}$ is a
connection on the principal $G$-bundle $\phi^{*}{\mathbb{E}}_{G}$ over the Lie
groupoid $\mathbb{X}$.

###### Proof.

Let $s$ and $s^{\prime}$ (respectively, $t$ and $t^{\prime}$) be the source
maps (respectively, target maps) of ${\mathbb{X}}$ and ${\mathbb{Y}}$
respectively. The two maps $s^{\prime}\circ\phi_{1}$ and $\phi_{0}\circ s$
from $X_{1}$ to $Y_{0}$ coincide. Similarly, the two maps
$t^{\prime}\circ\phi_{1}$ and $\phi_{0}\circ t$ from $X_{1}$ to $Y_{0}$ also
coincide.

Using this, it is straightforward to check that the connection
$\phi^{*}_{0}\nabla$ on $\phi^{*}E_{G}\,\longrightarrow\,X_{0}$ is a
connection on the principal $G$-bundle $\phi^{*}{\mathbb{E}}_{G}$ over the
groupoid $\mathbb{X}$. ∎

######  Example 5.7.

A connection on a principal $G$-bundle
${\mathbb{E}}_{G}\,=\,[Q\rightrightarrows E_{G}]$ over the Lie groupoid
${\mathbb{X}}\,=\,[X\rightrightarrows X]$ is the same as a connection on the
(ordinary) principal $G$-bundle $E_{G}\rightarrow X$ over the smooth space
$X$.

######  Example 5.8.

Let $\pi_{0}\colon\,E\rightarrow X_{0}$ be a vector bundle and $\pi_{G}\colon
E_{G}\rightarrow X_{0}$ a principal $G$-bundle over $X_{0}.$ Let
$\mathbb{X}:=[E\rightrightarrows X_{0}]$ be the Lie groupoid introduced in
Example 4.7. Then $E_{G}$ is an $\mathbb{X}$-space with respect to the map
$(v,p)\mapsto p,$ for all $(v,p)\in E\times E_{G}$ satisfying
$\pi_{0}(v)=s(v)=\pi_{G}(p).$ The action of $\mathbb{X}$ is obviously
compatible with the action of $G$ on $E_{G},$ and thus we obtain a principal
$G$-bundle over $\mathbb{X}.$ Then any connection on the (ordinary) principal
$G$-bundle $E_{G}\rightarrow X_{0}$ defines a connection on the principal
$G$-bundle over $\mathbb{X}.$

######  Example 5.9.

Let $E_{G}\rightarrow X_{0}$ be an $H$-equivariant principal $G$-bundle over
$X_{0}.$ Let ${\mathbb{X}}\,=\,[H\times X_{0}\rightrightarrows X_{0}]$ be the
transformation groupoid. As in Example 3.5, consider $E_{G}\rightarrow X_{0}$
to be a $G$-bundle over ${\mathbb{X}}.$ Then a connection $1$-form on the
(ordinary) principal $G$-bundle over $X_{0}$ is a connection on the $G$-bundle
over $\mathbb{X}$ if and only if it is $H$-invariant and vanishes on the
fundamental vector field generated by the action of $H$ on $E_{G}.$

######  Remark 5.10.

The definition of a connection on a principal $G$-bundle over a Lie groupoid
given in this article is less rigid then the one by Laurent-Gengoux, Tu and Xu
[LGTX, Def. 3.5]. the difference between our approach and the one in [LGTX]
becomes clear when comparing the associated de Rham complexes and Chern-Weil
maps, which is discussed in our follow-up article [BCKN, Sect. 5 & 6]. In
particular, it is not hard to see that the two definitions coincide in the
case of principal $G$-bundles over an étale Lie groupoid
${\mathbb{X}}\,=\,[X_{1}\rightrightarrows X_{0}]$ with integrable distribution
${\mathcal{H}}\,=\,TX_{1}$.

The space of connections on a principal $G$-bundle ${\mathbb{E}}_{G}$ over the
Lie groupoid ${\mathbb{X}}$ is an affine space for the space of all
$\text{ad}(E_{G})$-valued $1$-forms on the groupoid.

Henceforth, given a Lie groupoid ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows
X_{0}]$ we will assume that there exist a given integrable distribution
${\mathcal{H}}\,\subset\,TX_{1}$, in other words, we have given a flat
connection on the Lie groupoid ${\mathbb{X}}$.

Consider now again the Atiyah exact sequence as constructed in (5.6). The Lie
bracket of vector fields defines a Lie algebra structure on the sheaves of
sections of all three vector bundles. The Lie algebra structure on the sheaf
of sections of $\text{ad}(E_{G})$ is linear with respect to the multiplication
by functions on $X_{0}$, or in other words, the fibers of $\text{ad}(E_{G})$
are Lie algebras.

Let $\mathfrak{g}$ be again the Lie algebra for the Lie group $G$. Recall that
$\text{ad}(E_{G})\,=\,(E_{G}\times{\mathfrak{g}})/G$ is the vector bundle on
$X_{0}$ associated to principal $G$-bundle $E_{G}$ for the adjoint action of
$G$ on $\mathfrak{g}$. Since this adjoint action of $G$ preserves the Lie
algebra structure of $\mathfrak{g}$, it follows that the fibers of
$\text{ad}(E_{G})$ are Lie algebras identified with $\mathfrak{g}$ up to
conjugations.

Given any splitting of the Atiyah exact sequence (5.6)

$D\,:\,TX_{0}\,\longrightarrow\,\text{At}(E_{G})\,,$

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

$K(D)\,\in\,H^{0}(X_{0},\,\text{ad}(E_{G})\otimes\Omega^{2}_{X_{0}})\,,$

which is an $\text{ad}(E_{G})$-valued $2$-form on the groupoid ${\mathbb{X}}$.

######  Definition 5.11.

For a principal $G$-bundle ${\mathbb{E}}_{G}=[s^{*}E_{G}\rightrightarrows
E_{G}]$ over a Lie groupoid ${\mathbb{X}}=[X_{1}\rightrightarrows X_{0}]$ with
a connection $D$, the section $K(D)$ is called the curvature of the connection
$D$.

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

######  Theorem 5.12.

For any invariant form $\nu\,\in\,(\text{Sym}^{k}({\mathfrak{g}}^{*}))^{G}$,
the evaluation $\nu(K(D))$ on the curvature $K(D)$ is a closed $2k$-form on
the Lie groupoid ${\mathbb{X}}$.

###  5.2. Connections on principal $G$-bundles over differentiable stacks

Finally, we shall turn to the stacky picture and study connections for
principal $G$-bundles on a differentiable stack (compare also [BMW]).

######  Definition 5.13 (Connection on principal bundle over differentiable
stack).

Let ${\mathscr{E}}_{G}$ be a principal $G$-bundle on a differentiable stack
${\mathscr{X}}$. A connection $\nabla$ on ${\mathscr{E}}_{G}$ consists of the
data of a connection $\nabla_{u}$ on each principal $G$-bundle
${\mathscr{E}}_{G,u}$, where $u\,:\,U\,\longrightarrow\,{\mathscr{X}}$ is a
smooth atlas for ${\mathscr{X}}$, which pulls back naturally with respect to
each $2$-commutative diagram of the form

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{\varphi}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v}$$\textstyle{\mathscr{X}.}$

A connection $\nabla$ on ${\mathscr{E}}_{G}$ is flat or integrable if it is in
addition integrable on each principal $G$-bundle ${\mathscr{E}}_{G,u}$.

Let us unravel this definition with more details. We can realize the
connections for each atlas $u$ in terms of $\mathfrak{g}$-valued $1$-forms
such that
$\omega_{{\mathscr{E}}_{G,u}}\,\in\,H^{0}({{\mathscr{E}}_{G,u}},\,\Omega^{1}_{{\mathscr{E}}_{G,u}}\otimes{\mathfrak{g}})$
is the corresponding $1$-form for the connection $\nabla_{u}$ on
${\mathscr{E}}_{G,u}$. From the Cartesian diagram

$\textstyle{\varphi^{*}{\mathscr{E}}_{G,v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\varphi}}$$\textstyle{{\mathscr{E}}_{G,v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{V}$

we get a connection $\bar{\varphi}^{*}\omega_{{\mathscr{E}}_{G,v}}$ on
$\varphi^{*}{\mathscr{E}}_{G,v}$. The condition for the existence of a
connection is then given as follows

$\omega_{{\mathscr{E}}_{G,u}}\,=\,\theta_{\varphi,\alpha}^{*}\bar{\varphi}^{*}\omega_{{\mathscr{E}}_{G,v}}\,.$

From now on, we will assume that ${\mathscr{X}}$ is a Deligne–Mumford stack,
which in particular means that the tangent stack $T{\mathscr{X}}$ gives rise
to a vector bundle $T{\mathscr{X}}\,\longrightarrow\,{\mathscr{X}}$ over
${\mathscr{X}}$. For a general differentiable stack, this is not always the
case (see [Be], [LGTX], [Hp]).

Given a principal $G$-bundle ${\mathscr{E}}_{G}$ over $\mathscr{X}$, we define
the associated Atiyah bundle $\text{At}({\mathscr{E}}_{G})$ over
${\mathscr{X}}$ by setting

$\text{At}({\mathscr{E}}_{G})_{u}\,:=\,\text{At}({\mathscr{E}}_{G,u})\,.$

for any étale morphism $u\,:\,U\,\longrightarrow\,{\mathscr{X}}$. In addition,
for each $2$-commutative diagram of the form

$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\scriptstyle{\varphi}$$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v}$$\textstyle{\mathscr{X}}$
(5.11)

where $u,v.\varphi$ are étale morphisms, we can compose the isomorphism
$\text{At}({\mathscr{E}}_{G,u})\,\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\,\text{At}(\varphi^{*}{\mathscr{E}}_{G,v})$
with the isomorphism
$\text{At}(\varphi^{*}{\mathscr{E}}_{G,v})\,\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\,\varphi^{*}\text{At}({\mathscr{E}}_{G,u})$
to get an isomorphism

$\text{At}({\mathscr{E}}_{G})_{u}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\varphi^{*}\text{At}({\mathscr{E}}_{G})_{v}.$

Similarly, given a principal $G$-bundle ${\mathscr{E}}_{G}$ over
${\mathscr{X}}$, we can define the associated adjoint bundle
$\text{ad}({\mathscr{E}}_{G})$ over ${\mathscr{X}}$ arising from the adjoint
representation of $G$ by setting (compare [BMW, Sect. 1.4])

$\text{ad}({\mathscr{E}}_{G})\,:=\,{\mathscr{E}}_{G}\times^{G}\mathfrak{g}\,,$

where $\text{ad}({\mathscr{E}}_{G})_{u}\,=\,\text{ad}({\mathscr{E}}_{G,u}).$
We then obtain a commutative diagram of vector bundles

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{ad}({\mathscr{E}}_{G})_{u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{At}({\mathscr{E}}_{G})_{u}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{TU\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varphi^{*}\text{ad}({\mathscr{E}}_{G})_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varphi^{*}\text{At}({\mathscr{E}}_{G})_{v}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\varphi^{*}TV\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

which gives a well-defined short exact sequence of vector bundles over the
Deligne–Mumford stack ${\mathscr{X}}$, called the Atiyah exact sequence
associated to the principal $G$-bundle ${\mathscr{E}}_{G}$

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{ad}({\mathscr{E}}_{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{At}({\mathscr{E}}_{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T{\mathscr{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

######  Remark 5.14.

In the situation of a general differentiable stack ${\mathscr{X}}$, similarly
as for the tangent stack, $\text{At}({\mathscr{E}}_{G})$ and
$\text{ad}({\mathscr{E}}_{G})$ will generally not give vector bundles over
${\mathscr{X}}$.

From the constructions and considerations above, it follows now

######  Proposition 5.15.

A principal $G$-bundle ${\mathscr{E}}_{G}$ over a Deligne–Mumford stack
${\mathscr{X}}$ admits a connection if and only if its associated Atiyah exact
sequence has a splitting.

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

######  Theorem 5.16.

Giving a connection (respectively, flat connection) on a principal $G$-bundle
${\mathscr{E}}_{G}$ over a Deligne–Mumford stack ${\mathscr{X}}$ with étale
atlas $x\,:\,X_{0}\,\longrightarrow\,{\mathscr{X}}$ is equivalent to giving a
connection (respectively, flat connection) on the associated principal
$G$-bundle $\mathbb{E}_{G}$ over the groupoid
$\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$. Giving a connection
(respectively, flat connection) on a principal $G$-bundle
$\mathbb{E}_{G}\,=\,[s^{*}E_{G}\rightrightarrows E_{G}]$ over an étale Lie
groupoid $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ is equivalent to
giving a connection (respectively, flat connection) on the associated
principal $G$-bundle ${\mathscr{E}}_{G}$ of $\mathbb{E}_{G}$-torsors over the
classifying stack $\mathscr{B}\mathbb{X}$ of $\mathbb{X}$-torsors.

###### Proof.

This follows from the categorical equivalence of Theorem 3.7 between the
category ${\mathscr{B}}un_{G}({\mathscr{X}})$ of principal $G$-bundles over
${\mathscr{X}}$ and the category ${\mathbb{B}}un_{G}({\mathbb{X}})$ of
principal $G$-bundles over the associated Lie groupoid
$\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ and the fact that a splitting
of the Atiyah exact sequence of associated vector bundles over the stack
${\mathscr{X}}$

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{ad}({\mathscr{E}}_{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\text{At}({\mathscr{E}}_{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T{\mathscr{X}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

corresponds to a splitting of the Atiyah exact sequence of associated vector
bundles over the smooth space $X_{0}$

$0\,\longrightarrow\,\text{ad}(E_{G})\,\longrightarrow\,\text{At}(E_{G})\,{\longrightarrow}\,TX_{0}\,\longrightarrow\,0\,$

and vice versa involving the $2$-Cartesian diagram

$\textstyle{E_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathscr{E}}_{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathscr{X}}$

and unraveling the explicit constructions of associated bundles as given in
the proof of Theorem 3.7. ∎

Let us now consider the classifying stack ${\mathscr{B}}^{\nabla}G$ of
principal $G$-bundles with connections. The objects are triples
$(P,\,\omega,\,S)$ where $S$ is a smooth space of $\mathfrak{S}$ and $P$ is a
principal $G$-bundle over $S$ and
$\omega\,\in\,\Omega^{1}(S,\,\mathfrak{g})^{G}$ a connection $1$-form. A
morphisms
$(P,\,\omega,\,S)\,\longrightarrow\,(P^{\prime},\,\omega^{\prime},\,S^{\prime})$
is given by a commutative diagram

$\textstyle{P\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{P^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S^{\prime}}$

where $\varphi\,:\,P\,\longrightarrow\,P^{\prime}$ is a $G$-equivariant map
and $\varphi^{*}\omega^{\prime}=\omega$. Then ${\mathscr{B}}^{\nabla}G$
together with the projection functor

$\pi:{\mathscr{B}}^{\nabla}G\,\longrightarrow\,{\mathfrak{S}}\,,\
~{}(P,\,\omega,\,S)\,\longmapsto\,S$

is a groupoid fibration over $\mathfrak{S}$. We note that
${\mathscr{B}}^{\nabla}G$ 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 5.17.

It is not clear in general if and under which conditions
${\mathscr{B}}^{\nabla}G$ is actually a differentiable or Deligne–Mumford
stack and not just a stack over $\mathfrak{S}$.

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

######  Proposition 5.18.

Let ${\mathscr{X}}$ be a differentiable stack. Giving a principal $G$-bundle
with connection over ${\mathscr{X}}$ is equivalent to giving a morphism of
stacks ${\mathscr{X}}\,\longrightarrow\,{\mathscr{B}}^{\nabla}G$ and two
principal $G$-bundles with connections over ${\mathscr{X}}$ are isomorphic if
and only if the corresponding morphisms of stacks
${\mathscr{X}}\,\longrightarrow\,{\mathscr{B}}^{\nabla}G$ are $2$-isomorphic.

Similarly, as before, we can consider the groupoid fibration
${\mathscr{B}}un_{G}^{\nabla}({\mathscr{X}})$ over $\mathfrak{S}$ whose
objects over a smooth space $U$ are principal $G$-bundles ${\mathscr{E}}_{G}$
over ${\mathscr{X}}\times U$ with connections and whose morphisms are given by
pullback diagrams of principal $G$-bundles with connections as above. From
Proposition 5.18 we get therefore the following:

######  Proposition 5.19.

There is an equivalence of groupoid fibrations over $\mathfrak{S}$

${\mathscr{B}}un_{G}^{\nabla}({\mathscr{X}})\,\cong\,{\mathscr{H}}om({\mathscr{X}},\,{\mathscr{B}}^{\nabla}G)\,.$

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

## Acknowledgements

The authors would like to thank the anonymous referee for very useful comments
and suggestions. The first author acknowledges the support of a J. C. Bose
Fellowship. The second author acknowledges research support from SERB, DST,
Government of India grant MTR/2018/000528. The fourth author would like to
thank the Tata Institute of Fundamental Research (TIFR) in Mumbai and the
University of Leicester for financial support.

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