Lie groupoid

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, the source and target operations 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, the source and target operations are submersions, and all the category operations (source and target, composition, and identity-assigning map) are smooth. 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. Just as every Lie group has a Lie algebra, every Lie groupoid has a Lie algebroid. Beside isomorphism of groupoids there is a more coarse notation of equivalence, the so-called Morita equivalence. A quite general example is the Morita-morphism of the Čech groupoid which goes as follows. Let M be a smooth manifold and { U α } {displaystyle {U_{alpha }}} an open cover of M. Define G 0 := ⨆ α U α {displaystyle G_{0}:=igsqcup _{alpha }U_{alpha }} the disjoint union with the obvious submersion p : G 0 → M {displaystyle p:G_{0} o M} . In order to encode the structure of the manifold M define the set of morphisms G 1 := ⨆ α , β U α β {displaystyle G_{1}:=igsqcup _{alpha ,eta }U_{alpha eta }} where U α β = U α ∩ U β ⊂ M {displaystyle U_{alpha eta }=U_{alpha }cap U_{eta }subset M} . The source and target map are defined as the embeddings s : U α β → U α {displaystyle s:U_{alpha eta } o U_{alpha }} and t : U α β → U β {displaystyle t:U_{alpha eta } o U_{eta }} . And multiplication is the obvious one if we read the U α β {displaystyle U_{alpha eta }} as subsets of M (compatible points in U α β {displaystyle U_{alpha eta }} and U β γ {displaystyle U_{eta gamma }} actually are the same in M and also lie in U α γ {displaystyle U_{alpha gamma }} ). This Čech groupoid is in fact the pullback groupoid of M ⇒ M {displaystyle MRightarrow M} , i.e. the trivial groupoid over M, under p. That is what makes it Morita-morphism. In order to get the notion of an equivalence relation we need to make the construction symmetric and show that it is also transitive. In this sense we say that 2 groupoids G 1 ⇒ G 0 {displaystyle G_{1}Rightarrow G_{0}} and H 1 ⇒ H 0 {displaystyle H_{1}Rightarrow H_{0}} are Morita equivalent iff there exists a third groupoid K 1 ⇒ K 0 {displaystyle K_{1}Rightarrow K_{0}} together with 2 Morita morphisms from G to K and H to K. Transitivity is an interesting construction in the category of groupoid principal bundles and left to the reader. It arises the question of what is preserved under the Morita equivalence. There are 2 obvious things, one the coarse quotient/ orbit space of the groupoid G 0 / G 1 = H 0 / H 1 {displaystyle G_{0}/G_{1}=H_{0}/H_{1}} and secondly the stabilizer groups G p ≅ H q {displaystyle G_{p}cong H_{q}} for corresponding points p ∈ G 0 {displaystyle pin G_{0}} and q ∈ H 0 {displaystyle qin H_{0}} .