# Connections on Lie groupoids and Chern-Weil theory

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., Vithura, Kerala
695551, India<EMAIL_ADDRESS>,  Praphulla Koushik  School of
Mathematics, Indian Institute of Science Education and Research–Pune, Dr. Homi
Bhabha Road, Pashan, Pune, 411008, India<EMAIL_ADDRESS>and
Frank Neumann  School of Computing and Mathematical Sciences, University of
Leicester, University Road, Leicester LE1 7RH, England, UK<EMAIL_ADDRESS>
###### Abstract.

Let $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ be a Lie groupoid equipped
with a connection, given by a smooth distribution
$\mathcal{H}\,\subset\,TX_{1}$ transversal to the fibers of the source map.
Under the assumption that the distribution $\mathcal{H}$ is integrable, we
define a version of de Rham cohomology for the pair
$(\mathbb{X},\,\mathcal{H})$, and we study connections on principal
$G$-bundles over $(\mathbb{X},\,\mathcal{H})$ in terms of the associated
Atiyah sequence of vector bundles. We also discuss associated constructions
for differentiable stacks. Finally, we develop the corresponding Chern-Weil
theory and describe characteristic classes of principal $G$-bundles over a
pair $(\mathbb{X},\,\mathcal{H})$.

###### Key words and phrases:

principal bundles, Lie groupoids, differentiable stacks, connections, Chern-
Weil theory

###### 2010 Mathematics Subject Classification:

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

###### Contents

  1. 1 Introduction
  2. 2 Principal bundles over Lie groupoids
  3. 3 de Rham cohomology for Lie groupoids and differentiable stacks
  4. 4 Connections on principal bundles over Lie groupoids
  5. 5 Forms for connections on principal Lie groupoid bundles
  6. 6 Chern-Weil map and characteristic classes

##  1\. Introduction

The geometry of principal bundles over Lie groupoids and its applications in
geometry and physics have been a very active areas of research in recent
years. In particular, the concepts of a connection on Lie groupoids and
connections on principal bundles over Lie groupoids as well as its geometric,
analytic, and algebraic properties, have been discussed by several authors in
the process of extending concepts of differential geometry from smooth
manifolds to Lie groupoids [LGTX2, BX, Be2, CM, CLW, Ta, Sa, DE]. An important
ingredient of any such framework is the construction of the associated Chern-
Weil theory and a theory of characteristic classes via de Rham cohomology. For
example, Laurent-Gengoux, Tu and Xu [LGTX2, CM] study Chern-Weil theory for
principal bundles on Lie groupoids via de Rham cohomology defined by using
simplicial manifolds associated to the Lie groupoid nerves (compare also [Du,
FN]). Behrend in [Be2] investigates flat connections on Lie groupoids via the
theory of integrable distributions and cofoliations and studies the associated
de Rham cohomology. Tang in [Ta] defines in a similar fashion flat connections
for Lie groupoids, which he called étalifications and investigates its
applications in symplectic and Poisson geometry and for deformation
quantization (see also [Ha]). In [BX] Behrend and Xu define connections and
curvings and the associated Chern-Weil theory and theory of characteristic
classes in the general setting of groupoid $S^{1}$-central extensions. More
recently, Arias Abad and Crainic [AC] introduced more general and flexible
Ehresmann connections for any Lie groupoids. Trentinaglia [Tr] investigates
and describes the space of Cartan or multiplicative connections on proper Lie
groupoids. Furthermore, Herrera and Ortiz [HO] are currently developing
aspects of the geometry of principal 2-bundles over Lie groupoids involving
Atiyah LA-groupoids. Recently Connections and “higher gauge transformations”
[see, [MP1, MP2, MP3, Ma-Mi, BCKMM]] and the references therein for recent
developments in higher gauge theory] have been studied in terms Atiyah
sequences of principal 2-bundles over Lie groupoids in [CCK]. For a general
discussion and recent results on the geometry of Lie groupoids, we refer to
[dH, MM, Ma1, Ma2].

Another important reason for the interest in the geometry of Lie groupoids is
its association with the geometry of differentiable stacks. Differentiable
stacks have presentations by Lie groupoids, in fact, they are basically Morita
equivalence classes of Lie groupoids (see for example [He, LGSX, Me, CK, Ca,
Gi]). Many concepts from the geometry of Lie groupoids can be extended to
differentiable stacks via Morita invariance as long as they respect the
multiplicative groupoid structure.

The objective of this article is to provide a new and more flexible approach
for connections on principal bundles over Lie groupoids (with integrable
connections) in terms of Atiyah sequences of vector bundles associated to
transversal tangential distribution. A main application is the development of
the associated Chern-Weil theory and theory of characteristic classes for such
principal Lie groupoid bundles. Our approach is directly inspired by the
classical work of Atiyah [At] about connections on fiber bundles in complex
geometry. In the course of this article, the necessary framework for de Rham
cohomology on a Lie groupoid with integrable connection is developed and
studied. More precisely, given a Lie groupoid
$\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ equipped with an integrable
connection, defined by a distribution $\mathcal{H}\,\subset\,TX_{1}$ which is
transversal to the fibers of the source map, we define a new version of de
Rham cohomology for such a pair $(\mathbb{X},\,\mathcal{H})$ and study general
connections on principal $G$-bundles over $(\mathbb{X},\,\mathcal{H})$ in
terms of the associated Atiyah sequence of vector bundles. This approach
allows then for a natural way to develop the corresponding Chern-Weil theory
and construction of characteristic classes for principal $G$-bundles over a
given pair $(\mathbb{X},\,\mathcal{H})$. We will also discuss extensions of
our constructions respecting the multiplicative structures to differentiable
stacks, namely to the special class of Deligne-Mumford stacks, which are
presented by étale Lie groupoids. Étale Lie groupoids always admit a
connection. Special cases of Deligne-Mumford stacks also include orbifolds and
foliated spaces. We are exploring extensions of our constructions to more
general differentiable stacks also in a related article [BCKN1] (compare also
[BN]). Our approach here is also closely related to that of Behrend on
cofoliations for differentiable stacks [Be2].

The purpose here is to study connections in terms of splittings of the Atiyah
sequence and the corresponding Chern-Weil theory. In order to do so, we
introduce connections on Lie groupoids and develop the corresponding de Rahm
cohomology. While [LGTX2] discusses the Chern-Weil theory and de Rahm
cohomology for principal bundles over Lie groupoids, their approach uses
simplicial nerves of Lie groupoid, without involvement of any given Lie
groupoid connections. In comparison, in our article a given Lie groupoid
connection is a necessity to study the Atiyah sequence for a principal bundle
over a Lie groupoid. See the discussion around Equation 3.15 where we
distinguish the two approaches with an illustrating example. Nonetheless, a
simplicial approach can be adopted as well in our framework, as explained in
Subsection 3.3. However the observation is cursory and we have not pursued
this approach here in more detail. On the other hand [Be2] works with a given
connection on a Lie groupoid, but their definition is slightly more strict
than ours, and uses a particular de Rham cohomology associated with a flat Lie
groupoid connection. The main purpose of their paper is to study the
cofoliations of stacks with respect to flat Lie groupoid connections.

Outline and organization of the article. In the first section (Section 2) we
recall standard notions and constructions, such as the definitions of
principal $G$-bundles and vector bundles over a Lie groupoid
$\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ and set up our notations. For
a Lie group $G$, a principal $G$-bundle $(E_{G}\rightarrow
X_{0},\,\mathbb{X})$ over a Lie groupoid $\mathbb{X}$ is a $G$-bundle
$E_{G}\,\longrightarrow\,X_{0}$ with an action (compatible with the action of
$G$ on $E_{G}$) of $\mathbb{X}$ on $E_{G}$ (Definition 2.8). Likewise, a
vector bundle over $\mathbb{X}$ is a vector bundle $E\,\longrightarrow\,X_{0}$
with an action of $\mathbb{X}$ on $E$ inducing a linear map between fibers. In
the following section (Section 3) we introduce the notion of a connection on a
Lie groupoid $\mathbb{X}=[X_{1}\rightrightarrows X_{0}]$ as a smooth
distribution $\mathcal{H}\,\subset\,TX_{1}$ complementing the kernel of the
differential $ds$ for the source map $s$. A connection is called integrable or
flat if the corresponding distribution $\mathcal{H}$ is integrable. This
definition was originally introduced in [BN] (compare also [BCKN1]). Assuming
our Lie groupoid admits a connection, then a differential form on $X_{0}$ is
said to be a differential form on the Lie groupoid $\mathbb{X}$ if it
satisfies certain compatibility condition with respect to the source and
target map and the connection on the Lie groupoid (see Definition 3.1). We
show that, for an integrable connection, the exterior derivative of a
differential form on $\mathbb{X}$ is well defined. In turn, we obtain the
graded de Rham cohomology algebra $H_{\rm dR}^{*}({\mathbb{X},\,\mathcal{H}})$
of the pair $(\mathbb{X},\,\mathcal{H})$. It should be noted that the
integrability condition on $\mathcal{H}$ will be crucial here. In addition, we
give a brief outline of the construction of a simplicial version of de Rham
cohomology for a Lie groupoid with integrable connection and we extend our
constructions to Deligne-Mumford stacks. Given a principal $G$-bundle
$(E_{G}\rightarrow X_{0},\,\mathbb{X})$ over a Lie groupoid $\mathbb{X}$, we
obtain the associated Atiyah sequence

$0\,\longrightarrow\,(E_{G}\times\mathfrak{g})/G\,\longrightarrow\,(TE_{G})/G\,\longrightarrow\,TX_{0}\,\longrightarrow\,0$

of vector bundles over the manifold $X_{0}$ associated to the principal bundle
$E_{G}\,\longrightarrow\,X_{0}$. In Subsection 3.3 we will also compare our
framework with the ones developed by Behrend [Be1, Be2] and Behrend-Xu [BX].
We will also give a brief comparison between our approach here with that of
Laurent-Gengoux, Tu and Xu [LGTX2]. In Section 4 we prove a key result, namely
that for a Lie groupoid $\mathbb{X}$ with a connection $\mathcal{H}$ one can
define actions of $\mathbb{X}$ on $(E_{G}\times\mathfrak{g})/G,(TE_{G})/G$ and
$TX_{0}$, turning the Atiyah sequence into a sequence of vector bundles over
the Lie groupoid $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$. A connection
on $(E_{G}\rightarrow X_{0},\,\mathbb{X})$ is then defined as the splitting of
this sequence of vector bundles over the Lie groupoid $\mathbb{X}$. In the
following sections (Sections 5 and 6) we assume connections on a Lie groupoid
exist and to be integrable. In Section 5 we characterize connections on a
principal $G$-bundle $(E_{G}\rightarrow X_{0},\,\mathbb{X})$ over a given Lie
groupoid $\mathbb{X}$ in terms of $\mathfrak{g}$-valued differential forms on
the Lie groupoid. Theorem 6.1 in Section 6 presents the main result of this
article, namely the existence of a well defined Chern-Weil map for a principal
bundle $(E_{G}\rightarrow X_{0},\,\mathbb{X})$ over a Lie groupoid
$\mathbb{X}$ with integrable connection $\mathcal{H}$ and its independence
from the choice of a connection on $(E_{G}\rightarrow X_{0},\,\mathbb{X})$.
Finally, we also describe the associated theory of characteristic classes
within this framework.

It is worthwhile to further explore the relations between our approach and the
seemingly different approaches in the literature towards Chern-Weil theory for
Lie groupoids and differentiable stacks and its applications in geometry and
physics, like the construction of secondary characteristic classes and
multiplicative $K$-theory. In fact, our framework is very much adaptable for
the theory of secondary characteristic classes of Lie groupoids and
cofoliations, respectively relating to Chern-Simons theory and symplectic
geometry. Some of these themes will be the topic of our follow-up article
[BCKN2].

##  2\. Principal bundles over Lie groupoids

In this section, we will recall the notion and basic properties of a principal
bundle over a Lie groupoid. We refer also to [MM], [BX], [He] and [LGTX2] for
some of the material presented here; [BN] and [BCKN1] are also referred.

###  2.1. Smooth spaces, Lie groupoids, and principal bundles

We will work over the category of ${\mathcal{C}}^{\infty}$-manifolds and refer
to it also as the category $\mathfrak{S}$ of smooth spaces and smooth maps.
All manifolds considered here will be second countable but are not necessarily
Hausdorff spaces. We do not impose the Hausdorff condition to ensure that
certain interesting examples of non-Hausdorff groupoids, such as foliation
groupoids, are not left out.

A smooth space or smooth manifold will mean an object in $\mathfrak{S}$. The
tangent bundle of any smooth space $X$ will be denoted by $TX$. A smooth map
will refer to a morphism in $\mathfrak{S}$, 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
$\mathfrak{S}$ which is also a submersion. An étale map corresponds,
therefore, to the notion of a local diffeomorphism.

For any smooth space $X$, its structure sheaf will be denoted by
${\mathcal{O}}_{X}$. A vector bundle on $X$ will be identified with its sheaf
of sections, which is a 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 $T^{*}X$ is
$\Omega^{p}(X)\,:=\,\bigwedge^{p}T^{*}X$.

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 étale topology on $\mathfrak{S}$
(compare also [Vi]). If either of the two morphisms $U\rightarrow X$ and
$V\rightarrow X$ in $\mathfrak{S}$ is a submersion, then the fiber product
exists.

######  Remark 2.1.

As in [BCKN1] we could as well instead of working over the category of
${\mathcal{C}}^{\infty}$-manifolds consider the category of complex analytic
manifolds and the category of smooth schemes of finite type over the field of
complex numbers. We leave it to the reader to make the necessary amendments in
the definitions and constructions presented here.

######  Definition 2.2 (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 set
$X_{1}$ of arrows and the set $X_{0}$ of objects are objects of $\mathfrak{S}$
(so they are smooth manifolds) 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 Lie 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 Lie groupoid.

######  Definition 2.3 (Morphisms of Lie groupoids).

A morphism between Lie groupoids ${\mathbb{X}}\,=\,[X_{1}\rightrightarrows
X_{0}]$ and ${\mathbb{Y}}\,=\,[Y_{1}\rightrightarrows Y_{0}]$ is a functor
$\phi\,=\,(\phi_{1},\,\phi_{0})\,:\,{\mathbb{X}}\,\longrightarrow\,{\mathbb{Y}}$
such that

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

are smooth maps which commute with all structure morphisms of the groupoids. A
morphism $\phi\,:\,{\mathbb{X}}\,\longrightarrow\,{\mathbb{Y}}$ of Lie
groupoids is a Morita morphism 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 a pullback diagram, that is, $X_{1}\,\cong\,Y_{1}\times_{Y_{0}\times
Y_{0}}(X_{0}\times X_{0})$.

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

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

Now we will recall the general notions of principal Lie groupoid bundles over
smooth manifolds. First, we need to define Lie groupoid actions.

######  Definition 2.4 (Action of a Lie groupoid on a manifold).

Let $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ be a Lie groupoid. Let $P$
be a smooth manifold. A left action of the Lie groupoid $\mathbb{X}$ on the
manifold $P$ is given by a pair of smooth maps

$a\,\colon\,P\,\longrightarrow\,X_{0}\ \ \text{ and }\ \
\mu\,\colon\,X_{1}\times_{s,X_{0},a}P\,\longrightarrow\,P$

satisfying the following conditions:

  1. (1)

$\mu(1_{a(p)},\,p)\,=\,p$ for all $p\,\in\,P$,

  2. (2)

$a(\mu(\gamma,\,p))\,=\,t(\gamma)$ for all
$(\gamma,\,p)\,\in\,X_{1}\times_{s,X_{0},a}P$, and

  3. (3)

$\mu(\gamma^{\prime},\,\mu(\gamma,\,p))\,=\,\mu((\gamma^{\prime}\circ\gamma),\,p)$
for all
$(\gamma^{\prime},\,\gamma,\,p)\,\in\,X_{1}\times_{s,X_{0},t}X_{1}\times_{s,X_{0},a}P$.

For convenience, the map $\mu$ in Definition 2.4 will usually be denoted
simply by a dot “$\cdot$”.

A right action of the Lie groupoid $\mathbb{X}$ on the manifold $P$ is defined
likewise, with the action map instead given by

$\mu\,\colon\,P\times_{a,X_{0},t}X_{1}\,\longrightarrow\,P\,.$

######  Example 2.5.

Consider a Lie group $G$ as the Lie groupoid $[G\,\rightrightarrows\,*]$ with
singleton object set $*$ and morphism set $G$. The actions of
$[G\rightrightarrows*]$ on a smooth manifold $P$ are then precisely the smooth
actions of the Lie group $G$ on $P$.

######  Example 2.6.

Given a smooth manifold $M$, we get a Lie groupoid
$[M\,\rightrightarrows\,M]$, where both maps $s$ and $t$ are given by the
identity map of $M$. An action of $[M\rightrightarrows M]$ on a smooth
manifold $P$ is the same as a smooth map $P\,\longrightarrow\,M$.

######  Example 2.7.

Suppose a Lie group $G$ is acting on a smooth manifold $M$. Then we have a Lie
groupoid $[M\times G\,\rightrightarrows\,M]$, the action groupoid, with source
and target maps given by $(m,\,g)\,\longmapsto\,m$ and
$(m,\,g)\,\longmapsto\,m\cdot g$ respectively. An action of $[M\times
G\rightrightarrows M]$ on a manifold $P$ is then an action of the Lie group
$G$ on the manifold $P$ together with a $G$-equivariant smooth map
$P\,\longrightarrow\,M$.

######  Definition 2.8 (Principal $G$-bundle over a Lie groupoid).

Let $G$ be a Lie group, and let $\mathbb{X}\,=\,[X_{1}\rightrightarrows
X_{0}]$ be a Lie groupoid. A right principal $G$-bundle over $\mathbb{X}$ is a
(right) principal $G$-bundle $\pi\,\colon\,E_{G}\,\longrightarrow\,X_{0}$ over
the smooth manifold $X_{0}$ together with a left action

$(\pi:E_{G}\rightarrow X_{0},\ \mu\colon
X_{1}\times_{s,X_{0},\pi}E_{G}\rightarrow E_{G}),$

of the Lie groupoid $\mathbb{X}$ on the manifold $E_{G}$, such that
$(\gamma\cdot p)\cdot g\,=\,\gamma\cdot(p\cdot g)$ for all
$(\gamma,\,p,\,g)\,\in\,X_{1}\times_{X_{0}}E_{G}\times G$. A principal
$G$-bundle over $\mathbb{X}$ will be denoted as $\bigl{(}E_{G}\rightarrow
X_{0},\,\mathbb{X}\bigr{)}$. A morphism of right principal $G$-bundles from
$(E_{G}\rightarrow X_{0},\,\mathbb{X})$ to $(E_{G}^{\prime}\rightarrow
X_{0},\,\mathbb{X})$ is a morphism

$\psi\,\colon\,(E_{G},\,\pi,\,X_{0})\,\longrightarrow\,(E_{G}^{\prime},\,\pi^{\prime},\,X_{0})$

of the underlying principal bundles over the manifold $X_{0}$, such that

$\psi(\mu(\gamma,\,e))\,=\,\mu^{\prime}(\gamma,\,\psi(e))$

for all $(\gamma,\,e)\,\in\,X_{1}\times_{s,X_{0},a}E_{G}$. Similarly, we also
have the notion of a left principal $G$-bundle over $\mathbb{X}$ and morphisms
of left principal $G$-bundles.

######  Example 2.9.

Let $G,\,H$ be a pair of Lie groups. A principal $G$-bundle over the Lie
groupoid $[H\rightrightarrows*]$ is the same as a left-action of $H$ on $G$
that commutes with the right-translation action of $G$ on itself, meaning that
$h\cdot(gg^{\prime})\,=\,(h\cdot g)g^{\prime}$ for all $h\,\in\,H$ and
$g,\,g^{\prime}\,\in\,G$.

######  Example 2.10.

Let $G$ be a Lie group. Let $\mathbb{X}\,=\,[M\rightrightarrows M]$ be the Lie
groupoid associated to a smooth manifold $M$. Then, a principal $G$-bundle
over $\mathbb{X}$ is the same as a principal $G$-bundle
$P\,\longrightarrow\,M$ over the manifold $M$.

######  Example 2.11.

Let $G,\,H$ be a pair of Lie groups. Then an $H$-equivariant smooth principal
$G$-bundle $P\,\longrightarrow\,M$ over a smooth manifold $M$ defines a
principal $G$-bundle over the Lie groupoid $[M\times H\rightrightarrows M]$.

######  Example 2.12.

Let $\pi\,\colon\,P\,\longrightarrow\,M$ be a principal $G$-bundle over a
manifold $M.$ Consider the groupoid $[P\times P\rightrightarrows P]$ with
source and target maps as first and second projections respectively and with
the obvious multiplication. The Lie group $G$ acts on $P\times P$ by
$(p_{1},\,p_{2})\cdot g\,=\,(p_{1}\cdot g,\,p_{2}\cdot g).$ Under the quotient
by the action of $G$, the groupoid $[P\times P\rightrightarrows P]$ descends
to a Lie groupoid $\mathbb{P}_{\rm Gauge}:=\,[\frac{P\times
P}{G}\rightrightarrows M\,].$ This groupoid is often called the gauge groupoid
or Atiyah groupoid in the literature. It plays an important role in gauge
theory and the theory of Lie algebroids (see [Lo, KoSt]). We will show that
$\pi\colon P\rightarrow M$ is a $G$-bundle over ${\mathbb{P}}_{\rm Gauge}$. To
see this we first note that for any $\bigl{(}[(p_{1},p_{2})],q\bigr{)}\in
s^{*}{P}$ we have a $g\in G$ such that $q\cdot g=p_{1}.$ Of course $g$ depends
on the choice of the representative element $(p_{1},p_{2})\in[(p_{1},p_{2})].$
Now define a smooth map $\mu\colon s^{*}{P}\rightarrow P$ by
$\mu\bigl{(}[(p_{1},p_{2})],q\bigr{)}=p_{2}\cdot g^{-1}$. Observe that the map
is well defined for the following reason. Take any other representative
element $(p_{1}\cdot g^{\prime},p_{2}\cdot
g^{\prime})=(p^{\prime}_{1},p^{\prime}_{2}).$ Then
$q\cdot(gg^{\prime})=p^{\prime}_{1}$ and thus
$p^{\prime}_{2}\cdot(gg^{\prime})^{-1}=p_{2}\cdot(g^{\prime}g^{\prime-1}g^{-1})=p_{2}\cdot
g^{-1}.$ Moreover it is easily verified that
$\mu([p_{1},p_{2}],q)g=\mu([p_{1},p_{2}],qg)$. Thus $P\rightarrow M$ is a
principal $G$-bundle over the Lie groupoid ${\mathbb{P}}_{\rm Gauge}.$

Definition 2.8 also gives the definition of real and complex vector bundles
(of rank $k$) over a Lie groupoid $\mathbb{X}$, which are basically principal
$\mathrm{GL}(k,\mathbb{F})$-bundles over $\mathbb{X}$, where $\mathbb{F}$ is
the field of real or complex numbers.

######  Definition 2.13 (Vector bundle over a Lie groupoid).

Let $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ be a Lie groupoid. A (rank
$k$) vector bundle over the Lie groupoid $\mathbb{X}$ is a (rank $k$) vector
bundle $\pi\,\colon\,E\,\longrightarrow\,X_{0}$ over the manifold $X_{0}$
together with a left action

$\bigl{(}\pi\,:\,E\,\longrightarrow\,X_{0},\
\mu\,\colon\,X_{1}\times_{s,X_{0},\pi}E\,\longrightarrow\,E\bigr{)}$

of $\mathbb{X}$ on $E$ such that the map
$\mu_{\gamma}\,\colon\,E_{s(\gamma)}\,\longrightarrow\,E_{t(\gamma)}$,
$a\,\longmapsto\,\gamma\cdot a$, is linear for every $\gamma\,\in\,X_{1}$.

Morphisms of vector bundles over Lie groupoids are defined by imitating the
definition of morphisms of principal bundles.

We also will use the following general notion of a principal Lie groupoid
bundle (compare with [BX] and [LGSX]).

######  Definition 2.14 (Principal $\mathbb{X}$-bundle over a manifold).

Let $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ be a Lie groupoid. Let $M$
be a smooth manifold. A principal right $\mathbb{X}$-bundle over $M$ is a
surjective submersion $\pi\,\colon\,P\,\longrightarrow\,M$ together with a
right action

$\bigl{(}a\colon P\,\longrightarrow\,X_{0},\ \mu\colon
P\times_{a,X_{0},t}X_{1}\,\longrightarrow\,P\bigr{)}$

of the Lie groupoid $\mathbb{X}$ on the manifold $P$, such that following two
conditions are satisfied:

  1. (1)

$\pi(p\cdot h)\,=\,\pi(p)$ for all $(p,\,h)\,\in\,P\times_{X_{0}}X_{1}$, and

  2. (2)

the map $P\times_{a,X_{0},t}X_{1}\,\longrightarrow\,P\times_{\pi,M,\pi}P$
given by $(p,\,h)\,\longmapsto\,(p,\,p\cdot h)$ is a diffeomorphism.

Similarly, we also have the notion of a principal left $\mathbb{X}$-bundle
over $M$.

Let us present two important examples with more details.

######  Example 2.15.

Let $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ be a given Lie groupoid.
The source map $s\,\colon\,X_{1}\,\longrightarrow\,X_{0}$ defines a principal
$\mathbb{X}$-bundle over $X_{0}.$ Similarly, the target map
$t\,\colon\,X_{1}\,\longrightarrow\,X_{0}$ along with the composition
$X_{1}\times_{s,X_{0},t}X_{1}\,\longrightarrow\,X_{1}$ defines an action of
$\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ on the manifold $X_{1}.$

######  Example 2.16.

Given a Lie groupoid $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$, a
principal $\mathbb{X}$-bundle $\pi\,\colon\,P\,\longrightarrow\,M$ and a map
of smooth manifolds $f\,\colon\,N\,\longrightarrow\,M,$ we have the obvious
notion of a pullback $\mathbb{X}$-bundle $f^{*}P\,\longrightarrow\,N$. The
action of the Lie groupoid $\mathbb{X}$ on $f^{*}(P)$ is given by
${\widetilde{a}}\,=\,a\circ{\rm pr}_{1}$ and
$((p,\,n),\,\gamma)\,\longmapsto\,(\mu(p,\,\gamma),\,n),$ where $(\mu,\,a)$ is
the original action of $\mathbb{X}$ on $P$.

###  2.2. Fibered categories and differentiable stacks

We shall now recall the definition of differentiable stacks and the associated
differentiable stack of a Lie groupoid. For this, we first need the following
general definitions and constructions (compare [LGSX, Vi]). Given any category
$\mathcal{K}$ we will denote as usual by $\mathcal{K}_{0}$ the objects and by
$\mathcal{K}_{1}$ the morphisms of $\mathcal{K}$.

######  Definition 2.17.

Let $\mathcal{C}$ and $\mathscr{X}$ be categories and
$\pi_{\mathscr{X}}\,:\,\mathscr{X}\,\longrightarrow\,\mathcal{C}$ be a
functor. An arrow $\theta\,\colon\,\xi\,\longrightarrow\,\eta$ in
$\mathscr{X}$ is said to be a Cartesian arrow in $\mathscr{X}$ if for every
morphism $\theta^{\prime}\,\colon\,\xi^{\prime}\,\longrightarrow\,\eta$ in
$\mathscr{X}$, and a morphism
$h\,\colon\,\pi_{\mathscr{X}}(\xi^{\prime})\,\longrightarrow\,\pi_{\mathscr{X}}(\xi)$
in $\mathcal{C}$ with $\pi(\theta)\circ h\,=\,\pi(\theta^{\prime})$, there
exists a unique morphism $\Phi\,\colon\,\xi^{\prime}\,\longrightarrow\,\xi$ in
$\mathscr{X}$ such that $\theta\circ\Phi\,=\,\theta^{\prime}$ in $\mathscr{X}$
and $\pi_{\mathscr{X}}(\Phi)\,=\,h$ in $\mathcal{C}$. We visually represent a
Cartesian arrow by the following diagram:

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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ }}{ } {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{$\scriptstyle{\pi(\theta)}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>(2.1)

######  Definition 2.18 (Fibered category).

A functor $\pi_{\mathscr{X}}\,:\,\mathscr{X}\,\longrightarrow\,\mathcal{C}$ is
a fibered category over $\mathcal{C}$ if for every
$(\eta,f)\,\in\,\mathscr{X}_{0}\times_{\mathcal{C}_{0},t}\mathcal{C}_{1}$
there exists a Cartesian arrow $\theta\,\colon\,\xi\,\longrightarrow\,\eta$
with $\pi_{\mathscr{X}}(\theta)\,=\,f$. We call such $\xi$ to be a pullback of
$\eta$ along $f$.

A morphism of fibered categories over $\mathcal{C}$ from
$(\mathscr{X},\,\pi_{\mathscr{X}},\,\mathcal{C})$ to
$(\mathscr{Y},\,\pi_{\mathscr{Y}},\,\mathcal{C})$ is a functor
$F\,\colon\,\mathscr{X}\,\longrightarrow\,\mathscr{Y}$ that maps a Cartesian
arrow in $\mathscr{X}$ to a Cartesian arrow in $\mathscr{Y}$ satisfying
$\pi_{\mathscr{Y}}\circ F\,=\,\pi_{\mathscr{X}}$.

######  Example 2.19.

Given a Lie groupoid $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$, consider
the category ${\mathscr{B}}\mathbb{X}$ of principal $\mathbb{X}$-bundles. The
functor
$\pi_{\mathscr{B}\mathbb{X}}\,:\,\mathscr{B}\mathbb{X}\,\longrightarrow\,\mathfrak{S}$
that sends a principal $\mathbb{X}$-bundle $P\,\longrightarrow\,M$ to the
smooth manifold $M$ is a fibered category over $\mathfrak{S}$.

######  Example 2.20.

Let $M$ be a smooth manifold and ${\underline{M}}$ the comma category. The
functor ${\underline{M}}\,\longrightarrow\,\mathfrak{S}$ which sends an object
$N\,\longrightarrow\,M$ in ${\underline{M}}$ to the object $N$ in
$\mathfrak{S}$ is a fibered category over $\mathfrak{S}$.

Let $\pi_{\mathscr{X}}\,:\,\mathscr{X}\,\longrightarrow\,\mathcal{C}$ be a
fibered category. To an object $U$ of $\mathcal{C}$ we associate a category
$\mathscr{X}(U)$ with

$\displaystyle\text{Obj}(\mathscr{X}(U))\,=\,\\{\eta\in\text{Obj}(\mathscr{X})\,\mid\,\pi_{\mathscr{X}}(\eta)=U\\},$
$\displaystyle\text{Mor}_{\mathscr{X}(U)}(\eta,\,\eta^{\prime})\,=\,\\{f\in\text{Mor}(\mathscr{X})\,\mid\,\pi_{\mathscr{X}}(f)\,=\,1_{U}\\}\,.$

The category $\mathscr{X}(U)$ is called the fiber of $U$. In this fashion we
obtain a pseudo-functor associated to
$\pi_{\mathscr{X}}\,:\,\mathscr{X}\,\longrightarrow\,\mathcal{C}$,

$\mathscr{X}\,:\,\mathcal{C}^{\rm op}\,\longrightarrow\,\rm{Cat}$

that sends $U$ to the category $\mathscr{X}(U)$. It should be clarified that
by abuse of notation this pseudo-functor will be also denoted by
$\mathscr{X}.$

Let $\mathcal{C}$ be a category with a specified Grothendieck topology. We
refer to such a category as a site (compare [Vi]). In particular, by the site
${\mathfrak{S}}$ we mean the big étale site $\mathfrak{S}_{\text{\'{e}t}}$ as
defined before.

Let $\pi_{\mathscr{X}}\,:\,\mathscr{X}\,\longrightarrow\,\mathcal{C}$ be a
fibered category over a site $\mathcal{C}$. For the ease of exposition we will
also simplify our notation here: $U_{ij}$ will denote the fiber product
$U_{i}\times_{U}U_{j}$ in $\mathcal{C}$ and so on, while ${\rm pr}_{i},\,{\rm
pr}_{ij}$ etc. will denote various projection maps obtained from the pullback
diagrams; for an arrow $f\,:\,U\,\longrightarrow\,V$ in $\mathcal{C}$ the
functor
${\mathscr{X}}(f)\,:\,\mathscr{X}(V)\,\longrightarrow\,{\mathscr{X}}(U)$ will
be denoted by $f^{*}$.

Let $(\mathscr{X},\,\pi_{\mathscr{X}},\,\mathcal{C})$ be a fibered category
over a site ${\mathcal{C}}$, and let
$\\{\sigma_{i}\,\colon\,U_{i}\,\longrightarrow\,U\\}$ be a covering of an
object $U$ in $\mathcal{C}$. To the collection
$\\{U_{i}\,\longrightarrow\,U\\}$ we associate the following descent category
$\mathscr{X}(\\{U_{i}\rightarrow U\\})$. An object in
$\mathscr{X}(\\{U_{i}\rightarrow U\\})$ is a family of pairs
$(\\{\xi_{i}\\},\,\\{\phi_{ij}\\})$, where each $\xi_{i}$ is an object of
$\mathscr{X}(U_{i})$ and each
$\phi_{ij}\,\colon\,\rm{pr}_{2}^{*}(\xi_{j})\,\longrightarrow\,{\rm
pr}_{1}^{*}(\xi_{i})$ is an isomorphism in $\mathscr{X}(U_{ij})$ satisfying
the cocycle relation ${\rm pr}_{13}^{*}(\phi_{ik})\,=\,{\rm
pr}_{12}^{*}(\phi_{ij})\circ{\rm pr}_{23}^{*}(\phi_{jk})$ in
$\mathscr{X}(U_{ijk})$. An arrow

$\big{(}\\{\xi_{i}\\},\,\\{\phi_{ij}\\})\,\longrightarrow\,(\\{\eta_{i}\\},\,\\{\psi_{ij}\\}\big{)}$

is a collection $\\{\alpha_{i}\,\colon\,\xi_{i}\,\longrightarrow\,\eta_{i}\\}$
satisfying $\psi_{ij}\circ{\rm pr}_{2}^{*}\alpha_{j}\,=\,{\rm
pr}_{1}^{*}\alpha_{i}\circ\phi_{ij}$ for every pair of indices $i,\,j$. Here,
for any morphism $\sigma_{i}:U_{i}\rightarrow U$ in $\mathcal{C}$, we have the
associated functor
$\sigma_{i}^{*}:\mathscr{X}(U)\rightarrow\mathscr{X}(U_{i})$. So, for the
object $\xi$ in $\mathscr{X}(U)$, we have the object $\sigma_{i}^{*}(\xi)$ in
$\mathscr{X}(U_{i})$. Moreover we get a functor

$\mathscr{X}(U)\,\longrightarrow\,\mathscr{X}(\\{U_{i}\rightarrow U\\})$ (2.2)

that sends an object $\xi$ in $\mathscr{X}(U)$ to the object
$\\{\sigma_{i}^{*}(\xi),\,\phi_{ij}\\}$, where

$\phi_{ij}\,\colon\,{\rm
pr}_{2}^{*}(\sigma_{j}^{*}(\xi))\,\longrightarrow\,{\rm
pr}_{1}^{*}(\sigma_{i}^{*}(\xi))$

are the isomorphisms given by the universal property of a pullback.

######  Definition 2.21 (Stack).

Let $\mathcal{C}$ be a site. A fibered category
$(\mathscr{X},\,\pi_{\mathscr{X}},\,\mathcal{C})$ is said to be a stack (over
the site $\mathcal{C})$ if for each object $U$ of $\mathcal{C}$, and a
covering $\\{U_{\alpha}\,\longrightarrow\,U\\}$ of $U$, the functor defined in
(2.2) is an equivalence of categories. A morphism of stacks is a morphism of
the underlying fibered categories.

We are mainly interested here in categories fibered in groupoids; that is, a
fibered category $\mathscr{X}\,\longrightarrow\,\mathcal{C}$ such that each
fiber $\mathscr{X}(U)$ is a groupoid. Similarly, stack will mean here that the
underlying fibered category is a category fibered in groupoids.

The following standard and useful examples will be referred to time and again.

######  Example 2.22.

The fibered category ${\underline{M}}\,\longrightarrow\,\mathfrak{S}$
associated to the manifold $M$ in Example 2.20 is a stack over the site
$\mathfrak{S}$ with respect to the étale topology. Here and onwards, we denote
this stack simply by $M$.

######  Example 2.23.

Let $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ be a Lie groupoid. Then
the fibered category $\mathscr{B}\mathbb{X}\,\longrightarrow\,\mathfrak{S}$ in
Example 2.19 is a stack over the site $\mathfrak{S}$ with respect to the étale
topology. This is also known as the classifying stack associated to the Lie
groupoid $\mathbb{X}$.

Now we can introduce the main class of stacks, which are important for
geometry and analysis, generalizing smooth manifolds.

######  Definition 2.24 (Differentiable stack).

Let $\mathfrak{S}$ be the big étale site. A stack
$\mathscr{X}\,\longrightarrow\,\mathfrak{S}$ is called a differentiable stack
if there exists a Lie groupoid $\mathbb{X}$ such that there is an isomorphism
of stacks $\mathscr{B}\mathbb{X}\,\cong\,\mathscr{X}$.

The classifying stack $\mathscr{B}\mathbb{X}\,\longrightarrow\,\mathfrak{S}$
of a Lie groupoid $\mathbb{X}$ and the stack ${M}\rightarrow\mathfrak{S}$
associated to a smooth manifold $M$ are both examples of differentiable stacks
over the site $\mathfrak{S}_{\text{\'{e}t}}$. It is a known fact that a pair
of Morita equivalent Lie groupoids yields a pair of isomorphic stacks (see
[BX], [BCKN1]).

A useful equivalent definition of a differentiable stack is given by using the
notion of a ‘presentation’ or ‘atlas’ (see [BX], [He] and [BCKN1]).

######  Definition 2.25 (Representable surjective submersion).

A morphism of stacks $F\,:\,\mathscr{X}\,\longrightarrow\,{\mathscr{Y}}$ over
$\mathfrak{S}$ is called a representable surjective submersion if for every
smooth manifold $U$ and every morphism of stacks
$U\,\longrightarrow\,{\mathscr{Y}}$, the following two conditions hold:

  1. (1)

the $2$-fiber product of stacks $\mathscr{X}\times_{\mathscr{Y}}U$ is
representable, i.e., isomorphic to a smooth manifold, and

  2. (2)

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

A stack $\mathscr{X}$ over $\mathfrak{S}$ is differentiable if there exists an
object $X$ of $\mathfrak{S}$ together with a representable surjective
submersion $x\,:\,X\,\longrightarrow\,{\mathscr{X}}$ (see [BX], [He]). The map
of stacks $x\,:\,X\,\longrightarrow\,{\mathscr{X}}$ is called an atlas or
presentation for $\mathscr{X}$.

######  Example 2.26.

Let $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ be a Lie groupoid. Let
$t\,\colon\,X_{1}\,\longrightarrow\,X_{0}$ be the associated principal
$\mathbb{X}$-bundle, as in Example 2.15. Then we have a functor $x_{0}\colon
X_{0}\,\longrightarrow\,{\mathscr{B}}\mathbb{X}$ which sends an object
$f\colon N\,\longrightarrow\,X_{0}$ to the pullback principal
$\mathbb{X}$-bundle $f^{*}X_{1}\,\longrightarrow\,N$. This functor in fact
defines an atlas.

Conversely given an atlas $x\,:\,X\,\longrightarrow\,{\mathscr{X}}$ for the
differentiable stack $\mathscr{X},$ we can produce a Lie groupoid
$\mathbb{X}\,=\,[X\times_{\mathscr{X}}X\rightrightarrows X].$ Then we have an
isomorphism $\mathscr{B}\mathbb{X}\,\cong\,\mathscr{X}.$ If
$y\,:\,Y\,\longrightarrow\,{\mathscr{X}}$ is any other atlas for
$\mathscr{X}$, then $\mathbb{X}$ and
$\mathbb{Y}\,=\,[Y\times_{\mathscr{X}}Y\rightrightarrows Y]$ are Morita
equivalent Lie groupoids.

######  Definition 2.27 (Deligne-Mumford stack).

A differentiable stack $\mathscr{X}\,\longrightarrow\,\mathfrak{S}$ will be
called a (proper) Deligne-Mumford stack if there exists a (proper) étale
presentation $x\,:\,X\,\longrightarrow\,{\mathscr{X}}.$

######  Example 2.28.

Proper Deligne-Mumford stacks correspond to smooth orbifolds and are presented
by proper étale Lie groupoids (see [Le]).

Note that by the second part of Example 2.26 for any Deligne-Mumford stack
$\mathscr{X}$ there exists an étale Lie groupoid
$\mathbb{X}\,=\,[X\times_{\mathcal{D}}X\rightrightarrows X]$ such that
$\mathscr{B}\mathbb{X}\,\cong\,\mathscr{X}.$ However the property of being
étale is not preserved under Morita equivalence. Thus there may exist a non-
étale Lie groupoid presenting the same Deligne-Mumford stack.

##  3\. de Rham cohomology for Lie groupoids and differentiable stacks

In this section, we study connections and de Rham cohomology for Lie groupoids
and Deligne-Mumford stacks. The existence of an integrable connection will
hereby play an important role.

###  3.1. Connections and integrable connections on Lie groupoids

In this section, we will define the notion of an (integrable) connection
$\mathcal{H}$ on a Lie groupoid $\mathbb{X}\,=\,[X_{1}\rightrightarrows
X_{0}]$ and introduce the de Rham cohomology ring of the pair
$(\mathbb{X},\,\mathcal{H})$. We begin by recalling the Atiyah sequence in the
classical set-up for smooth manifolds.

Let $M$ be a smooth manifold, $G$ a Lie group and
$\pi\,\colon\,P\,\longrightarrow\,M$ a principal $G$-bundle. The Lie algebra
of $G$ will be denoted by $\mathfrak{g}$. Consider the diagonal action of $G$
on $P\times{\mathfrak{g}}$ induced by the adjoint action of $G$ on
$\mathfrak{g}$. Then the quotient

$\text{ad}(P)\,:=\,(P\times\mathfrak{g})/G\,\longrightarrow\,P/G\,=\,M$

becomes a vector bundle over $M$; it is known as the adjoint bundle of $P$. On
the other hand, the action of $G$ on $P$ induces an action of $G$ on the
tangent bundle $TP$ of $P$. The quotient

$\text{At}(P)\,:=\,(TP)/G\,\longrightarrow\,P/G\,=\,M$

is a vector bundle, which is known as the Atiyah bundle (see [At]). The action
of $G$ on $P$ identifies the trivial bundle
$P\times\mathfrak{g}\,\longrightarrow\,P$ with the relative tangent bundle for
the projection $\pi$, and hence $\text{ad}(P)$ is a subbundle of
$\text{At}(P)$. Consequently, we obtain a short exact sequence of vector
bundles on $M$

$0\,\longrightarrow\,\text{ad}(P)\,\stackrel{{\scriptstyle
j^{P}}}{{\longrightarrow}}\,\text{At}(P)\,\stackrel{{\scriptstyle
d\pi}}{{\longrightarrow}}\,TM\,\longrightarrow\,0$ (3.1)

which is called the Atiyah sequence associated to the principal $G$-bundle $P$
(see [At]). Note that the projection $d\pi$ in (3.1) gives the quotient by $G$
of the differential $d\pi\,:\,TP\,\longrightarrow\,\pi^{*}TM$ of the map
$\pi$.

A connection on a principal $G$-bundle $P$ is then defined to be a splitting
of the Atiyah sequence. In other words, a connection on $P$ is a homomorphism

$D\,:\,TM\,\longrightarrow\,\text{At}(P)$

such that $(d\pi)\circ D\,=\,\text{Id}_{TM}$, where $d\pi$ is the homomorphism
in (3.1).

Now consider a principal $G$-bundle
$(E_{G}\,\rightarrow\,X_{0},\,\,[X_{1}\,\rightrightarrows\,X_{0}])$ over a Lie
groupoid $\mathbb{X}=[X_{1}\,\rightrightarrows\,X_{0}]$. The underlying
(classical) principal $G$-bundle $E_{G}\,\longrightarrow\,X_{0}$ yields the
Atiyah sequence of vector bundles over $X_{0}$,

$0\,\longrightarrow\,\text{ad}(E_{G})\,\xrightarrow{j^{P}}\,\text{At}(E_{G})\,\xrightarrow{d\pi}\,TX_{0}\,\longrightarrow\,0$
(3.2)

(see (3.1)). Our aim is to interpret (3.2) as an exact sequence of vector
bundles over the Lie groupoid $\mathbb{X}=[X_{1}\rightrightarrows X_{0}]$. For
that we need an action of the Lie groupoid $\mathbb{X}$ on the tangent space
manifold $TX_{0}$. We already have the tangent bundle projection map
$TX_{0}\,\longrightarrow\,X_{0}$. It remains to provide the smooth map

$\mu\,\colon\,X_{1}\times_{X_{0}}TX_{0}\,\longrightarrow\,TX_{0}$ (3.3)

satisfying the conditions in Definition 2.4. As
$TX_{0}\,\longrightarrow\,X_{0}$ is surjective, (by condition (2) in
Definition 2.4) the map $\mu$ in (3.3) should produce a linear map
$T_{s(\gamma)}X_{0}\,\longrightarrow\,T_{t(\gamma)}X_{0}$ for every
$\gamma\,\in\,X_{1}$. Precisely for this purpose we introduce below the notion
of a connection on the Lie groupoid $\mathbb{X}$.

Let $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ be a Lie groupoid. Let
$\mathcal{H}\,\subset\,TX_{1}$ be a distribution on the manifold $X_{1}$ which
is a complement of the kernel of the differential
$ds\,\colon\,TX_{1}\,\longrightarrow\,TX_{0}$ of the source map $s$; so

$\mathcal{H}_{\gamma}\oplus\ker(ds)_{\gamma}\,=\,T_{\gamma}X_{1}$ (3.4)

for every $\gamma\,\in\,X_{1}$. Let

$P_{\mathcal{H}_{\gamma}}\,:\,T_{\gamma}X_{1}\,\longrightarrow\,\mathcal{H}_{\gamma}\,\subset\,T_{\gamma}X_{1}$
(3.5)

be the projection constructed using the decomposition in (3.4). For any
$v\,\in\,T_{\gamma}X_{1}$,

$P_{\mathcal{H}_{\gamma}}(v)\,\in\,\mathcal{H}_{\gamma}$

is called the horizontal component of $v$.

Since $s$ is a submersion, the decomposition in (3.4) yields an isomorphism
$\mathcal{H}_{\gamma}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,T_{s(\gamma)}X_{0}$.
In fact, we have an isomorphism of vector bundles

$\mathcal{H}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,s^{*}TX_{0}$

over $X_{1}$. The following compositions of homomorphisms

$s^{*}TX_{0}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,\mathcal{H}\,\hookrightarrow\,TX_{1}\,\stackrel{{\scriptstyle
dt}}{{\longrightarrow}}\,t^{*}TX_{0}$ (3.6)

over $X_{1}$ will be denoted by $\theta$, where
$dt\,:\,TX_{1}\,\longrightarrow\,t^{*}TX_{0}$ is the differential of the map
$t$.

We now recall the definition of a connection on Lie groupoids from [BN, Def.
3.1] (see also [BCKN1, 4.1.])

######  Definition 3.1 (Connection on a Lie groupoid).

A connection $\mathcal{H}$ on a Lie groupoid
$\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ is a distribution
$\mathcal{H}\,\subset\,TX_{1}$ satisfying the following three conditions:

  1. (1)

$\mathcal{H}$ is a complement of kernel of $ds$,

  2. (2)

$(de)_{x}(T_{x}X_{0})\,=\,\mathcal{H}_{e(x)}X_{1}$ for all $x\,\in\,X_{0}$,
and

  3. (3)

$\theta_{\gamma}\circ\theta_{\gamma^{\prime}}\,=\theta_{\gamma\circ\gamma^{\prime}}$
for every composable pair $\gamma,\,\gamma^{\prime}\,\in\,X_{1}$, where
$\theta$ is the homomorphism in (3.6).

A connection $\mathcal{H}$ on $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$
is said to be integrable (or flat) if the distribution
$\mathcal{H}\,\subset\,TX_{1}$ is integrable.

######  Remark 3.2.

Behrend [Be2] defines a connection on a Lie groupoid
$\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ as a subbundle
$E\,\subset\,TX_{1}$ such that $[E\rightrightarrows TX_{0}]$ is a subgroupoid
of the tangent groupoid $[TX_{1}\rightrightarrows TX_{0}]$, which is
integrable (or flat) if $E\subset TX_{1}$ is an integrable distribution. Tang
[Ta] called these integrable connections also étalifications and studies them
in the context of deformation quantization. These distributions $E\,\ \subset
TX_{1}$ are also referred to in the literature as multiplicative distributions
[DE].

Let $\mathcal{H}\,\subset\,TX_{1}$ be a connection on the Lie groupoid
$\mathbb{X}$. Consider a principal $G$-bundle $(E_{G}\rightarrow X_{0},\
\mathbb{X})$ over $\mathbb{X}$. Then

${{\rm
pr}_{1}}\,:\,s^{*}E_{G}\,=\,X_{1}\times_{s}E_{G}\,\longrightarrow\,X_{1}$

is a principal $G$-bundle over $X_{1}$. We get an associated Lie groupoid
$[s^{*}E_{G}\rightrightarrows E_{G}]$ whose source map
$s^{*}E_{G}\,\longrightarrow\,E_{G}$ is the second projection
$(a,\,\gamma)\,\longmapsto\,a$ and the target map
$s^{*}E_{G}\,\longrightarrow\,E_{G}$ is the action
$(a,\,\gamma)\,\longmapsto\,a\cdot\gamma$ (the map $\mu$ in Definition 2.8).
Let

$d{\rm pr}_{1}\,\colon\,Ts^{*}E_{G}\,\longrightarrow\,TX_{1}$ (3.7)

be the differential of the projection ${\rm
pr}_{1}\,:\,s^{*}E_{G}\,\longrightarrow\,X_{1}$,
$(a,\,\gamma)\,\longmapsto\,\gamma$.

The following lemma is then evident.

######  Lemma 3.3.

Let $\mathcal{H}\,\subset\,TX_{1}$ be a connection on the Lie groupoid
$\mathbb{X}=[X_{1}\rightrightarrows X_{0}]$. Then

$\widetilde{\mathcal{H}}\,:=\,({d\rm
pr}_{1})^{-1}(\mathcal{H})\,\subset\,Ts^{*}E_{G}$

is a connection on the Lie groupoid $[s^{*}E_{G}\rightrightarrows E_{G}]$,
where ${d\rm pr}_{1}$ is the map in (3.7). If the connection $\mathcal{H}$ is
integrable, then so is $\widetilde{\mathcal{H}}$.

The connection $\widetilde{\mathcal{H}}$ in Lemma 3.3 will be called the
pullback of $\mathcal{H}$. For the purpose of future reference we define here
the map $\widetilde{\theta}$ (see (3.6)) to be given by the following
compositions of homomorphisms

$\begin{split}\widetilde{\theta}:\,\,&{{\rm
pr}_{2}}^{*}TE_{G}\,\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\,\widetilde{\mathcal{H}}\,\hookrightarrow\,Ts^{*}E_{G}\,\stackrel{{\scriptstyle
d\,\mu}}{{\longrightarrow}}\,{\mu}^{*}TE_{G},\\\ &\qquad
w\,\longmapsto\,{\widetilde{\theta}}_{\gamma,a}(w)\,=\,{d\mu}_{(\gamma,a)}(P_{\mathcal{H}_{\gamma}}(v),\,w)\end{split}$
(3.8)

for any $v\in T_{\gamma}X_{1},\,w\,\in\,T_{a}E_{G}$ satisfying
$ds(\gamma,\,v)\,=\,d\pi(a,\,w)$ (in other words, for any
$(v,\,w)\,\in\,T_{(\gamma,a)}s^{*}E_{G}$). It is immediate that the map
$\widetilde{\theta}$ is well defined since $P_{\mathcal{H}_{\gamma}}$ is
actually trivial on ${\ker}(ds).$

######  Example 3.4.

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

######  Example 3.5.

Let $M$ be a smooth manifold equipped with an action of a Lie group $G$. Let
$\mathbb{X}=[M\times G\rightrightarrows M]$ be the associated action Lie
groupoid. As the source map $s\,\colon\,M\times G\,\longrightarrow\,M$ is the
projection map, the differential $(ds)_{(m,g)}\,\colon\,T_{(m,g)}(M\times
G)\,\longrightarrow\,T_{m}M$ is given by $(v,\,A)\,\longmapsto\,v$ for all
$v\,\in\,T_{m}M,\ A\,\in\,T_{g}G$. Thus,

$\ker((ds)_{(m,g)})\,=\,\\{(v,\,A)\,\in\,T_{m}M\times
T_{g}G\,\mid\,v\,=\,0\\}\,=\,T_{g}G$

for each $(m,\,g)\,\in\,M\times G$. Now defining $\mathcal{H}_{(m,g)}(M\times
G)\,=\,T_{m}M\,\subset\,T_{(m,g)}(M\times G)$, we obtain a connection
$\mathcal{H}_{\mathbb{X}}\,\subset\,T(M\times G)$ on the action groupoid
$\mathbb{X}$. This connection is clearly integrable.

######  Example 3.6.

Let $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ be an étale Lie groupoid,
meaning the map $s\,\colon\,X_{1}\,\longrightarrow\,X_{0}$ is a local
diffeomorphism. Then the differential
$ds\,\colon\,TX_{1}\,\longrightarrow\,s^{*}TX_{0}$ is an isomorphism and the
distribution $\mathcal{H}_{\mathbb{X}}=\,TX_{1}$ is a connection on the Lie
groupoid $\mathbb{X}$. This connection is again integrable.

######  Example 3.7.

Let $\pi\,\colon\,E\,\longrightarrow\,M$ be a finite rank vector bundle over
the manifold $M.$ We can view $\mathbb{X}=[E\rightrightarrows M]$ as a
groupoid with both the source and target maps both being $\pi$ (thus, a pair
of composable morphisms will belong to the same fiber), and composition is
simply the addition of vectors on the corresponding fiber. Then a connection
on the vector bundle $\pi\,\colon\,E\,\longrightarrow\,M$ defines a connection
$\mathcal{H}_{\mathbb{X}}$ on $\mathbb{X}$ by smoothly splitting $E$ into the
horizontal component complementing $\ker{d\pi}.$ Obviously, if one connection
is integrable, then so is the other.

######  Example 3.8.

Let $\pi\,\colon\,P\,\longrightarrow\,M$ be a principal $G$-bundle over a
manifold $M$ and $\mathbb{P}_{\rm Gauge}:=\,[\frac{P\times
P}{G}\rightrightarrows M\,]$ the gauge groupoid of Example 2.12. Let $A$ be a
connection on the principal bundle $\pi\,\colon\,P\,\longrightarrow\,M$. The
horizontal distribution $\mathcal{H}\subset TP$ defined by the connection $A$
induces a connection $\mathcal{H}_{\mathbb{P}_{\rm Gauge}}$ on the Lie
groupoid $\mathbb{P}_{\rm Gauge}$ as follows. Define

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

where $(p,\,q)\cdot G\,\subset\,P\times P$ is the orbit of the element
$(p,\,q)\,\in\,P\times P.$ Now since $\mathcal{H}$ complements $\ker{d\pi}$
and it is $G$-invariant, ${\mathcal{H}}_{[p,q]}$ defines a connection
$\mathcal{H}_{\mathbb{P}_{\rm Gauge}}$ on $\mathbb{P}_{\rm Gauge}.$ If the
connection $A$ is integrable, then so is the induced connection
$\mathcal{H}_{\mathbb{P}_{\rm Gauge}}$ on the gauge groupoid $\mathbb{P}_{\rm
Gauge}.$

We refer to [Be2, Section 2.3] and [Ta, Section 2.1.1] for other interesting
examples of connections on Lie groupoids.

###  3.2. Differential forms and de Rham cohomology of a pair
$(\mathbb{X},\mathcal{H})$

Let us fix a pair $(\mathbb{X},\mathcal{H})$ consisting of a Lie groupoid
$\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ together with an existing
connection $\mathcal{H}\,\subset\,TX_{1}$ on $\mathbb{X}$. We recall from [BN]
the notion of a differential $k$-form on $\mathbb{X}$.

######  Definition 3.9 (Horizontal component of a differential form).

Let $\varphi\,:\,X_{1}\,\longrightarrow\,\bigwedge^{k}T^{*}X_{1}$ be a
differential $k$-form on the manifold $X_{1}$. The horizontal component of
$\varphi$ (with respect to the connection $\mathcal{H}$) is the differential
$k$-form

$\widehat{\mathcal{H}}(\varphi)(\gamma)(v_{1},\,\cdots,\,v_{k})\,=\,\varphi(\gamma)(P_{\mathcal{H}_{\gamma}}(v_{1}),\,\cdots,\,P_{\mathcal{H}_{\gamma}}(v_{k})),$

for all $\gamma\,\in\,X_{1}$ and $v_{i}\,\in\,T_{\gamma}X_{1}$,
$1\,\leq\,i\,\leq\,k$, where $P_{\mathcal{H}_{\gamma}}$ is the projection in
(3.5).

######  Definition 3.10 (Differential forms on a Lie groupoid).

A differential $k$-form
$\varphi\,\colon\,X_{0}\,\longrightarrow\,\bigwedge^{k}T^{*}X_{0}$ on the
manifold $X_{0}$ is said to be a differential $k$-form on $\mathbb{X}$ (with
respect to the connection $\mathcal{H}$) if

$\widehat{\mathcal{H}}(s^{*}\varphi)\,=\,\widehat{\mathcal{H}}(t^{*}\varphi)\,,$

where $\widehat{\mathcal{H}}$ is constructed as in Definition 3.9. We will
denote the vector space of differential $k$-forms of the pair
$(\mathbb{X},\,\mathcal{H})$ by $\Omega^{k}(\mathbb{X},\,\mathcal{H})$.

The following lemma, which was stated in [BN] without proof, shows that
differential forms on a Lie groupoid with connection are closed with respect
to exterior derivation under the assumption that the connection on the Lie
groupoid is integrable.

######  Lemma 3.11.

Let $\mathcal{H}\,\subset\,TX_{1}$ be an integrable connection on a Lie
groupoid $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$. Let
$\varphi\,:\,X_{0}\,\longrightarrow\,\bigwedge^{k}T^{*}X_{0}$ be a
differential $k$-form on $\mathbb{X}$. Then
$d\varphi\,:\,X_{0}\,\longrightarrow\,\bigwedge^{k+1}T^{*}X_{0}$ is a
differential $(k+1)$-form on $\mathbb{X}$.

###### Proof.

It is enough to prove the case $k\,=\,1$. Indeed, for an arbitrary $k$, the
proof runs almost verbatim.

Let $\varphi$ be a differential $1$-form on
$\mathbb{X}=[X_{1}\rightrightarrows X_{0}]$. This means that we have

$(s^{*}\varphi)(\gamma)(P_{\mathcal{H}_{\gamma}}(v))\,=\,(t^{*}\varphi)(\gamma)(P_{\mathcal{H}_{\gamma}}(v))$

for all $\gamma\,\in\,X_{1}$ and $v\,\in\,T_{\gamma}X_{1}$. Hence we get

$(s^{*}\varphi)(\gamma)(v)\,=\,(t^{*}\varphi)(\gamma)(v)$ (3.9)

for all $\gamma\,\in\,X_{1}$ and $v\,\in\,\mathcal{H}_{\gamma}$.

To prove the lemma we need to show that

$d(s^{*}\varphi)(\gamma)(P_{\mathcal{H}_{\gamma}}(v_{1}),\,P_{\mathcal{H}_{\gamma}}(v_{2}))\,=\,d(t^{*}\varphi)(\gamma)(P_{\mathcal{H}_{\gamma}}(v_{1}),\,P_{\mathcal{H}_{\gamma}}(v_{2}))$
(3.10)

for all $\gamma\,\in\,X_{1}$ and $v_{1},\,v_{2}\,\in\,T_{\gamma}X_{1}$.

Let $v_{1},\,v_{2}\,\in\,\mathcal{H}_{\gamma}$. Let $Z_{1}$ (respectively,
$Z_{2}$) be a section of $\mathcal{H}$ defined around $\gamma$ such that
$Z_{1}(\gamma)\,=\,v_{1}$ (respectively, $Z_{2}(\gamma)\,=\,v_{2}$). We have

$\displaystyle
d(s^{*}\varphi)(\gamma)(Z_{1},\,Z_{2})\,=\,Z_{1}((s^{*}\varphi)(Z_{2}))-Z_{2}((s^{*}\varphi)(Z_{1}))-(s^{*}\varphi)([Z_{1},\,Z_{2}])$
$\displaystyle
d(t^{*}\varphi)(\gamma)(Z_{1},\,Z_{2})\,=\,Z_{1}((t^{*}\varphi)(Z_{2}))-Z_{2}((t^{*}\varphi)(Z_{1}))-(t^{*}\varphi)([Z_{1},\,Z_{2}])$

Now, since $[Z_{1},\,Z_{2}]$ is a section of $\mathcal{H}$, using this and
(3.9) we conclude that (3.10) holds. ∎

Now let us introduce morphisms of Lie groupoids with connections.

######  Definition 3.12.

Let $\bigl{(}\mathbb{X}\,=\,[X_{1}\rightrightarrows
X_{0}],\,\mathcal{H}_{\mathbb{X}}\bigr{)}$ and
$\bigl{(}\mathbb{Y}\,=\,[Y_{1}\rightrightarrows
Y_{0}],\,\mathcal{H}_{\mathbb{Y}}\bigr{)}$ be a pair of Lie groupoids equipped
with connections. A morphism of Lie groupoids
$(F,\,f)\,\colon\,[X_{1}\rightrightarrows
X_{0}]\,\longrightarrow\,[Y_{1}\rightrightarrows Y_{0}]$ will be called a
morphism of Lie groupoids with connections if

$dF(\mathcal{H}_{\mathbb{X}})\,=\,\mathcal{H}_{\mathbb{Y}}\,,$

where $dF$ is the differential of the map $F$. We shall employ the notation

$(F,\,f)\,\colon\,\bigl{(}[X_{1}\rightrightarrows
X_{0}],\,\mathcal{H}_{\mathbb{X}}\bigr{)}\,\longrightarrow\,\bigl{(}[Y_{1}\rightrightarrows
Y_{0}],\,\mathcal{H}_{\mathbb{Y}}\bigr{)}$

for such a morphism.

The following lemma is a straight-forward consequence of the definition.

######  Lemma 3.13.

Let $(F,\,f)\,\colon\,\bigl{(}\mathbb{X}=[X_{1}\rightrightarrows
X_{0}],\,\mathcal{H}_{\mathbb{X}}\bigr{)}\,\longrightarrow\,\bigl{(}\mathbb{Y}=[Y_{1}\rightrightarrows
Y_{0}],\,\mathcal{H}_{\mathbb{Y}}\bigr{)}$ be a morphism of Lie groupoids with
connections. Let $\varphi\,:\,Y_{0}\,\longrightarrow\,\bigwedge^{k}T^{*}Y_{0}$
be a differential form on the Lie groupoid $\mathbb{Y}$. Then
$f^{*}\varphi\,:\,X_{0}\,\longrightarrow\,\bigwedge^{k}T^{*}X_{0}$ is a
differential form on the Lie groupoid $\mathbb{X}$.

When the morphism $F$ in Lemma 3.13 is a surjective submersion, the following
converse also holds.

######  Lemma 3.14.

Let $(F,\,f)\,\colon\,\bigl{(}\mathbb{X}=[X_{1}\rightrightarrows
X_{0}],\,\mathcal{H}_{\mathbb{X}}\bigr{)}\,\longrightarrow\,\bigl{(}\mathbb{Y}=[Y_{1}\rightrightarrows
Y_{0}],\,\mathcal{H}_{\mathbb{Y}}\bigr{)}$ be a morphism of Lie groupoids with
connection such that $F$ is a surjective submersion. Let $\varphi$ be a
differential $k$-form on $Y_{0}$ such that $f^{*}\varphi$ is a differential
$k$-form on the Lie groupoid $\mathbb{X}$. Then $\varphi$ is a differential
form on the Lie groupoid $\mathbb{Y}$.

###### Proof.

Since $F$ is a surjective submersion, for any two differential $k$-forms
$\omega_{1}$ and $\omega_{2}$ on $Y_{1}$, if
$F^{*}\omega_{1}\,=\,F^{*}\omega_{2}$, then $\omega_{1}\,=\,\omega_{2}$. The
lemma now follows from this. ∎

Let $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ now be a Lie groupoid
equipped with an integrable connection $\mathcal{H}\,\subset\,TX_{1}$. Let
${\Omega}^{k}(\mathbb{X},\mathcal{H})$ denote the vector space of differential
$k$-forms on the Lie groupoid $\mathbb{X}$, with respect to $\mathcal{H}$. In
view of Lemma 3.11 we have a cochain complex
$\Omega^{\bullet}(\mathbb{X},\,\mathcal{H})$ given by

$\cdots\,\longrightarrow\,{\Omega}^{k-1}(\mathbb{X},\,\mathcal{H})\,\longrightarrow\,{\Omega}^{k}(\mathbb{X},\,\mathcal{H})\,\longrightarrow\,{\Omega}^{k+1}(\mathbb{X},\,\mathcal{H})\,\longrightarrow\,\cdots.$

The $k^{th}$-cohomology group of this cochain complex, denoted by
$H^{k}_{dR}(\mathbb{X},\mathcal{H})$, is called the $k^{th}$-de Rham
cohomology of the pair $(\mathbb{X},\,\mathcal{H})$ and we have the de Rham
cohomology ring of the pair $(\mathbb{X},\,\mathcal{H})$ by setting

$H_{\rm dR}^{*}(\mathbb{X},\,\mathcal{H})\,:=\,\bigoplus_{k=0}^{\dim
X_{0}}H^{k}_{\rm dR}(\mathbb{X},\,\mathcal{H})\,.$ (3.11)

Also, there is a natural homomorphism $H_{\rm
dR}^{*}(\mathbb{X},\,\mathcal{H})\,\longrightarrow\,H_{\rm
dR}^{*}(X_{0},\,{\mathbb{R}})$ which is, in general, neither injective nor
surjective.

######  Remark 3.15.

It should be noted that differential forms are defined on any Lie groupoid
equipped with a connection. However, the connection needs to be integrable in
order to be able to define the exterior derivative. A different definition of
the de Rham cohomology of a Lie groupoid was introduced by Laurent-Gengoux, Tu
and Xu, [LGTX2], using the simplicial manifold given by the nerve associated
to the Lie groupoid (compare also with [Du, Be2, BX, FN]).

######  Example 3.16.

Let $M$ be a smooth manifold and $\mathbb{X}\,=\,[M\rightrightarrows M]$ be
the associated Lie groupoid. Fix the connection $\mathcal{H}_{m}M\,=\,T_{m}M$
as in Example 3.4. Then the $k$-th de Rham cohomology group of the pair
$(\mathbb{X},\,\mathcal{H})$ is isomorphic to the $k$-th de Rham cohomology
group $H^{k}_{\rm dR}(M)$ of $M$ and so we have $H^{k}_{\rm
dR}(\mathbb{X},\,\mathcal{H})\,\cong\,H^{k}_{\rm dR}(M)$.

######  Example 3.17.

Let $G$ be a Lie group acting on a smooth manifold $M$. Let
$\mathbb{X}\,=\,[M\times G\rightrightarrows M]$ be the associated action Lie
groupoid. In particular, the source and target maps respectively are given by
${\rm pr}_{1}\,\colon\,(m,\,g)\,\longmapsto\,m$ and
$\mu\,\colon\,(m,\,g)\,\longmapsto\,m\cdot g.$ Define a connection
$\mathcal{H}_{(m,g)}(M\times G)\,=\,T_{m}M$ as in Example 3.5. Then, a
differential $k$-form $\omega$ on $\mathbb{X}$ is a differential form on $M$
satisfying the condition $\widehat{\mathcal{H}}({\rm
pr}_{1}^{*}\omega)(m,\,g)=\widehat{\mathcal{H}}(\mu^{*}\omega)(m,\,g)$ for all
$(m,g)\in M\times G$. That means, $\omega(m)\,=\,({R_{g}}^{*}\omega)(m),$
where ${R_{g}}^{*}$ is the induced action of $G$ on the space of differential
forms. Thus the space of differential forms of the Lie groupoid
$\mathbb{X}\,=\,[M\times G\rightrightarrows M]$ with connection
$\mathcal{H}_{(m,g)}\,=\,T_{m}M$ is the space of $G$-invariant differential
forms on $M$. In turn, we have $H^{k}_{\rm
dR}(\mathbb{X},\,\mathcal{H})\,=\,H^{k}(\Omega^{\bullet}(M)^{G})$, where
$H^{k}(\Omega^{\bullet}(M)^{G})$ denotes the cohomology corresponding to the
$G$-invariant differential forms. In particular if $G$ is compact and
connected then, by a classical theorem of E. Cartan, the inclusion map
$\Omega^{\bullet}(M)^{G}\,\longrightarrow\,\Omega^{\bullet}(M)$ induces an
isomorphism $H^{*}(\Omega^{\bullet}(M)^{G})\,\longrightarrow\,H^{*}(M)$ of
cohomology rings and thus we have $H^{k}_{\rm
dR}(\mathbb{X},\,\mathcal{H})\,\cong\,H^{k}(M)$. Note that the simplicial de
Rham cohomology associated to the simplicial nerve of the action Lie groupoid
has been computed in [Be1, BX]. In contrast, this simplicial de Rham
cohomology of $\mathbb{X}$ is isomorphic to the whole $G$-equivariant
cohomology $H^{*}_{G}(M)$ of $M$. See Subsection 3.3 for a brief comparison
between our approach to de Rham cohomology of Lie groupoids with connections
and that of Behrend [Be1] and Behrend-Xu [BX].

######  Example 3.18.

Let $\pi\,\colon\,\,P\,\longrightarrow\,M$ be a principal bundle and $A$ a
connection on it. Consider the Lie groupoid ${\mathbb{P}_{\rm
Gauge}}\,=\,[\frac{P\times P}{G}\rightrightarrows M]$ and the connection
$\mathcal{H}$ on it, as described in Example 3.8. It is not difficult to
verify that an $\alpha\,\in\,\Omega^{k}(M),\ 0\,\leq\,k$, is a differential
form on ${\mathbb{P}_{\rm Gauge}}$ if it satisfies the condition
$\pi^{*}\alpha(p)\,=\,\pi^{*}\alpha(q)$ for all $p,\,q\,\in\,P.$ In particular
if $\alpha\,\in\,\Omega^{0}(M),$ the condition says that
$\pi^{*}\alpha\,=\,\alpha\circ\pi$ is a constant function (and hence by
surjectivity of $\pi,$ $\alpha$ as well). For $k\,>\,0$ we have
$\Omega^{k}({\mathbb{P}_{\rm
Gauge}},\,\mathcal{H})\,=\,\\{\alpha\,\in\,\Omega^{k}(M)\,\mid\,\pi^{*}\alpha\,=\,0\\}.$
In turn when the connection is integrable, we get

$\begin{split}&H^{0}_{\rm dR}(\mathbb{P}_{\rm
Gauge},\,\mathcal{H})\,=\mathbb{R},\\\ &H^{1}_{\rm dR}(\mathbb{P}_{\rm
Gauge},\,\mathcal{H})\,=\\{\alpha\in\Omega^{1}(M)\,\mid\,\pi^{*}\alpha\,=\,0,\
d\alpha\,=\,0\\},\end{split}$

and so on for the higher cohomology groups.

We end this section with a result regarding the cohomology ring homomorphism
associated with a morphism of Lie groupoids with connection. The following
proposition is deduced using Lemma 3.11.

######  Proposition 3.19.

Let $\mathcal{H}_{\mathbb{X}}$ and $\mathcal{H}_{\mathbb{Y}}$ be integrable
connections on the Lie groupoids $\mathbb{X}\,=\,[X_{1}\rightrightarrows
X_{0}]$ and $\mathbb{Y}\,=\,[Y_{1}\rightrightarrows Y_{0}]$ respectively. Then
a morphism of Lie groupoids with connections

$(F,\,f)\,\colon\,\bigl{(}\mathbb{X}=[X_{1}\rightrightarrows
X_{0}],\,\mathcal{H}_{\mathbb{X}}\bigr{)}\,\longrightarrow\,\bigl{(}\mathbb{Y}=[Y_{1}\rightrightarrows
Y_{0}],\,\mathcal{H}_{\mathbb{Y}}\bigr{)}$

induces a homomorphism of de Rham cohomology groups

$H^{k}_{\rm
dR}(\mathbb{Y},\,\mathcal{H}_{\mathbb{Y}})\,\longrightarrow\,H^{k}_{\rm
dR}(\mathbb{X},\,\mathcal{H}_{\mathbb{X}})$

that sends any $[\alpha]$ to $[f^{*}\alpha]$. These maps produce a morphism of
graded $\mathbb{R}$-algebras $H^{*}_{\rm
dR}(\mathbb{Y},\,\mathcal{H}_{\mathbb{Y}})\,\longrightarrow\,H^{*}_{\rm
dR}(\mathbb{X},\,\mathcal{H}_{\mathbb{X}})$.

###### Proof.

Let $[\alpha]\,\in\,H^{k}_{\rm dR}(\mathbb{Y},\,\mathcal{H}_{\mathbb{Y}}).$ In
particular, that means that $\alpha\,\in\,\Omega^{k}(\mathbb{Y})$ and
$0\,=\,d\alpha\,\in\,\Omega^{k+1}(\mathbb{Y}).$ Then, using Lemma 3.13 we have
that $f^{*}\alpha\,\in\,\Omega^{k}(\mathbb{X})$ and
$0\,=\,df^{*}\alpha\,\in\,\Omega^{k+1}(\mathbb{X}).$ Also Lemma 3.11 implies
that $df^{*}\theta\,\in\,\Omega^{k}(\mathbb{Y})$ for any
$\theta\,\in\,\Omega^{k-1}(\mathbb{X})$. Thus we have
$[f^{*}\alpha]\,=\,[f^{*}(\alpha+d\theta)].$ ∎

###  3.3. Simplicial de Rham cohomology of a Lie groupoid

In [BX] Behrend and Xu introduced the notion of de Rham cohomology for Lie
groupoids, combining the construction of Čech cohomology and the de Rham
cohomology of differential forms, relying on the simplicial manifold
associated to the groupoid nerve. Our definition (and the motivation behind
it) of the de Rham cohomology for Lie groupoids with connections is very much
different. First of all, we introduce the notion of a connection on a Lie
groupoid to study Atiyah sequences over a Lie groupoid (see the discussion
around (3.2)). A natural consequence for the existence of a connection on a
Lie groupoid is the definition of differential forms as introduced in
Definition 3.10, and the associated de Rham cohomology. In Section 5 this
definition will allow us to describe connections on principal bundles over Lie
groupoids as differential forms and subsequently to develop the associated
Chern-Weil theory and theory of characteristic classes (see Section 6). More
succinctly, one may say the groupoid structure is exploited in our model by
taking into account the connection on the Lie groupoid (see condition 3 in
Definition 3.1), not through the groupoid nerves. However, our framework can
be adopted for the set-up of simplicial smooth manifolds, which will be
pursued in detail in follow-up work [BCKN2]. Here we will give a brief outline
of the constructions. For this, let us recall the constructions in [BX] and
compare them with ours.

Let $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ be a Lie groupoid. Since
the source and target maps are both smooth submersions, for each $i\geq 0$ we
obtain a smooth manifold $X_{i}$ by setting

$X_{i}\,:=\,\underbrace{X_{1}\times_{X_{0}}\cdots\times_{X_{0}}X_{1}}_{i-{\rm
times}}.$

For each $i\geq 1$ there are $i+1$ canonical maps ${\rm pr}_{i,\alpha}\colon
X_{i}\rightarrow X_{i-1}$ given by

$\begin{split}&{\rm
pr}_{i,1}\,\colon\,(\phi_{1},\cdots,\phi_{i})\,\longmapsto\,(\phi_{2},\cdots,\phi_{i}),\\\
&{\rm
pr}_{i,2}\,\colon\,(\phi_{1},\cdots,\phi_{i})\,\longmapsto\,(\phi_{1},\cdots,\phi_{i-1}),\\\
&{\rm
pr}_{i,\alpha}\,\colon\,(\phi_{1},\cdots,\phi_{i})\,\longmapsto\,(\phi_{1},\cdots,\phi_{\alpha-2}\circ\phi_{\alpha-1},\cdots,\phi_{i}),\,\
3\leq\alpha\leq i+1.\end{split}$ (3.12)

In turn we get a simplicial manifold

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
23.51796pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X_{\bullet}\,=\,\biggl{\\{}\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
47.51796pt\raise-6.45831pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
47.51796pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
47.51796pt\raise 6.45831pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
47.51796pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
47.51796pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
89.38737pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
89.38737pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
89.38737pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
89.38737pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
131.25679pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
131.25679pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
131.25679pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X_{0}\,}$}}}}}}}\ignorespaces}}}}\ignorespaces\biggr{\\}}.$

By defining the differentials

$\partial_{i}\,:=\,\sum_{\alpha=1}^{i+2}(-1)^{\alpha+1}{\rm
pr}^{*}_{{i+1},\alpha}\colon\Omega^{p}(X_{i})\rightarrow\Omega^{p}(X_{i+1}),$
(3.13)

we obtain a cochain complex

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
19.5101pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Omega^{p}(X_{0})\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\,}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
19.51012pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
25.65195pt\raise 6.075pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.78612pt\hbox{$\scriptstyle{\partial_{0}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 43.5101pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
43.5101pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Omega^{p}(X_{1})\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\,}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
82.53032pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{}}$}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.78612pt\hbox{$\scriptstyle{\partial_{1}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 106.5303pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
106.5303pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\Omega^{p}(X_{2})\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\,}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
145.55052pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 169.5505pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{$\textstyle{\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}$}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
169.5505pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ldots}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

and then the usual exterior derivatives of differential forms yield the
following double complex

$\begin{matrix}\vdots&{}&\vdots&{}&\vdots&{}&{}\\\
\Omega^{p}(X_{0})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&\Omega^{p}(X_{1})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&\Omega^{p}(X_{2})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&\cdots\\\
\Big{\downarrow}{d}&{}&\Big{\downarrow}{d}&{}&\Big{\downarrow}{d}&{}&{}\\\
\Omega^{p+1}(X_{0})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&\Omega^{p+1}(X_{1})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&\Omega^{p+1}(X_{2})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&\cdots\\\
\vdots&{}&\vdots&{}&\vdots&{}&{}\end{matrix}$ (3.14)

With the differential $\delta\omega\,=\,\partial\omega+(-1)^{i}d\omega$ for
any $\omega\,\in\,\Omega^{p}(X_{i}),$ we obtain the total complex

$C^{n}(\mathbb{X})\,=\,\bigoplus_{p+i=n}\Omega^{p}(X_{i}).$ (3.15)

The total complex in (3.15) and the corresponding cohomology have been
respectively called de Rham complex and de Rham cohomology of $\mathbb{X}$ in
[BX]. Here to avoid any conflict of terminologies we will refer to them
respectively as simplicial de Rham complex and simplicial de Rham cohomology
of $\mathbb{X}$. The contrast between the two approaches can be further
highlighted by considering Example 3.17 again. We have computed the de Rham
cohomology of the action Lie groupoid $\mathbb{X}\,=\,[M\times
G\rightrightarrows M]$ for the connection $\mathcal{H}_{(m,g)}\,=\,T_{m}M,$
which is given as $H^{k}(\Omega^{\bullet}(M)^{G})$, the cohomology
corresponding to the $G$-invariant differential forms. The simplicial de Rham
cohomology of the action Lie groupoid $\,[M\times G\rightrightarrows M]$
instead turns out to be the $G$-equivariant cohomology of the smooth manifold
$M$ (compare for example [Be1], [BX]).

Now suppose the Lie groupoid $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$
admits an integrable connection $\mathcal{H}\,\subset\,TX_{1}.$ Thus we have a
splitting $\mathcal{H}\oplus\ker(ds)\,=\,TX_{1}$ as in (3.4). Let
$T\mathbb{X}\,=\,[TX_{1}\rightrightarrows TX_{0}]$ be the associated tangent
groupoid. Furthermore (by $\mathrm{(3.4)}$ and Definition 3.9) $\mathcal{H}$
defines a smooth splitting of $\Omega^{p}{(X_{1})}$ into ‘horizontal’ and
‘vertical’ components,

$\Omega^{p}{(X_{1})}\,=\,\Omega_{\rm Hor}^{p}{(X_{1})}\oplus\Omega_{\rm
Ver}^{p}{(X_{1})}.$

We identify $T(X_{1}\times_{X_{0}}\times\cdots\times_{X_{0}}X_{1})$ with
$TX_{1}\times_{TX_{0}}\cdots\times_{TX_{0}}TX_{1}$ and get a pullback diagram

$\textstyle{{TX_{1}\times_{TX_{0}}TX_{1}}{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
22.76228ptd{\rm pr}_{1}}$$\scriptstyle{d{\rm
pr}_{2}}$$\textstyle{TX_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{dt}$$\textstyle{TX_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{ds}$$\textstyle{TX_{0}.}$

Similarly, $\Omega^{p}(X_{1}\times_{X_{0}}\times\cdots\times_{X_{0}}X_{1})$ is
identified with
$\Omega^{p}X_{1}\oplus_{\Omega^{p}X_{0}}\cdots\oplus_{\Omega^{p}X_{0}}\Omega^{p}X_{1}$
given by the pushout diagrams

$\textstyle{\Omega^{p}X_{0}{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s^{*}}$$\scriptstyle{t^{*}}$$\textstyle{\Omega^{p}X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\textstyle{\Omega^{p}X_{1}{\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}$$\scriptstyle{i_{2}}$$\textstyle{{\Omega^{p}(X_{1})\oplus_{\Omega^{p}(X_{0})}\Omega^{p}(X_{1})},}$

where we set

$\Omega^{p}(X_{1})\oplus_{\Omega^{p}(X_{0})}\Omega^{p}(X_{1})\,:=\,(\Omega^{p}(X_{1})\oplus\Omega^{p}(X_{1}))/\sim$

with $(\alpha,\,\beta)\,\sim\,(\alpha+s^{*}(\theta),\,\beta-t^{*}(\theta))$
for any $\theta\,\in\,{\Omega^{p}(X_{0})}.$

Consider the subspace

${\ker(ds)}\times_{TX_{0}}{\ker(ds)}\,\subset\,{TX_{1}\times_{TX_{0}}TX_{1}}\,\simeq\,T(X_{1}\times_{X_{0}}X_{1})\,=\,TX_{2}$

and the differentials of the projection maps ${\rm
pr}_{2,\alpha}\,\colon\,X_{2}\,\longrightarrow\,X_{1}$

$\begin{split}&{\rm
pr}_{2,1}\,\colon\,(\phi_{1},\,\phi_{2})\,\longmapsto\,\phi_{1},\\\ &{\rm
pr}_{2,2}\,\colon\,(\phi_{1},\,\phi_{2})\,\longmapsto\,\phi_{2},\\\ &{\rm
pr}_{3,3}\,\colon\,(\phi_{1},\,\phi_{2})\,\longmapsto\,\phi_{1}\circ\phi_{2},\end{split}$
(3.16)

restricted to the subspace ${\ker(ds)}\times_{TX_{0}}{\ker(ds)}.$ Note that
since $ds(u\circ v)\,=\,ds(v)$, the kernel ${\ker(ds)}\subset{TX_{1}}$ is
closed under composition, that is, if $(u,\,v)\,\in\,{\ker
ds}\times_{TX_{0}}{\ker(ds)}$, then $u\circ v\,\in\,{\ker(ds)}.$ This means
that if we restrict the differentials of each maps in (3.16) to the subspace
${\ker(ds)}\times_{TX_{0}}{\ker(ds)}$, then we get their images in
${\ker(ds)}$, so for each $1\,\leq\,\alpha\,\leq\,i+1$ we have

$d{\rm
pr}_{2,\alpha}\,\colon\,{\ker(ds)}\times_{TX_{0}}{\ker(ds)}\,\longrightarrow\,{\ker(ds)}.$

The above argument evidently generalizes by iteration to

$d{\rm
pr}_{i,\alpha}\,\colon\,\underbrace{\ker(ds)\times_{TX_{0}}\cdots\times_{TX_{0}}\ker(ds)}_{i-{\rm
times}}\,\longrightarrow\,\underbrace{\ker(ds)\times_{TX_{0}}\cdots\times_{TX_{0}}\ker(ds)}_{(i-1)-{\rm
times}}$

for all $i\geq 2$ and $1\leq\alpha\leq i+1$. For each $p\geq 0,$ a dual
argument gives the maps

${\rm pr}^{*}_{i,\alpha}\,\colon\,\underbrace{\Omega_{\rm
Ver}^{p}{(X_{1})}\oplus_{\Omega^{p}(X_{0})}\cdots\oplus_{\Omega^{p}(X_{0})}\Omega_{\rm
Ver}^{p}{(X_{1})}}_{(i-1)-{\rm
times}}\,\longrightarrow\,\underbrace{\Omega_{\rm
Ver}^{p}{(X_{1})}\oplus_{\Omega^{p}(X_{0})}\cdots\oplus_{\Omega^{p}(X_{0})}\Omega_{\rm
Ver}^{p}{(X_{1})}}_{i-{\rm times}}$

for all $i\geq 2$ and $1\leq\alpha\leq i+1$. We introduce the notation

$\underbrace{\Omega_{\rm
Ver}^{p}{(X_{1})}\oplus_{\Omega^{p}(X_{0})}\cdots\oplus_{\Omega^{p}(X_{0})}\Omega_{\rm
Ver}^{p}{(X_{1})}}_{i-{\rm times}}:=\Omega_{\rm
Ver}^{p}{(X_{i})}\subset\Omega^{p}(X_{i}),\ i\,\geq\,1.$

This means that the maps defined in (3.13) preserve the ‘vertical spaces’,

$\partial_{i-1}\,\colon\,\Omega_{\rm
Ver}^{p}{(X_{i-1})}\,\longrightarrow\,\Omega_{\rm Ver}^{p}{(X_{i})}$

for all $i\,\geq\,2.$ Now observe that by Definition 3.10, the map

$\partial_{0}\,=\,s^{*}-t^{*}\,\colon\,\Omega^{p}(X_{0})\,\longrightarrow\,\Omega^{p}(X_{1})$

restricted to $\Omega^{p}(\mathbb{X})\,\subset\,\Omega^{p}(X_{0})$ gives the
map $\partial_{0}\,\colon\,\Omega^{p}(\mathbb{X})\,\longrightarrow\Omega_{\rm
Ver}^{p}(X_{1}).$ For purely notational convenience here we set

$\Omega^{p}(\mathbb{X})\,:=\,\Omega_{\rm Ver}^{p}(X_{0}).$

Moreover, since the connection $\mathcal{H}$ on $\mathbb{X}$ is integrable,
the exterior differentials
$d\colon\Omega^{p}(X_{i})\rightarrow\Omega^{p+1}(X_{i})$ preserves the
‘vertical spaces’ (see Lemma 3.11 and the proof there),

$d\,\colon\,\Omega_{\rm Ver}^{p}(X_{i})\,\longrightarrow\,\Omega_{\rm
Ver}^{p+1}(X_{i}).$

Thus we deduce the following.

######  Proposition 3.20.

Let $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ be a Lie groupoid with an
integrable connection $\mathcal{H}.$ Then $\mathcal{H}$ induces a subcomplex
of the double complex (3.14),

$\begin{matrix}\vdots&{}&\vdots&{}&\vdots&{}&{}\\\ \Omega^{p}_{\rm
Ver}(X_{0})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&\Omega_{\rm
Ver}^{p}(X_{1})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&\Omega_{\rm
Ver}^{p}(X_{2})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&\cdots\\\
\Big{\downarrow}{d}&{}&\Big{\downarrow}{d}&{}&\Big{\downarrow}{d}&{}&{}\\\
\Omega_{\rm
Ver}^{p+1}(X_{0})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&\Omega_{\rm
Ver}^{p+1}(X_{1})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&\Omega_{\rm
Ver}^{p+1}(X_{2})&\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}&\cdots\\\
\vdots&{}&\vdots&{}&\vdots&{}&{}\end{matrix}$ (3.17)

Then one may consider again the associated total complex

$C_{\rm Ver}^{n}(\mathbb{X})\,=\,\bigoplus_{p+i=n}\Omega_{\rm Ver}^{p}(X_{i})$

and the corresponding cohomology respectively as simplicial de Rham complex
and simplicial de Rham cohomology $H_{dR}^{*}(X_{\bullet},\,\mathcal{H})$ of a
Lie groupoid with an integrable connection. This set-up will be explored in
more details in our follow-up article [BCKN2].

###  3.4. Forms and de Rham cohomology for Deligne-Mumford stacks

Here we extend our construction of de Rham cohomology from Lie groupoids with
connections to Deligne-Mumford stacks (see Definition 2.27) using the fact
that a Deligne-Mumford stack is presented by an étale Lie groupoids
$\mathbb{X}=[X_{1}\rightrightarrows X_{0}]$, which always admits the
connection $\mathcal{H}=TX_{1}$.

Let $\mathscr{X}$ be any differentiable stack and
$x\,\colon\,X\,\longrightarrow\,\mathscr{X}$ a presentation. Set
$X_{0}\,=\,X.$ Since $x\,\colon\,X\,\longrightarrow\,\mathscr{X}$ is a
surjective representable submersion, we obtain a family of smooth manifolds
$\\{X_{i}\\}_{i\geq 0}$, where

$X_{i}\,:=\,\underbrace{X\times_{\mathscr{X}}\cdots\times_{\mathscr{X}}X}_{(i+1)-{\rm
times}}.$

For each $i\,\geq\,1$ there are $i+1$ canonical projection maps $\\{{\rm
pr}_{i,\alpha}\,\colon\,X_{i}\,\longrightarrow\,X_{i-1}\\}_{\alpha}$ defining
a simplicial manifold (see (3.12))

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84.94304pt\raise 4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
84.94304pt\raise-4.30554pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
84.94304pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
126.81245pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
126.81245pt\raise 2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
126.81245pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X_{0}\,}$}}}}}}}\ignorespaces}}}}\ignorespaces\biggr{\\}}.$
(3.18)

In particular, $\mathbb{X}\,:=\,[X_{1}\rightrightarrows X_{0}]$ is the Lie
groupoid associated to the atlas with source and target maps respectively
given by ${\rm pr}_{1,1}$ and ${\rm pr}_{1,2}$ (see Example 2.26) and
$X_{\bullet}$ the simplicial manifold given by the nerve of $\mathbb{X}$. Let
$\Omega^{p}(X_{i})$ be the vector space of differential $p$-forms on the
smooth manifold $X_{i}.$ Then using the pullbacks along the projections we get
a simplicial vector space

$\textstyle{\Omega^{p}(X_{\bullet})\colon\,\,\,\Omega^{p}(X_{0})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{p}(X_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{p}(X_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots}$
(3.19)

Define for each $i\,\geq\,0$ the differential

$\partial_{i}\,:=\,\sum_{\alpha=1}^{i+2}(-1)^{\alpha+1}{\rm
pr}^{*}_{{i+1},\alpha}\,\colon\,\Omega^{p}(X_{i})\,\longrightarrow\,\Omega^{p}(X_{i+1}).$

In turn, we get a cochain complex

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Let

$h^{i}(\Omega^{p}(X_{\bullet}))$

be the corresponding cohomology groups. In particular this means

$h^{0}(\Omega^{p}(X_{\bullet}))=\ker{\partial_{0}}=\ker({\rm
pr}^{*}_{1,1}-{\rm pr}^{*}_{1,2}).$ (3.20)

For any other atlas $y\colon Y\,\rightarrow\,\mathscr{X}$ we conclude using
Corollary 3.1 in [BX],

$h^{i}(\Omega^{p}(X_{\bullet}))\cong h^{i}(\Omega^{p}(Y_{\bullet})).$ (3.21)

We refer to [Be1, BX] for a more extensive discussion on the above
constructions.

Now suppose $\mathscr{X}$ is a Deligne-Mumford stack and $x\colon
X\,\longrightarrow\,\mathscr{X}$ an étale presentation. Consider the étale Lie
groupoid $\mathbb{X}=[X_{1}\rightrightarrows X_{0}].$ As we have seen in
Example 3.6, then $\mathcal{H}=\,TX_{1}$ is an integrable connection on the
Lie groupoid $\mathbb{X}$. For this connection the set of differential forms
on $(\mathbb{X},TX_{1})$ (see Definition 3.10) reduces to a much simpler form,

$\Omega^{p}(\mathbb{X},\,TX_{1})\,=\,\ker({\rm pr}^{*}_{1,1}-{\rm
pr}^{*}_{1,2}).$ (3.22)

Then using (3.20) and (3.21) we arrive at the following result.

######  Proposition 3.21.

Let $\mathscr{X}$ be a Deligne-Mumford stack, and let
$x\,\colon\,X\,\longrightarrow\,\mathscr{X}$ and
$y\,\colon\,Y\,\longrightarrow\,\mathscr{X}$ a pair of étale presentations.
Set $X_{1}\,:=\,X\times_{\mathscr{X}}X$ and
$Y_{1}\,:=\,Y\times_{\mathscr{X}}Y$. Let
$\mathbb{X}\,=\,[X_{1}\rightrightarrows X]$ and
$\mathbb{Y}\,=\,[Y_{1}\rightrightarrows Y]$ be the associated étale Lie
groupoids. Then there is an isomorphism

$\Omega^{p}(\mathbb{X},\,TX_{1})\,\cong\,\Omega^{p}(\mathbb{Y},\,TY_{1}).$

It is now straightforward to verify that the above isomorphism descends to an
isomorphism of de Rham cohomologies (see 3.11) of the associated Lie groupoids
as well.

######  Corollary 3.22.

Let the conditions and notations be as in Proposition 3.21. Then

$H^{k}_{\rm dR}(\mathbb{X},\,TX_{1})\,\cong\,H^{k}_{\rm
dR}(\mathbb{Y},\,TY_{1}).$

With these results in mind, we can define differential forms and de Rham
cohomology for a Deligne-Mumford stack.

######  Definition 3.23.

Let $\mathscr{X}$ be a Deligne-Mumford stack. Choose an étale presentation
$x\,\colon\,X\,\longrightarrow\,\mathscr{X}$. A differential form on
$\mathscr{X}$ is a differential form on the étale Lie groupoid with connection
$\bigl{(}\mathbb{X}=[X_{1}\rightrightarrows X],TX_{1}\bigr{)}$, where
$X_{1}:=X\times_{\mathscr{X}}X$. We obtain a cochain complex
$\Omega^{\bullet}(\mathscr{X})$ by setting

$\Omega^{p}(\mathscr{X})\,:=\,\Omega^{p}(\mathbb{X},\,TX_{1}).$

The $k$-th de Rham cohomology group of $\mathscr{X}$ is the $k$-th de Rham
cohomology group of the pair $\bigl{(}\mathbb{X}\,=\,[X_{1}\rightrightarrows
X],\ TX_{1}\bigr{)}$ and we write

$H^{k}_{\rm dR}(\mathscr{X})\,:=\,H^{k}_{\rm dR}(\mathbb{X},\,TX_{1}).$

Corollary 3.22 in particular shows that this is well-defined and does not
depend on a choice of a specific étale presentation for the Deligne-Mumford
stack $\mathscr{X}$.

######  Example 3.24.

Let $G$ be a discrete group (a $0$-dimensional Lie group) acting on a smooth
manifold $M$. Then the action Lie groupoid $\mathbb{X}\,=\,[M\times
G\rightrightarrows M]$ is étale. In turn, the associated classifying stack
$\mathscr{B}\mathbb{X}$ (see Example 2.19) of the action Lie groupoid is then
a Deligne-Mumford stack, namely isomorphic to the quotient stack $[M/G]$.
Furthermore the discreteness of $G$ implies for the connection in Example 3.17
that we have $\mathcal{H}_{(m,g)}\,=\,T_{m}M\,\simeq\,T_{(m,g)}(M\times G)$.
As we have seen in Example 3.17, the differential $k$-forms on $[M\times
G\rightrightarrows M]$ are the $G$-invariant differential forms on $M$ and
$H^{k}_{\rm dR}(\mathbb{X},\,\mathcal{H})\,=\,H^{k}(\Omega^{\bullet}(M)^{G})$.
where $H^{k}(\Omega^{\bullet}(M)^{G})$ is again the cohomology corresponding
to the $G$-invariant differential forms. Thus the de Rham cohomology of the
Deligne-Mumford stack $\mathscr{B}\mathbb{X}$ just gives the cohomology
corresponding to the $G$-invariant differential forms. In particular, if the
manifold $M$ is just a point with trivial $G$-action, the differentiable stack
$\mathscr{B}\mathbb{X}$ is then just the classifying stack $\mathscr{B}G$ of
the discrete group $G$ and its de Rham cohomology therefore gives the group
cohomology $H^{k}(G,\,\mathbb{R})$ of $G$.

##  4\. Connections on principal bundles over Lie groupoids

Let $\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$ be a Lie groupoid and
$(E_{G}\,\longrightarrow\,X_{0},\ [X_{1}\rightrightarrows X_{0}])$ a principal
$G$-bundle over $\mathbb{X}$. Let

$0\,\longrightarrow\,\text{ad}(E_{G})\,\longrightarrow\,\text{At}(E_{G})\,\longrightarrow\,TX_{0}\,\longrightarrow\,0$
(4.1)

be the associated Atiyah sequence corresponding to the principal $G$-bundle
$E_{G}$ (see (3.1)). In this section we will interpret (4.1) as a short exact
sequence of vector bundles over $\mathbb{X}$. This will facilitate the
definition of a connection on
$(E_{G}\,\longrightarrow\,X_{0},\,[X_{1}\rightrightarrows X_{0}])$.

First we explain the action of the Lie groupoid $\mathbb{X}$ on $TX_{0},\,{\rm
At}(E_{G})\,=\,(TE_{G})/G$ and ${\rm
ad}(E_{G})\,=\,(E_{G}\times\mathfrak{g})/G$. For this fix a connection

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

on $\mathbb{X}$.

###  4.1. Action of $\mathbb{X}$ on $TX_{0}$

Consider the homomorphism $\theta$ in (3.6). Using it we define the map

$\begin{split}\mu_{\rm
tan}\,\colon\,&X_{1}\times_{X_{0}}TX_{0}\,\longrightarrow\,TX_{0}\\\
&(\gamma,\,v)\,\longmapsto\,(t(\gamma),\,\theta_{\gamma}(v)).\end{split}$
(4.2)

Let $\tau\,\colon\,TX_{0}\,\longrightarrow\,X_{0}$ be the natural projection.
Then the pair

$(\tau:\,TX_{0}\,\to\,X_{0},\ \mu_{\rm tan})$

defines an action of $\mathbb{X}$ on $TX_{0}$ (see Definition 2.4).

###  4.2. Action of $\mathbb{X}$ on ${\rm At}(E_{G})$

As seen in Lemma 3.3, the connection $\mathcal{H}$ induces a connection
$\widetilde{\mathcal{H}}\,:=\,({\rm
pr}_{1*})^{-1}(\mathcal{H})\,\subset\,Ts^{*}E_{G}$ on
$[s^{*}E_{G}\,\rightrightarrows\,E_{G}]$. Substituting
$([s^{*}E_{G}\,\rightrightarrows\,E_{G}],\,\widetilde{\mathcal{H}})$ in place
of $(\mathbb{X},\,\mathcal{H})$ in (4.2), we get a map

$\widetilde{\mu}\,:\,s^{*}E_{G}\times_{E_{G}}TE_{G}\,\longrightarrow\,TE_{G}\,.$

Taking the quotient with respect to the action of $G$, we get a map

$\mu_{\rm
at}\,\colon\,X_{1}\,\times_{X_{0}}\,(TE_{G})/G\,=\,X_{1}\,\times_{X_{0}}\,\text{At}(E_{G})\,\longrightarrow\,(TE_{G})/G\,=\,\text{At}(E_{G})\,.$
(4.3)

Now $\mu_{\rm at}$ and the natural projection $\pi_{\rm at}\,:\,{\rm
At}(E_{G})\,\longrightarrow\,X_{0}$ together define an action of $\mathbb{X}$
on ${\rm At}(E_{G})$.

###  4.3. Action of $\mathbb{X}$ on ${\rm ad}(E_{G})$

Consider the restriction of the map $\widetilde{\mu}$ in Subsection 4.2 to
$s^{*}E_{G}\times_{E_{G}}\text{ker}(d\pi)\,\subset\,s^{*}E_{G}\times_{E_{G}}TE_{G}$,
where $d\pi$ is the differential of the natural projection
$\pi\,:\,E_{G}\,\longrightarrow\,X_{0}$. The image of this restricted map is
clearly $\text{ker}(d\pi)\,\subset\,TE_{G}$. We recall that
$\text{ad}(E_{G})\,=\,\text{ker}(d\pi)/G\,\subset\,(TE_{G})/G\,=\,\text{At}(E_{G})$.
Consequently, the action $(\mu_{\rm at},\,\pi_{\rm at})$ of $\mathbb{X}$ on
${\rm At}(E_{G})$ in Subsection 4.2 produces an action of $\mathbb{X}$ on
${\rm ad}(E_{G})$.

Then the exactness of the Atiyah sequence in (4.1) readily implies the
following:

######  Proposition 4.1.

Let $\mathcal{H}$ be a connection on the Lie groupoid
$\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$. Then $TX_{0}$, ${\rm
At}(E_{G})$ and ${\rm ad}(E_{G})$ are vector bundles over $\mathbb{X}$.
Moreover, the homomorphisms in the short exact sequence in (4.1) are
compatible with the actions of $\mathbb{X}$ on $TX_{0}$, ${\rm At}(E_{G})$ and
${\rm ad}(E_{G})$.

Now we define connections on a principal bundle over a given Lie groupoid.

######  Definition 4.2 (Connections on principal bundles over Lie groupoids).

Let $\mathcal{H}\,\subset\,TX_{1}$ be a connection on the Lie groupoid
$\mathbb{X}\,=\,[X_{1}\rightrightarrows X_{0}]$. Let
$\bigl{(}E_{G}\,\rightarrow\,X_{0},\,\,\mathbb{X}\bigr{)}$ be a principal
$G$-bundle over $\mathbb{X}$. A connection $\mathcal{D}$ on
$(E_{G}\,\longrightarrow\,X_{0},\,\,\mathbb{X})$ is a splitting of the Atiyah
sequence in (4.1) of vector bundles over $\mathbb{X}$.

######  Remark 4.3.

Note that a connection on $(E_{G}\,\longrightarrow\,X_{0},\,\,\mathbb{X})$
automatically gives a connection on the underlying principal $G$-bundle
$E_{G}\,\longrightarrow\,X_{0}$; however, the converse is not true in general.

##  5\. Forms for connections on principal Lie groupoid bundles

Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra. The vector space of
$\mathfrak{g}$-valued differential $k$-forms on a manifold $Y$ will be denoted
by $\Omega^{k}(Y,\,\mathfrak{g}).$

Let $P$ be a principal $G$-bundle over a manifold $M$. Then, there is a
bijective correspondence between the following two sets:

  1. (1)

The set of all splittings of the Atiyah sequence for $P$ (see (3.1)).

  2. (2)

The set of $\mathfrak{g}$-valued $1$-forms $A$ on $P$ satisfying the following
conditions:

     * •

the map $A\,:\,TP\,\longrightarrow\,\mathfrak{g}$ is $G$-equivariant for the
adjoint action of $G$ on $\mathfrak{g}$, and

     * •

the restriction of $A$ to every fiber of the projection
$P\,\longrightarrow\,M$ coincides with the Maurer-Cartan form for the action
of $G$ on the fiber.

Given a splitting homomorphism $\rho\,:\,TM\,\longrightarrow\,\text{At}(P)$
for the Atiyah sequence for $P$, the corresponding $\mathfrak{g}$-valued
$1$-form $A$ on $P$ is uniquely determined by the condition that the quotient
by $G$ of the kernel of $A$ coincides with the image of $\rho$. Conversely,
given a form $TP\,\longrightarrow\,P\times\mathfrak{g}$ satisfying the above
two conditions, after taking the quotient by the action of $G$, we get a
homomorphism $\text{At}(P)\,\longrightarrow\,\text{ad}(P)$. This homomorphism
gives a splitting of the Atiyah sequence for $P$.

The above bijective correspondence provides an alternative definition of a
connection on $P$. For more details on the above correspondence and for the
Atiyah sequence approach to connections on principal bundles, we refer also to
[Ma2, Appendix A].

It was observed in Proposition 4.1 that given a Lie groupoid $\mathbb{X}$
equipped with a connection $\mathcal{H}$, and a principal $G$-bundle $E_{G}$
on $\mathbb{X}$, there is an Atiyah sequence on $\mathbb{X}$ whose splittings
correspond to connections on $E_{G}$. On the other hand, in Section 3.2 we
developed the associated theory of differential forms on a Lie groupoid
$\mathbb{X}$ with connection $\mathcal{H}$ and the corresponding de Rham
cohomology for the pair $(\mathbb{X},\mathcal{H})$. In this section, our aim
is to describe a connection $\mathcal{D}$ on a principal bundle
$(E_{G}\rightarrow X_{0},\mathbb{X})$ as a differential $1$-form. Throughout
this section, we will always assume the connection $\mathcal{H}$ on
$\mathbb{X}$ to be integrable.

###  5.1. Connections and forms for principal bundles over Lie groupoids

Assume $\mathcal{D}\,:\,TX_{0}\,\longrightarrow\,\text{At}(E_{G})$ is a
connection on the principal $G$-bundle $(E_{G}\longrightarrow
X_{0},\,\mathbb{X}=[X_{1}\rightrightarrows X_{0}])$ given by a splitting of
the Atiyah sequence of the associated vector bundles

$0\,\longrightarrow\,\text{ad}(E_{G})\,\stackrel{{\scriptstyle
j}}{{\longrightarrow}}\,\text{At}(E_{G})\,\stackrel{{\scriptstyle
d\pi}}{{\longrightarrow}}\,TX_{0}\,\longrightarrow\,0$

over $\mathbb{X}$. Consider the connection on $E_{G}\,\longrightarrow\,X_{0}$
given by $\mathcal{D}$ (see Remark 4.3). As observed at the beginning of this
section, this gives a $\mathfrak{g}$-valued differential $1$-form $\omega$ on
the manifold $E_{G}$. In Lemma 5.2 we will see that this differential $1$-form
$\omega\,\in\,\Omega^{1}(E_{G},\,\mathfrak{g})$ on the manifold $E_{G}$ is in
fact a differential $1$-form on the Lie groupoid $[s^{*}E_{G}\rightrightarrows
E_{G}]$. For that, we first note the following property of differential forms
on the Lie groupoids $[s^{*}E_{G}\rightrightarrows E_{G}]$ and $\mathbb{X}.$

######  Lemma 5.1.

Let $(E_{G}\xrightarrow{\pi}X_{0},\ \mathbb{X}\,=\,[X_{1}\rightrightarrows
X_{0}])$ be a principal $G$-bundle over a Lie groupoid $\mathbb{X}$. Let
$\mathcal{H}\,\subset\,TX_{1}$ be a connection on $\mathbb{X}$, and let
$\widetilde{\mathcal{H}}\,\subset\,T(s^{*}E_{G})$ be the pullback connection
on the Lie groupoid $[s^{*}E_{G}\rightrightarrows E_{G}]$ (see Lemma 3.3). Let
$\tau$ be a differential $k$-form on the manifold $X_{0}$, and let
$\pi^{*}\tau$ be the pullback form on the manifold $E_{G}$. Then, $\tau$ is a
differential form on the Lie groupoid $\mathbb{X}$ if and only if
$\pi^{*}\tau$ is a differential $k$-form on the Lie groupoid
$[s^{*}E_{G}\rightrightarrows E_{G}]$.

###### Proof.

Consider the morphism of Lie groupoids

$({\rm
pr}_{1},\,\pi)\,\colon\,[s^{*}E_{G}\,\rightrightarrows\,E_{G}]\,\longrightarrow\,\mathbb{X}\,.$

As $\pi\,\colon\,E_{G}\,\longrightarrow\,X_{0}$ is a surjective submersion,
its pullback ${\rm pr}_{1}\,\colon\,s^{*}E_{G}\,\longrightarrow\,X_{1}$ is a
surjective submersion as well. Take any $(\gamma,\,a)\,\in\,s^{*}E_{G}$ and
$(v,\,w)\,\in\,T_{(\gamma,a)}s^{*}E_{G}$. We have

$\displaystyle({\rm
pr}_{1})_{*,(\gamma,a)}\bigl{(}P_{\widetilde{\mathcal{H}}_{(\gamma,a)}}(v,\,w)\bigr{)}$
$\displaystyle\,=\,({\rm
pr}_{1})_{*,(\gamma,a)}\bigl{(}P_{\mathcal{H}_{\gamma}}(v),w\bigr{)}$
$\displaystyle\,=\,P_{\mathcal{H}_{\gamma}}(v)$
$\displaystyle\,=\,P_{\mathcal{H}_{{\rm pr}_{1}(\gamma,a)}}\bigl{(}({\rm
pr}_{1})_{*,(\gamma,a)}(v,\,w)\bigr{)}.$

Now the lemma follows immediately from Lemma 3.13 and Lemma 3.14. ∎

The following lemma classifies connections on the pair
$(E_{G}\xrightarrow{\pi}X_{0},\,\mathbb{X})$ in terms of $\mathfrak{g}$-valued
differential $1$-forms on the Lie groupoid
$[s^{*}E_{G}\,\rightrightarrows\,E_{G}]$.

######  Lemma 5.2.

Let $(E_{G}\xrightarrow{\pi}X_{0},\,\mathbb{X}\,=\,[X_{1}\rightrightarrows
X_{0}])$ be a principal $G$-bundle over a Lie groupoid with integrable
connection $\bigl{(}\mathbb{X},\,\mathcal{H}\bigr{)}$. Let $\omega$ be a
connection $1$-form on the principal $G$-bundle
$E_{G}\,\longrightarrow\,X_{0}$. Let $\mathcal{D}\,\colon\,{{\rm
At}(E_{G})}\,\longrightarrow\,{\rm ad}(E_{G})$ be the corresponding
homomorphism of vector bundles over $X_{0}$. Then $\mathcal{D}$ defines a
connection on $(E_{G}\,\xrightarrow{\pi}\,X_{0},\,\mathbb{X})$ if and only if
$\omega$ is a $\mathfrak{g}$-valued $1$-form on the Lie groupoid
$[s^{*}E_{G}\,\rightrightarrows\,E_{G}]$.

###### Proof.

Let
$\overline{\mathcal{D}}\,\colon\,TE_{G}\,\longrightarrow\,E_{G}\times\mathfrak{g}$
be the homomorphism of vector bundles given by the $1$-form
$\omega\,\in\,\Omega^{1}(E_{G},\,\mathfrak{g})$. Then,
$\overline{\mathcal{D}}$ descends to a homomorphism
$\mathcal{D}\,\colon\,\text{At}(E_{G})\,\longrightarrow\,\text{ad}(E_{G})$ of
vector bundles over $X_{0}$.

Suppose that the above homomorphism $\mathcal{D}$ is a connection on the
principal $G$-bundle $(E_{G}\xrightarrow{\pi}X_{0},\,\mathbb{X})$. We have to
show that $\omega$ is a differential form on $[s^{*}E_{G}\rightrightarrows
E_{G}]$.

Since $\mathcal{D}$ is a connection on
$(E_{G}\xrightarrow{\pi}X_{0},\,\mathbb{X})$, we know that $\mathcal{D}$ is a
morphism of vector bundles over the Lie groupoid $\mathbb{X}$, and hence
$\mathcal{D}(\gamma\cdot[r])\,=\,\gamma\cdot\mathcal{D}([r])$ for all
$(\gamma,\,[r])\,\in\,X_{1}\times_{X_{0}}(TE_{G})/G$; equivalently, the
homomorphism $\overline{\mathcal{D}}$ satisfies the condition that
$\overline{\mathcal{D}}\bigl{(}\gamma\cdot(a,\,v)\bigr{)}\,=\,\gamma\cdot\overline{\mathcal{D}}(a,\,v)$
for all
$\bigl{(}\gamma,\,(a,\,v)\bigr{)}\,\in\,X_{1}\times_{X_{0}}TE_{G}\,=\,s^{*}E_{G}$.

Let $\widetilde{\theta}$ be the map in (3.6) corresponding to the pullback
connection ${\widetilde{\mathcal{H}}}$ on
$s^{*}E_{G}\,\rightrightarrows\,E_{G}.$ Recall that
$\gamma\cdot(a,\,v)\,=\,\widetilde{\theta}_{(\gamma,a)}(v)$ and
$\overline{\mathcal{D}}\bigl{(}(a,\,v)\bigr{)}\,=\,\bigl{(}a,\omega(a)(v)\bigr{)}$
(see (3.8)). Thus we conclude that

$\begin{split}\bigl{(}\gamma\cdot a,\,\omega(\gamma\cdot
a)(\widetilde{\theta}_{(\gamma,a)}(v)\bigr{)}&\,=\,\bigl{(}\gamma\cdot
a,\,\omega(a)(v)\bigr{)}\\\ \Rightarrow\,\omega(\gamma\cdot
a)(\widetilde{\theta}_{(\gamma,a)}(v))&\,=\,\omega(a)(v).\end{split}$

Consequently, for any $(v,\,r)\,\in\,T_{(\gamma,\,a)}s^{*}E_{G}$,

$\displaystyle\bigl{(}\widehat{\widetilde{\mathcal{H}}}(\mu^{*}\omega)(\gamma,\,a)\bigr{)}(v,\,r)$
$\displaystyle\,=\,\omega(\gamma.a)(\widetilde{\theta}_{(\gamma,a)}(v))$
$\displaystyle\,=\,\omega(a)(v)$
$\displaystyle\,=\,\bigl{(}\widehat{\widetilde{\mathcal{H}}}({\rm
pr}_{2}^{*}\omega)(\gamma,a)\bigr{)}(v,\,r).$

The last equality establishes that $\omega$ is a differential $1$-form on the
Lie groupoid $[s^{*}E_{G}\rightrightarrows E_{G}]$.

To prove the converse, suppose that
$\omega\,\in\,\Omega^{1}(E_{G},\,\mathfrak{g})$ is a connection $1$-form on
the underlying principal $G$-bundle $E_{G}\,\longrightarrow\,X_{0}$. Assume
$\omega$ is a $\mathfrak{g}$-valued $1$-form on the Lie groupoid
$[s^{*}E_{G}\rightrightarrows E_{G}]$.

Let $\mathcal{D}\,\colon\,\text{At}(E_{G})\,\longrightarrow\,\text{ad}(E_{G})$
be the homomorphism of vector bundles given by $\omega$. To prove the converse
it suffices to show that the morphism $\mathcal{D}$ is a morphism of vector
bundles over the Lie groupoid $\mathbb{X}$.

Since $\omega$ is a differential $1$-form on $[s^{*}E_{G}\rightrightarrows
E_{G}]$, we have,

$\bigl{(}\widehat{\widetilde{\mathcal{H}}}(\mu^{*}\omega)(\gamma,\,a)\bigr{)}(v,\,r)\,=\,\bigl{(}\widehat{\widetilde{\mathcal{H}}}({\rm
pr}_{2}^{*}\omega)(\gamma,\,a)\bigr{)}(v,\,r)$

for $(\gamma,\,a)\,\in\,s^{*}E_{G}$ and
$(v,\,r)\,\in\,T_{(\gamma,a)}(s^{*}E_{G})$. Observe that the two equations

$\bigl{(}\widehat{\widetilde{\mathcal{H}}}(\mu^{*}\omega)(\gamma,a)\bigr{)}(v,r)\,=\,\omega(\gamma\cdot
a)(\widetilde{\theta}_{(\gamma,\,a)}(v)),\
\bigl{(}\widehat{\widetilde{\mathcal{H}}}({\rm
pr}_{2}^{*}\omega)(\gamma,\,a)\bigr{)}(v,\,r)\,=\,\bigl{(}\omega(a)\bigr{)}(v)$

together imply that

$\bigl{(}\omega(\gamma\cdot
a)\bigr{)}(\widetilde{\theta}_{(\gamma,\,a)}(v))\,=\,\bigl{(}\omega(a)\bigr{)}(v).$

After plugging the last equality into the equation

$\overline{\mathcal{D}}(\gamma\cdot(a,\,v))\,=\,(\gamma\cdot
a,\omega(\gamma\cdot a)(\widetilde{\theta}_{(\gamma,\,a)}(v)))\,,$

and comparing with
$\gamma\cdot\overline{\mathcal{D}}((a,\,v))\,=\,(\gamma\cdot a,\omega(a)(v))$,
we obtain that

$\displaystyle\\{\overline{\mathcal{D}}(\gamma\cdot
r)\,=\,\gamma\cdot\overline{\mathcal{D}}(r)\\}\,\Longrightarrow\,\\{{\mathcal{D}}(\gamma\cdot[r])\,=\,\gamma\cdot{\mathcal{D}}([r])\\}.$

Thus, $\mathcal{D}$ gives a connection on the principal $G$-bundle
$(E_{G}\rightarrow X_{0},\ \mathbb{X})$. This completes the proof. ∎

We now recall the Lie bracket operation on the Lie algebra valued differential
forms on a smooth manifold $Y$. Let $\mathfrak{g}$ be a Lie algebra. Let
$\omega$ and $\eta$ be $\mathfrak{g}$-valued differential $k$-form and
$l$-form respectively on $Y$. Then, the Lie bracket $[\omega,\,\eta]$ is a
$\mathfrak{g}$-valued differential $(k+l)$-form on $Y$ which is constructed as
follows:

$[\omega,\,\eta](a)(v_{1},\,\cdots,\,v_{k+l})$ (5.1)
$=\,\frac{1}{(k+l)!}\sum_{\sigma\in\Sigma_{k+l}}\text{sgn}(\sigma)[\omega(a)(v_{\sigma(1)},\cdots,v_{\sigma(k)}),\omega(a)(v_{\sigma(k+1)},\cdots,v_{\sigma(k+l)})]$

for all $a\,\in\,Y$ and $v_{i}\,\in\,T_{a}Y$, $1\,\leq\,i\,\leq\,k+l$ and
where $\Sigma_{n}$ denotes the symmetric group of order $n!$.

We then have the following basic property of connection forms on Lie
groupoids.

######  Lemma 5.3.

Let $\bigl{(}\mathbb{X}\,=\,[X_{1}\rightrightarrows
X_{0}],\,\mathcal{H}\bigr{)}$ be a Lie groupoid with an integrable connection,
and let
$(E_{G}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}X_{0},\,\mathbb{X})$ be
a principal $G$-bundle over $\mathbb{X}$. Let
$\widetilde{\mathcal{H}}\,\subset\,T(s^{*}E_{G})$ be the pullback of the
connection $\mathcal{H}$ to the Lie groupoid $[s^{*}E_{G}\rightrightarrows
E_{G}]$. Let $\mathcal{D}$ be a connection on the principal $G$-bundle $E_{G}$
on $\mathbb{X}$, and let
$\omega\,\colon\,E_{G}\,\,\longrightarrow\,T^{*}E_{G}\otimes\mathfrak{g}$ be
the connection $1$-form corresponding to $\mathcal{D}$ (see Lemma 5.2). Then
the Lie bracket $[\omega,\,\omega]\,\in\,\Omega^{2}(E_{G},\,{\mathfrak{g}})$
is a differential $2$-form on the Lie groupoid with connection
$\bigl{(}[s^{*}E_{G}\rightrightarrows
E_{G}],\,{\widetilde{\mathcal{H}}}\bigr{)}$.

###### Proof.

By definition,
$[\omega,\omega](a)(v_{1},\,v_{2})\,=\,[\omega(a)(v_{1}),\,\omega(a)(v_{2})]$
for all $a\,\in\,E_{G}$ and $v_{1},\,v_{2}\,\in\,T_{a}E_{G}$. For
$(\gamma,a)\,\in\,s^{*}E_{G}$ and
$(v_{1},\,r_{1}),\,(v_{2},\,r_{2})\,\in\,T_{(\gamma,a)}(s^{*}E_{G})$, we
observe that

$\displaystyle\widehat{\widetilde{\mathcal{H}}}({\rm
pr}_{2}^{*}([\omega,\omega]))(\gamma,a)$
$\displaystyle((v_{1},\,r_{1}),\,(v_{2},\,r_{2}))\,=\,[\omega(a)(v_{1}),\,\omega(a)(v_{2})],$
$\displaystyle\widehat{\widetilde{\mathcal{H}}}(\mu^{*}([\omega,\,\omega]))(\gamma,\,a)$
$\displaystyle((v_{1},\,r_{1}),(v_{2},\,r_{2}))\,=\,[\omega(\gamma\cdot
a)(\widetilde{\theta}_{(\gamma,a)}(v_{1})),\,\omega(\gamma\cdot
a)(\widetilde{\theta}_{(\gamma,\,a)}(v_{2}))].$

As it was seen in the proof of Lemma 5.2, $\omega(\gamma\cdot
a)(\widetilde{\theta}_{(\gamma,\,a)}(v,\,r))\,=\,\omega(a)(v)$ for all
$(\gamma,\,a)\,\in\,s^{*}E_{G}$ and for all
$(v,\,r)\,\in\,T_{(\gamma,a)}(s^{*}E_{G})$. In particular,

$\displaystyle\widehat{\widetilde{\mathcal{H}}}(\mu^{*}([\omega,\,\omega]))(\gamma,a)((v_{1},\,r_{1}),(v_{2},\,r_{2}))$
$\displaystyle=\,[\omega(\gamma\cdot
a)(\widetilde{\theta}_{(\gamma,a)}(v_{1})),\,\omega(\gamma\cdot
a)(\widetilde{\theta}_{(\gamma,a)}(v_{2}))]$
$\displaystyle=\,[\omega(a)(r_{1}),\,\omega(a)(r_{2})]$
$\displaystyle=\,\bigl{(}\widehat{\widetilde{\mathcal{H}}}({\rm
pr}_{2}^{*}([\omega,\,\omega]))(\gamma,\,a)\bigr{)}\bigl{(}(v_{1},\,r_{1}),(v_{2},\,r_{2})\bigr{)}.$

Thus, we have established that

$\widehat{\widetilde{\mathcal{H}}}(\mu^{*}([\omega,\omega]))\,=\,\widehat{\widetilde{\mathcal{H}}}({\rm
pr}_{2}^{*}([\omega,\,\omega]))\,.$

So $[\omega,\,\omega]\,\in\,\Omega^{2}(E_{G},\,{\mathfrak{g}})$ is a
differential $2$-form on the Lie groupoid