In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Lie groupoid is a
groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
* '' Group'' with a partial fu ...
where the set
of
objects and the set
of
morphism
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...
s are both
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...
s, all the
category
Category, plural categories, may refer to:
General uses
*Classification, the general act of allocating things to classes/categories Philosophy
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
* Category ( ...
operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
:
are
submersions.
A Lie groupoid can thus be thought of as a "many-object generalization" of a
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
, just as a groupoid is a many-object generalization of a
group. Accordingly, while Lie groups provide a natural model for (classical)
continuous symmetries, Lie groupoids are often used as model for (and arise from) generalised, point-dependent symmetries. Extending the
correspondence between Lie groups and Lie algebras, Lie groupoids are the global counterparts of
Lie algebroid
In mathematics, a Lie algebroid is a vector bundle A \rightarrow M together with a Lie bracket on its space of sections \Gamma(A) and a vector bundle morphism \rho: A \rightarrow TM, satisfying a Leibniz rule. A Lie algebroid can thus be thought ...
s.
Lie groupoids were introduced by
Charles Ehresmann
Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory.
He was an early member of the Bourbaki group, and is known for his work on the differentia ...
under the name ''differentiable groupoids''.
Definition and basic concepts
A Lie groupoid consists of
* two smooth manifolds
and
* two
surjective
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
submersions (called, respectively, source and target projections)
* a map
(called multiplication or composition map), where we use the notation
* a map
(called unit map or object inclusion map), where we use the notation
* a map
(called inversion), where we use the notation
such that
* the composition satisfies
and
for every
for which the composition is defined
*the composition is
associative
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
, i.e.
for every
for which the composition is defined
*
works as an
identity, i.e.
for every
and
and
for every
*
works as an
inverse, i.e.
and
for every
.
Using the language of
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, a Lie groupoid can be more compactly defined as a
groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
* '' Group'' with a partial fu ...
(i.e. a
small category
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...
where all the morphisms are invertible) such that the sets
of objects and
of morphisms are manifolds, the maps
,
,
,
and
are smooth and
and
are submersions. A Lie groupoid is therefore not simply a
groupoid object In category theory, a branch of mathematics, a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when the multiplication is only partially defined.
Defini ...
in the
category of smooth manifolds: one has to ask the additional property that
and
are submersions.
Lie groupoids are often denoted by
, where the two arrows represent the source and the target. The notation
is also frequently used, especially when stressing the simplicial structure of the associated
nerve
A nerve is an enclosed, cable-like bundle of nerve fibers (called axons). Nerves have historically been considered the basic units of the peripheral nervous system. A nerve provides a common pathway for the Electrochemistry, electrochemical nerv ...
.
In order to include more natural examples, the manifold
is not required in general to be
Hausdorff or
second countable (while
and all other spaces are).
Alternative definitions
The original definition by Ehresmann required
and
to possess a smooth structure such that only
is smooth and the maps
and
are subimmersions (i.e. have locally constant
rank). Such definition proved to be too weak and was replaced by Pradines with the one currently used.
While some authors introduced weaker definitions which did not require
and
to be submersions, these properties are fundamental to develop the entire Lie theory of groupoids and algebroids.
First properties
The fact that the source and the target map of a Lie groupoid
are smooth submersions has some immediate consequences:
* the
-fibres
, the
-fibres
, and the set of composable morphisms
are
submanifold
In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...
s;
* the inversion map
is a
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable.
Definit ...
;
*the unit map
is a
smooth embedding;
*the
isotropy groups are
Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
s;
* the orbits
are
immersed submanifolds;
* the
-fibre
at a point
is a
principal -bundle over the orbit
at that point.
Subobjects and morphisms
A Lie subgroupoid of a Lie groupoid
is a
subgroupoid (i.e. a
subcategory
In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...
of the category
) with the extra requirement that
is an immersed submanifold. As for a subcategory, a (Lie) subgroupoid is called wide if
. Any Lie groupoid
has two canonical wide subgroupoids:
* the unit/identity Lie subgroupoid
;
* the inner subgroupoid
, i.e. the bundle of isotropy groups (which however may fail to be smooth in general).
A normal Lie subgroupoid is a wide Lie subgroupoid
inside
such that, for every
with
, one has
. The isotropy groups of
are therefore
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
s of the isotropy groups of
.
A Lie groupoid morphism between two Lie groupoids
and
is a groupoid morphism
(i.e. a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
between the categories
and
), where both
and
are smooth. The
kernel of a morphism between Lie groupoids over the same base manifold is automatically a normal Lie subgroupoid.
The
quotient has a natural groupoid structure such that the projection
is a groupoid morphism; however, unlike
quotients of Lie groups,
may fail to be a Lie groupoid in general. Accordingly, the
isomorphism theorems
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship among quotients, homomorphisms, and subobjects. Versions of the theorems exist f ...
for groupoids cannot be specialised to the entire category of Lie groupoids, but only to special classes.
A Lie groupoid is called abelian if its isotropy Lie groups are
abelian. For similar reasons as above, while the definition of
abelianisation of a group extends to set-theoretical groupoids, in the Lie case the analogue of the quotient
may not exist or be smooth.
Bisections
A bisection of a Lie groupoid
is a smooth map
such that
and
is a diffeomorphism of
. In order to overcome the lack of symmetry between the source and the target, a bisection can be equivalently defined as a submanifold
such that
and
are diffeomorphisms; the relation between the two definitions is given by
.
The set of bisections forms a
group, with the multiplication
defined as
and inversion defined as
Note that the definition is given in such a way that, if
and
, then
and
.
The group of bisections can be given the
compact-open topology
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory ...
, as well as an (infinite-dimensional) structure of
Fréchet manifold compatible with the group structure, making it into a Fréchet-Lie group.
A local bisection
is defined analogously, but the multiplication between local bisections is of course only partially defined.
Examples
Trivial and extreme cases
*Lie groupoids
with one object are the same thing as Lie groups.
*Given any manifold
, there is a Lie groupoid
called the pair groupoid, with precisely one morphism from any object to any other.
*The two previous examples are particular cases of the trivial groupoid
, with structure maps
,
,
,
and
.
*Given any manifold
, there is a Lie groupoid
called the unit groupoid, with precisely one morphism from one object to itself, namely the identity, and no morphisms between different objects.
*More generally, Lie groupoids with
are the same thing as bundle of Lie groups (not necessarily locally trivial). For instance, any vector bundle is a bundle of abelian groups, so it is in particular a(n abelian) Lie groupoid.
Constructions from other Lie groupoids
*Given any Lie groupoid
and a surjective submersion
, there is a Lie groupoid
, called its pullback groupoid or induced groupoid, where
contains triples
such that
and
, and the multiplication is defined using the multiplication of
. For instance, the pullback of the pair groupoid of
is the pair groupoid of
.
*Given any two Lie groupoids
and
, there is a Lie groupoid
, called their
direct product, such that the groupoid morphisms
and
are surjective submersions.
*Given any Lie groupoid
, there is a Lie groupoid
, called its tangent groupoid, obtained by considering the
tangent bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is ...
of
and
and the
differential of the structure maps.
*Given any Lie groupoid
, there is a Lie groupoid
, called its cotangent groupoid obtained by considering the
cotangent bundle of
, the
dual of the Lie algebroid
(see below), and suitable structure maps involving the differentials of the left and right translations.
*Given any Lie groupoid
, there is a Lie groupoid
, called its jet groupoid, obtained by considering the
k-jets of the local bisections of
(with smooth structure inherited from the
jet bundle of
) and setting
,
,
,
and
.
Examples from differential geometry
*Given a submersion
, there is a Lie groupoid
, called the submersion groupoid or fibred pair groupoid, whose structure maps are induced from the pair groupoid
(the condition that
is a submersion ensures the smoothness of
). If
is a point, one recovers the pair groupoid.
*Given a Lie group
acting on a manifold
, there is a Lie groupoid
, called the action groupoid or translation groupoid, with one morphism for each triple
with
.
*Given any
vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...
, there is a Lie groupoid
, called the general linear groupoid, with morphisms between
being linear isomorphisms between the fibres
and
. For instance, if
is the trivial vector bundle of rank
, then
is the action groupoid.
*Any
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equ ...
with structure group ''
'' defines a Lie groupoid
, where ''
'' acts on the pairs
componentwise, called the gauge groupoid. The multiplication is defined via compatible representatives as in the pair groupoid.
*Any
foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an topological manifold, ''n''-manifold, the equivalence classes being connected, injective function, injectively immersed submanifolds, all of the same dimension ...
on a manifold
defines two Lie groupoids,
(or
) and
, called respectively the monodromy/homotopy/fundamental groupoid and the holonomy groupoid of
, whose morphisms consist of the
homotopy
In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. ...
, respectively
holonomy, equivalence classes of paths entirely lying in a leaf of
. For instance, when
is the trivial foliation with only one leaf, one recovers, respectively, the fundamental groupoid and the pair groupoid of
. On the other hand, when
is a simple foliation, i.e. the foliation by (connected) fibres of a submersion
, its holonomy groupoid is precisely the submersion groupoid
but its monodromy groupoid may even fail to be Hausdorff, due to a general criterion in terms of vanishing cycles. In general, many elementary foliations give rise to monodromy and holonomy groupoids which are not Hausdorff.
*Given any
pseudogroup , there is a Lie groupoid
, called its germ groupoid, endowed with the sheaf topology and with structure maps analogous to those of the jet groupoid. This is another natural example of Lie groupoid whose arrow space is not Hausdorff nor second countable.
Important classes of Lie groupoids
Note that some of the following classes make sense already in the category of set-theoretical or
topological groupoids.
Transitive groupoids
A Lie groupoid is transitive (in older literature also called connected) if it satisfies one of the following equivalent conditions:
* there is only one orbit;
* there is at least a morphism between any two objects;
* the map
(also known as the anchor of
) is surjective.
Gauge groupoids constitute the prototypical examples of transitive Lie groupoids: indeed, any transitive Lie groupoid is isomorphic to the gauge groupoid of some principal bundle, namely the
-bundle
, for any point
. For instance:
* the trivial Lie groupoid
is transitive and arise from the trivial principal
-bundle
. As particular cases, Lie groups
and pair groupoids
are trivially transitive, and arise, respectively, from the principal
-bundle
, and from the principal
-bundle
;
* an action groupoid
is transitive if and only if the group action is
transitive, and in such case it arises from the principal bundle
with structure group the isotropy group (at an arbitrary point);
* the general linear groupoid of
is transitive, and arises from the
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(E) associated with any vector bundle ''E''. The fiber of F(E) over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E_x''. The general linear group acts naturally on ...
;
*pullback groupoids, jet groupoids and tangent groupoids of
are transitive if and only if
is transitive.
As a less trivial instance of the correspondence between transitive Lie groupoids and principal bundles, consider the
fundamental groupoid of a (connected) smooth manifold
. This is naturally a topological groupoid, which is moreover transitive; one can see that
is isomorphic to the gauge groupoid of the
universal cover
In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphism ...
of
. Accordingly,
inherits a smooth structure which makes it into a Lie groupoid.
Submersions groupoids
are an example of non-transitive Lie groupoids, whose orbits are precisely the fibres of
.
A stronger notion of transitivity requires the anchor
to be a surjective submersion. Such condition is also called local triviality, because
becomes locally isomorphic (as Lie groupoid) to a trivial groupoid over any open
(as a consequence of the local triviality of principal bundles).
When the space
is second countable, transitivity implies local triviality. Accordingly, these two conditions are equivalent for many examples but not for all of them: for instance, if
is a transitive pseudogroup, its germ groupoid
is transitive but not locally trivial.
Proper groupoids
A Lie groupoid is called proper if
is a
proper map. As a consequence
* all isotropy groups of
are
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
;
* all orbits of
are closed submanifolds;
* the orbit space
is
Hausdorff.
For instance:
* a Lie group is proper if and only if it is compact;
* pair groupoids are always proper;
*unit groupoids are always proper;
* an action groupoid is proper if and only if the action is
proper;
*the fundamental groupoid is proper if and only if the fundamental groups are
finite.
As seen above, properness for Lie groupoids is the "right" analogue of compactness for Lie groups. One could also consider more "natural" conditions, e.g. asking that the source map
is proper (then
is called s-proper), or that the entire space
is compact (then
is called compact), but these requirements turns out to be too strict for many examples and applications.
Étale groupoids
A Lie groupoid is called étale if it satisfies one of the following equivalent conditions:
* the dimensions of
and
are equal;
*
is a
local diffeomorphism In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below.
Form ...
;
* all the
-fibres are
discrete
Discrete may refer to:
*Discrete particle or quantum in physics, for example in quantum theory
* Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit
* Discrete group, ...
As a consequence, also the
-fibres, the isotropy groups and the orbits become discrete.
For instance:
* a Lie group is étale if and only if it is discrete;
*pair groupoids are never étale;
*unit groupoids are always étale;
*an action groupoid is étale if and only if
is discrete;
* germ groupoids of pseudogroups are always étale.
Effective groupoids
An étale groupoid is called effective if, for any two local bisections
, the condition
implies
. For instance:
* Lie groups are effective if and only if are trivial;
*unit groupoids are always effective;
*an action groupoid is effective if the
-action is
free and
is discrete.
In general, any effective étale groupoid arise as the germ groupoid of some pseudogroup. However, a (more involved) definition of effectiveness, which does not assume the étale property, can also be given.
Source-connected groupoids
A Lie groupoid is called
-connected if all its
-fibres are
connected. Similarly, one talks about
-simply connected groupoids (when the
-fibres are
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
) or source-k-connected groupoids (when the
-fibres are
k-connected
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concep ...
, i.e. the first
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
s are trivial).
Note that the entire space of arrows
is not asked to satisfy any connectedness hypothesis. However, if
is a source-
-connected Lie groupoid over a
-connected manifold, then
itself is automatically
-connected.
For instanceː
* Lie groups are source
-connected if and only if they are
-connected;
* a pair groupoid is source
-connected if and only if
is
-connected;
* unit groupoids are always source
-connected;
* action groupoids are source
-connected if and only if
is
-connected;
* monodromy groupoids (hence also fundamental groupoids) are source simply connected;
* a gauge groupoid associated to a principal bundle
is source
-connected if and only if the total space
is.
Further related concepts
Actions and principal bundles
Recall that an action of a groupoid
on a set
along a function
is defined via a collection of maps
for each morphism
between
. Accordingly, an action of a Lie groupoid
on a manifold
along a smooth map
consists of a groupoid action where the maps
are smooth. Of course, for every
there is an induced smooth action of the isotropy group
on the fibre
.
Given a Lie groupoid
, a principal
-bundle consists of a
-space
and a
-invariant surjective submersion
such that
is a diffeomorphism. Equivalent (but more involved) definitions can be given using
-valued cocycles or local trivialisations.
When
is a Lie groupoid over a point, one recovers, respectively, standard
Lie group action In differential geometry, a Lie group action is a group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable.
__TOC__
Definition
Let \sigma: G \times M \to M, (g, x) \mapsto g \cdot ...
s and
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equ ...
s.
Representations
A representation of a Lie groupoid
consists of a Lie groupoid action on a vector bundle
, such that the action is fibrewise linear, i.e. each bijection
is a linear isomorphism. Equivalently, a representation of
on
can be described as a Lie groupoid morphism from
to the general linear groupoid
.
Of course, any fibre
becomes a representation of the isotropy group
. More generally, representations of transitive Lie groupoids are uniquely determined by representations of their isotropy groups, via the construction of the
associated vector bundle.
Examples of Lie groupoids representations include the following:
* representations of Lie groups
recover standard
Lie group representations
* representations of pair groupoids
are trivial vector bundles
* representations of unit groupoids
are vector bundles
* representations of action groupoid
are
-
equivariant vector bundles
* representations of fundamental groupoids
are vector bundles endowed with
flat connections
The set
of isomorphism classes of representations of a Lie groupoid
has a natural structure of
semiring
In abstract algebra, a semiring is an algebraic structure. Semirings are a generalization of rings, dropping the requirement that each element must have an additive inverse. At the same time, semirings are a generalization of bounded distribu ...
, with direct sums and tensor products of vector bundles.
Differentiable cohomology
The notion of differentiable cohomology for Lie groups generalises naturally also to Lie groupoids: the definition relies on the
simplicial structure of the
nerve
A nerve is an enclosed, cable-like bundle of nerve fibers (called axons). Nerves have historically been considered the basic units of the peripheral nervous system. A nerve provides a common pathway for the Electrochemistry, electrochemical nerv ...
of
, viewed as a category.
More precisely, recall that the space
consists of strings of
composable morphisms, i.e.
and consider the map
.
A differentiable ''
''-cochain of
with coefficients in some representation
is a smooth section of the pullback vector bundle
. One denotes by
the space of such ''
''-cochains, and considers the differential
, defined as
Then
becomes a
cochain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel ...
and its cohomology, denoted by
, is called the differentiable cohomology of
with coefficients in
. Note that, since the differential at degree zero is
, one has always
.
Of course, the differentiable cohomology of
as a Lie groupoid coincides with the standard differentiable cohomology of
as a Lie group (in particular, for
discrete group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and ...
s one recovers the usual
group cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology ...
). On the other hand, for any ''proper'' Lie groupoid
, one can prove that
for every
.
The Lie algebroid of a Lie groupoid
Any Lie groupoid
has an associated
Lie algebroid
In mathematics, a Lie algebroid is a vector bundle A \rightarrow M together with a Lie bracket on its space of sections \Gamma(A) and a vector bundle morphism \rho: A \rightarrow TM, satisfying a Leibniz rule. A Lie algebroid can thus be thought ...
, obtained with a construction similar to the one which associates a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
to any Lie groupː
* the vector bundle
is the vertical bundle with respect to the source map, restricted to the elements tangent to the identities, i.e.
;
* the Lie bracket is obtained by identifying
with the left-invariant vector fields on
, and by transporting their Lie bracket to
;
* the anchor map
is the differential of the target map
restricted to
.
The
Lie group–Lie algebra correspondence
In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are Isomorphism, isomorphic to each other have Lie algebra ...
generalises to some extends also to Lie groupoids: the first two Lie's theorem (also known as the subgroups–subalgebras theorem and the homomorphisms theorem) can indeed be easily adapted to this setting.
In particular, as in standard Lie theory, for any s-connected Lie groupoid
there is a unique (up to isomorphism) s-simply connected Lie groupoid
with the same Lie algebroid of
, and a local diffeomorphism
which is a groupoid morphism. For instance,
* given any connected manifold
its pair groupoid
is s-connected but not s-simply connected, while its fundamental groupoid
is. They both have the same Lie algebroid, namely the tangent bundle
, and the local diffeomorphism
is given by
.
* given any foliation
on
, its holonomy groupoid
is s-connected but not s-simply connected, while its monodromy groupoid
is. They both have the same Lie algebroid, namely the foliation algebroid
, and the local diffeomorphism
is given by