In mathematics, a Lie groupoid is a groupoid where the set Ob {\displaystyle \operatorname {Ob} } of objects and the set Mor {\displaystyle \operatorname {Mor} } of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations
s , t : Mor → Ob {\displaystyle s,t:\operatorname {Mor} \to \operatorname {Ob} }
are submersions. A Lie groupoid can thus be thought of as a "many-object generalization" of a Lie group, 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 algebroids. Lie groupoids were introduced by Charles Ehresmann under the name differentiable groupoids.
Definition and basic concepts A Lie groupoid consists of
two smooth manifolds G {\displaystyle G} and M {\displaystyle M}
two surjective submersions s , t : G → M {\displaystyle s,t:G\to M} (called, respectively, source and target projections) a map m : G ( 2 ) := { ( g , h ) ∣ s ( g ) = t ( h ) } → G {\displaystyle m:G^{(2)}:=\{(g,h)\mid s(g)=t(h)\}\to G} (called multiplication or composition map), where we use the notation g h := m ( g , h ) {\displaystyle gh:=m(g,h)}
a map u : M → G {\displaystyle u:M\to G} (called unit map or object inclusion map), where we use the notation 1 x := u ( x ) {\displaystyle 1_{x}:=u(x)}
a map i : G → G {\displaystyle i:G\to G} (called inversion), where we use the notation g − 1 := i ( g ) {\displaystyle g^{-1}:=i(g)}
such that
the composition satisfies s ( g h ) = s ( h ) {\displaystyle s(gh)=s(h)} and t ( g h ) = t ( g ) {\displaystyle t(gh)=t(g)} for every g , h ∈ G {\displaystyle g,h\in G} for which the composition is defined the composition is associative, i.e. g ( h l ) = ( g h ) l {\displaystyle g(hl)=(gh)l} for every g , h , l ∈ G {\displaystyle g,h,l\in G} for which the composition is defined
u {\displaystyle u} works as an identity, i.e. s ( 1 x ) = t ( 1 x ) = x {\displaystyle s(1_{x})=t(1_{x})=x} for every x ∈ M {\displaystyle x\in M} and g 1 s ( g ) = g {\displaystyle g1_{s(g)}=g} and 1 t ( g ) g = g {\displaystyle 1_{t(g)}g=g} for every g ∈ G {\displaystyle g\in G} for which the respective compositions are defined
i {\displaystyle i} works as an inverse, i.e. g − 1 g = 1 s ( g ) {\displaystyle g^{-1}g=1_{s(g)}} and g g − 1 = 1 t ( g ) {\displaystyle gg^{-1}=1_{t(g)}} for every g ∈ G {\displaystyle g\in G} . Using the language of category theory, a Lie groupoid can be more compactly defined as a groupoid (i.e. a small category where all the morphisms are invertible) such that the sets M {\displaystyle M} of objects and G {\displaystyle G} of morphisms are manifolds, the maps s {\displaystyle s} , t {\displaystyle t} , m {\displaystyle m} , i {\displaystyle i} and u {\displaystyle u} are smooth and s {\displaystyle s} and t {\displaystyle t} are submersions. A Lie groupoid is therefore not simply a groupoid object in the category of smooth manifolds: one has to ask the additional property that s {\displaystyle s} and t {\displaystyle t} are submersions. Lie groupoids are often denoted by G ⇉ M {\displaystyle G\rightrightarrows M} , where the two arrows represent the source and the target. The notation G 1 ⇉ G 0 {\displaystyle G_{1}\rightrightarrows G_{0}} is also frequently used, especially when stressing the simplicial structure of the associated nerve. In order to include more natural examples, the manifold G {\displaystyle G} is not required in general to be Hausdorff or second countable (while M {\displaystyle M} and all other spaces are).
Alternative definitions The original definition by Ehresmann required G {\displaystyle G} and M {\displaystyle M} to possess a smooth structure such that only m {\displaystyle m} is smooth and the maps g ↦ 1 s ( g ) {\displaystyle g\mapsto 1_{s(g)}} and g ↦ 1 t ( g ) {\displaystyle g\mapsto 1_{t(g)}} 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 s {\displaystyle s} and t {\displaystyle t} 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 G ⇉ M {\displaystyle G\rightrightarrows M} are smooth submersions has some immediate consequences:
the s {\displaystyle s} -fibres s − 1 ( x ) ⊆ G {\displaystyle s^{-1}(x)\subseteq G} , the t {\displaystyle t} -fibres t − 1 ( x ) ⊆ G {\displaystyle t^{-1}(x)\subseteq G} , and the set of composable morphisms G ( 2 ) ⊆ G × G {\displaystyle G^{(2)}\subseteq G\times G} are submanifolds; the inversion map i {\displaystyle i} is a diffeomorphism; the unit map u {\displaystyle u} is a smooth embedding; the isotropy groups G x {\displaystyle G_{x}} are Lie groups; the orbits O x ⊆ M {\displaystyle {\mathcal {O}}_{x}\subseteq M} are immersed submanifolds; the s {\displaystyle s} -fibre s − 1 ( x ) {\displaystyle s^{-1}(x)} at a point x ∈ M {\displaystyle x\in M} is a principal G x {\displaystyle G_{x}} -bundle over the orbit O x {\displaystyle {\mathcal {O}}_{x}} at that point.
Subobjects and morphisms A Lie subgroupoid of a Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} is a subgroupoid H ⇉ N {\displaystyle H\rightrightarrows N} (i.e. a subcategory of the category G {\displaystyle G} ) with the extra requirement that H ⊆ G {\displaystyle H\subseteq G} is an immersed submanifold. As for a subcategory, a (Lie) subgroupoid is called wide if N = M {\displaystyle N=M} . Any Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} has two canonical wide subgroupoids:
the unit/identity Lie subgroupoid u ( M ) = { 1 x ∈ G ∣ x ∈ M } {\displaystyle u(M)=\{1_{x}\in G\mid x\in M\}} ; the inner subgroupoid I G := { g ∈ G ∣ s ( g ) = t ( g ) } {\displaystyle IG:=\{g\in G\mid s(g)=t(g)\}} , 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 H ⊆ G {\displaystyle H\subseteq G} inside I G {\displaystyle IG} such that, for every h ∈ H , g ∈ G {\displaystyle h\in H,g\in G} with s ( h ) = t ( h ) = s ( g ) {\displaystyle s(h)=t(h)=s(g)} , one has g h g − 1 ∈ H {\displaystyle ghg^{-1}\in H} . The isotropy groups of H {\displaystyle H} are therefore normal subgroups of the isotropy groups of G {\displaystyle G} . A Lie groupoid morphism between two Lie groupoids G ⇉ M {\displaystyle G\rightrightarrows M} and H ⇉ N {\displaystyle H\rightrightarrows N} is a groupoid morphism F : G → H , f : M → N {\displaystyle F:G\to H,f:M\to N} (i.e. a functor between the categories G {\displaystyle G} and H {\displaystyle H} ), where both F {\displaystyle F} and f {\displaystyle f} are smooth. The kernel ker ( F ) := { g ∈ G ∣ F ( g ) = 1 s ( g ) } {\displaystyle \ker(F):=\{g\in G\mid F(g)=1_{s(g)}\}} of a morphism between Lie groupoids over the same base manifold is automatically a normal Lie subgroupoid. The quotient G / ker ( F ) ⇉ M {\displaystyle G/\ker(F)\rightrightarrows M} has a natural groupoid structure such that the projection G → G / ker ( F ) {\displaystyle G\to G/\ker(F)} is a groupoid morphism; however, unlike quotients of Lie groups, G / ker ( F ) {\displaystyle G/\ker(F)} may fail to be a Lie groupoid in general. Accordingly, the isomorphism theorems 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 G a b = G / ( I G , I G ) {\displaystyle G^{ab}=G/(IG,IG)} may not exist or be smooth.
Bisections A bisection of a Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} is a smooth map b : M → G {\displaystyle b:M\to G} such that s ∘ b = i d M {\displaystyle s\circ b=id_{M}} and t ∘ b {\displaystyle t\circ b} is a diffeomorphism of M {\displaystyle M} . In order to overcome the lack of symmetry between the source and the target, a bisection can be equivalently defined as a submanifold B ⊆ G {\displaystyle B\subseteq G} such that s ∣ B : B → M {\displaystyle s_{\mid B}:B\to M} and t ∣ B : B → M {\displaystyle t_{\mid B}:B\to M} are diffeomorphisms; the relation between the two definitions is given by B = b ( M ) {\displaystyle B=b(M)} . The set of bisections forms a group, with the multiplication b 1 ⋅ b 2 {\displaystyle b_{1}\cdot b_{2}} defined as ( b 1 ⋅ b 2 ) ( x ) := b 1 ( b 2 ( x ) ) b 2 ( x ) . {\displaystyle (b_{1}\cdot b_{2})(x):=b_{1}(b_{2}(x))b_{2}(x).} and inversion defined as b 1 − 1 ( x ) := i ∘ b 1 ( ( t ∘ b 2 ) − 1 ( x ) ) {\displaystyle b_{1}^{-1}(x):=i\circ b_{1}\left((t\circ b_{2})^{-1}(x)\right)} Note that the definition is given in such a way that, if t ∘ b 1 = ϕ 1 {\displaystyle t\circ b_{1}=\phi _{1}} and t ∘ b 2 = ϕ 2 {\displaystyle t\circ b_{2}=\phi _{2}} , then t ∘ ( b 1 ⋅ b 2 ) = ϕ 1 ∘ ϕ 2 {\displaystyle t\circ (b_{1}\cdot b_{2})=\phi _{1}\circ \phi _{2}} and t ∘ b 1 − 1 = ϕ 1 − 1 {\displaystyle t\circ b_{1}^{-1}=\phi _{1}^{-1}} . The group of bisections can be given the compact-open topology, 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 b : U ⊆ M → G {\displaystyle b:U\subseteq M\to G} is defined analogously, but the multiplication between local bisections is of course only partially defined.
Examples
Trivial and extreme cases Lie groupoids G ⇉ ∗ {\displaystyle G\rightrightarrows {*}} with one object are the same thing as Lie groups. Given any manifold M {\displaystyle M} , there is a Lie groupoid M × M ⇉ M {\displaystyle M\times M\rightrightarrows M} 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 M × G × M ⇉ M {\displaystyle M\times G\times M\rightrightarrows M} , with structure maps s ( x , g , y ) = y {\displaystyle s(x,g,y)=y} , t ( x , g , y ) = x {\displaystyle t(x,g,y)=x} , m ( ( x , g , y ) , ( y , h , z ) ) = ( x , g h , z ) {\displaystyle m((x,g,y),(y,h,z))=(x,gh,z)} , u ( x ) = ( x , 1 , x ) {\displaystyle u(x)=(x,1,x)} and i ( x , g , y ) = ( y , g − 1 , x ) {\displaystyle i(x,g,y)=(y,g^{-1},x)} . Given any manifold M {\displaystyle M} , there is a Lie groupoid u ( M ) ⇉ M {\displaystyle u(M)\rightrightarrows M} 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 s = t {\displaystyle s=t} 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 G ⇉ M {\displaystyle G\rightrightarrows M} and a surjective submersion μ : N → M {\displaystyle \mu :N\to M} , there is a Lie groupoid μ ∗ G ⇉ N {\displaystyle \mu ^{*}G\rightrightarrows N} , called its pullback groupoid or induced groupoid, where μ ∗ G ⊆ N × G × N {\displaystyle \mu ^{*}G\subseteq N\times G\times N} contains triples ( x , g , y ) {\displaystyle (x,g,y)} such that s ( g ) = μ ( y ) {\displaystyle s(g)=\mu (y)} and t ( g ) = μ ( x ) {\displaystyle t(g)=\mu (x)} , and the multiplication is defined using the multiplication of G {\displaystyle G} . For instance, the pullback of the pair groupoid of M {\displaystyle M} is the pair groupoid of N {\displaystyle N} . Given any two Lie groupoids G 1 ⇉ M 1 {\displaystyle G_{1}\rightrightarrows M_{1}} and G 2 ⇉ M 2 {\displaystyle G_{2}\rightrightarrows M_{2}} , there is a Lie groupoid G 1 × G 2 ⇉ M 1 × M 2 {\displaystyle G_{1}\times G_{2}\rightrightarrows M_{1}\times M_{2}} , called their direct product, such that the groupoid morphisms G 1 × G 2 → p r M 1 ∗ G 1 {\displaystyle G_{1}\times G_{2}\to \mathrm {pr} _{M_{1}}^{*}G_{1}} and G 1 × G 2 → p r M 2 ∗ G 2 {\displaystyle G_{1}\times G_{2}\to \mathrm {pr} _{M_{2}}^{*}G_{2}} are surjective submersions. Given any Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} , there is a Lie groupoid T G ⇉ T M {\displaystyle TG\rightrightarrows TM} , called its tangent groupoid, obtained by considering the tangent bundle of G {\displaystyle G} and M {\displaystyle M} and the differential of the structure maps. Given any Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} , there is a Lie groupoid T ∗ G ⇉ A ∗ {\displaystyle T^{*}G\rightrightarrows A^{*}} , called its cotangent groupoid obtained by considering the cotangent bundle of G {\displaystyle G} , the dual of the Lie algebroid A {\displaystyle A} (see below), and suitable structure maps involving the differentials of the left and right translations. Given any Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} , there is a Lie groupoid J k G ⇉ M {\displaystyle J^{k}G\rightrightarrows M} , called its jet groupoid, obtained by considering the k-jets of the local bisections of G {\displaystyle G} (with smooth structure inherited from the jet bundle of s : G → M {\displaystyle s:G\to M} ) and setting s ( j x k b ) = x {\displaystyle s(j_{x}^{k}b)=x} , t ( j x k b ) = t ( b ( x ) ) {\displaystyle t(j_{x}^{k}b)=t(b(x))} , m ( j t ( b ( x ) ) k b 1 , j x k b 2 ) = j x k ( b 1 ⋅ b 2 ) {\displaystyle m(j_{t(b(x))}^{k}b_{1},j_{x}^{k}b_{2})=j_{x}^{k}(b_{1}\cdot b_{2})} , u ( x ) = j x k u {\displaystyle u(x)=j_{x}^{k}u} and i ( j x k b ) = j t ( b ( x ) ) k b − 1 {\displaystyle i(j_{x}^{k}b)=j_{t(b(x))}^{k}b^{-1}} .
Examples from differential geometry Given a submersion μ : M → N {\displaystyle \mu :M\to N} , there is a Lie groupoid M × μ M := { ( x , y ) ∈ M × M ∣ μ ( x ) = μ ( y ) } ⇉ M {\displaystyle M\times _{\mu }M:=\{(x,y)\in M\times M\mid \mu (x)=\mu (y)\}\rightrightarrows M} , called the submersion groupoid or fibred pair groupoid, whose structure maps are induced from the pair groupoid M × M ⇉ M {\displaystyle M\times M\rightrightarrows M} (the condition that μ {\displaystyle \mu } is a submersion ensures the smoothness of M × μ M {\displaystyle M\times _{\mu }M} ). If N {\displaystyle N} is a point, one recovers the pair groupoid. Given a Lie group G {\displaystyle G} acting on a manifold M {\displaystyle M} , there is a Lie groupoid G × M ⇉ M {\displaystyle G\times M\rightrightarrows M} , called the action groupoid or translation groupoid, with one morphism for each triple g ∈ G , x , y ∈ M {\displaystyle g\in G,x,y\in M} with g x = y {\displaystyle gx=y} . Given any vector bundle E → M {\displaystyle E\to M} , there is a Lie groupoid G L ( E ) ⇉ M {\displaystyle GL(E)\rightrightarrows M} , called the general linear groupoid, with morphisms between x , y ∈ M {\displaystyle x,y\in M} being linear isomorphisms between the fibres E x {\displaystyle E_{x}} and E y {\displaystyle E_{y}} . For instance, if E = M × R n {\displaystyle E=M\times \mathbb {R} ^{n}} is the trivial vector bundle of rank k {\displaystyle k} , then G L ( E ) ⇉ M {\displaystyle GL(E)\rightrightarrows M} is the action groupoid. Any principal bundle P → M {\displaystyle P\to M} with structure group G {\displaystyle G} defines a Lie groupoid ( P × P ) / G ⇉ M {\displaystyle (P\times P)/G\rightrightarrows M} , where G {\displaystyle G} acts on the pairs ( p , q ) ∈ P × P {\displaystyle (p,q)\in P\times P} componentwise, called the gauge groupoid. The multiplication is defined via compatible representatives as in the pair groupoid. Any foliation F {\displaystyle {\mathcal {F}}} on a manifold M {\displaystyle M} defines two Lie groupoids, M o n ( F ) ⇉ M {\displaystyle \mathrm {Mon} ({\mathcal {F}})\rightrightarrows M} (or Π 1 ( F ) ⇉ M {\displaystyle \Pi _{1}({\mathcal {F}})\rightrightarrows M} ) and H o l ( F ) ⇉ M {\displaystyle \mathrm {Hol} ({\mathcal {F}})\rightrightarrows M} , called respectively the monodromy/homotopy/fundamental groupoid and the holonomy groupoid of F {\displaystyle {\mathcal {F}}} , whose morphisms consist of the homotopy, respectively holonomy, equivalence classes of paths entirely lying in a leaf of F {\displaystyle {\mathcal {F}}} . For instance, when F {\displaystyle {\mathcal {F}}} is the trivial foliation with only one leaf, one recovers, respectively, the fundamental groupoid and the pair groupoid of M {\displaystyle M} . On the other hand, when F {\displaystyle {\mathcal {F}}} is a simple foliation, i.e. the foliation by (connected) fibres of a submersion μ : M → N {\displaystyle \mu :M\to N} , its holonomy groupoid is precisely the submersion groupoid M × μ M {\displaystyle M\times _{\mu }M} 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 Γ ⊆ D i f f l o c ( M ) {\displaystyle \Gamma \subseteq \mathrm {Diff} _{loc}(M)} , there is a Lie groupoid G = G e r m ( Γ ) ⇉ M {\displaystyle G=\mathrm {Germ} (\Gamma )\rightrightarrows M} , 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 ( s , t ) : G → M × M {\displaystyle (s,t):G\to M\times M} (also known as the anchor of G ⇉ M {\displaystyle G\rightrightarrows M} ) 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