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In [[mathematics]], a '''Lie groupoid''' is a [[groupoid]] where the set <math>\operatorname{Ob}</math> of [[object (category theory)|object]]s and the set <math>\operatorname{Mor}</math> of [[morphism]]s are both [[manifold]]s, all the [[category (mathematics)|category]] operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations :<math>s,t : \operatorname{Mor} \to \operatorname{Ob} </math> are [[submersion (mathematics)|submersion]]s. 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 (mathematics)|group]]. Accordingly, while Lie groups provide a natural model for (classical) [[Continuous symmetry|continuous symmetries]], Lie groupoids are often used as model for (and arise from) generalised, point-dependent symmetries.<ref>{{Cite journal |last=Weinstein |first=Alan |author-link=Alan Weinstein |date=1996-02-03 |title=Groupoids: unifying internal and external symmetry |url=https://www.ams.org/notices/199607/weinstein.pdf |journal=Notices of the American Mathematical Society |volume=43 |pages=744–752 |doi= |arxiv=math/9602220 |bibcode=<!---->}}</ref> Extending the [[Lie group–Lie algebra correspondence|correspondence]] between Lie groups and Lie algebras, Lie groupoids are the global counterparts of [[Lie algebroid]]s. Lie groupoids were introduced by [[Charles Ehresmann]]<ref>{{Cite journal |last=Ehresmann |first=Charles |author-link=Charles Ehresmann |date=1959 |title=Catégories topologiques et categories différentiables |trans-title=Topological categories and differentiable categories |url=https://ncatlab.org/nlab/files/EhresmannCategoriesTopologiques.pdf |journal=Colloque de Géométrie différentielle globale |language=French |publisher=CBRM, Bruxelles |pages=137–150}}</ref><ref>{{Cite journal |last=Ehresmann |first=Charles |author-link=Charles Ehresmann |date=1963 |title=Catégories structurées |trans-title=Structured categories |url=http://www.numdam.org/item/?id=ASENS_1963_3_80_4_349_0 |journal=Annales scientifiques de l'École Normale Supérieure |language=fr |volume=80 |issue=4 |pages=349–426 |doi=10.24033/asens.1125 |doi-access=free}}</ref> under the name ''differentiable groupoids''. == Definition and basic concepts == A '''Lie groupoid''' consists of * two smooth manifolds <math>G</math> and <math>M</math> * two [[Surjective function|surjective]] [[Submersion (mathematics)|submersions]] <math>s,t : G \to M </math> (called, respectively, '''source''' and '''target''' projections) * a map <math>m: G^{(2)}:= \{ (g,h) \mid s(g)=t(h) \} \to G </math> (called '''multiplication''' or composition map), where we use the notation <math>gh := m (g,h) </math> * a map <math>u: M \to G </math> (called '''unit''' map or object inclusion map), where we use the notation <math>1_x:= u(x) </math> * a map <math>i: G \to G </math> (called '''inversion'''), where we use the notation <math>g^{-1} := i(g) </math> such that * the composition satisfies <math>s(gh) = s(h) </math> and <math>t(gh) = t(g) </math> for every <math>g,h \in G</math> for which the composition is defined *the composition is [[Associative property|associative]], i.e. <math>g(h l) = (gh) l </math> for every <math>g,h,l \in G</math> for which the composition is defined * <math>u </math> works as an [[Identity element|identity]], i.e. <math>s(1_x) = t(1_x) = x </math> for every <math>x \in M</math> and <math>g 1_{s(g)} = g </math> and <math>1_{t(g)} g = g </math> for every <math>g \in G</math> * <math>i </math> works as an [[Inverse element|inverse]], i.e. <math>g^{-1} g = 1_{s(g)} </math> and <math>g g^{-1} = 1_{t(g)} </math> for every <math>g \in G</math>. 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 <math>M</math> of objects and <math>G</math> of morphisms are manifolds, the maps <math>s</math>, <math>t</math>, <math>m</math>, <math>i</math> and <math>u</math> are smooth and <math>s</math> and <math>t</math> are submersions. A Lie groupoid is therefore not simply a [[groupoid object]] in the [[Category of manifolds|category of smooth manifolds]]: one has to ask the additional property that <math>s</math> and <math>t</math> are submersions. Lie groupoids are often denoted by <math>G \rightrightarrows M </math>, where the two arrows represent the source and the target. The notation <math>G_1 \rightrightarrows G_0 </math> is also frequently used, especially when stressing the simplicial structure of the associated [[Nerve (category theory)|nerve]]. In order to include more natural examples, the manifold <math>G</math> is not required in general to be [[Hausdorff space|Hausdorff]] or [[Second-countable space|second countable]] (while <math>M</math> and all other spaces are). === Alternative definitions === The original definition by Ehresmann required <math>G</math> and <math>M</math> to possess a smooth structure such that only <math>m</math> is smooth and the maps <math>g \mapsto 1_{s(g)} </math> and <math>g \mapsto 1_{t(g)} </math> are subimmersions (i.e. have locally constant [[Rank (differential topology)|rank]]). Such definition proved to be too weak and was replaced by Pradines with the one currently used.<ref>{{Cite journal |last=Pradines |first=Jean |date=1966 |title=Théorie de Lie pour les groupoïdes dif́férentiables. Relations entre propriétés locales et globales |trans-title=Lie theory for differentiable groupoids. Relations between local and global properties |url=https://gallica.bnf.fr/ark:/12148/bpt6k6248451z/f49.image |journal=C. R. Acad. Sci. Paris |language=French |volume=263 |pages=907–910 |via=[[Gallica]]}}</ref> While some authors<ref>{{Cite book|last1=Kumpera|first1=Antonio|url=https://www.degruyter.com/document/doi/10.1515/9781400881734/html|title=Lie Equations, Vol. I|last2=Spencer|first2=Donald Clayton|date=2016-03-02|publisher=Princeton University Press|isbn=978-1-4008-8173-4|language=en|doi=10.1515/9781400881734}}</ref> introduced weaker definitions which did not require <math>s</math> and <math>t</math> 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 <math>G \rightrightarrows M</math> are smooth submersions has some immediate consequences: * the <math>s</math>-fibres <math>s^{-1}(x) \subseteq G</math>, the <math>t</math>-fibres <math>t^{-1}(x) \subseteq G</math>, and the set of composable morphisms <math>G^{(2)} \subseteq G \times G</math> are [[submanifold]]s; * the inversion map <math>i</math> is a [[diffeomorphism]]; *the unit map <math>u</math> is a [[Embedding#Differential topology|smooth embedding]]; *the [[Groupoid#Vertex groups and orbits|isotropy groups]] <math>G_x</math> are [[Lie group]]s; * the orbits <math>\mathcal{O}_x \subseteq M</math> are [[immersed submanifold]]s; * the <math>s</math>-fibre <math>s^{-1}(x)</math> at a point <math>x \in M</math> is a [[Principal bundle|principal <math>G_x</math>-bundle]] over the orbit <math>\mathcal{O}_x</math> at that point. === Subobjects and morphisms === A '''Lie subgroupoid''' of a Lie groupoid <math>G \rightrightarrows M</math> is a [[Groupoid#Subgroupoids and morphisms|subgroupoid]] <math>H \rightrightarrows N</math> (i.e. a [[subcategory]] of the category <math>G</math>) with the extra requirement that <math>H \subseteq G</math> is an immersed submanifold. As for a subcategory, a (Lie) subgroupoid is called '''wide''' if <math>N = M</math>. Any Lie groupoid <math>G \rightrightarrows M</math> has two canonical wide subgroupoids: * the unit/identity Lie subgroupoid <math>u(M) = \{ 1_x \in G \mid x \in M \}</math>; * the inner subgroupoid <math>IG := \{ g \in G \mid s(g)=t(g) \}</math>, 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 <math>H \subseteq G</math> inside <math>IG</math> such that, for every <math>h \in H, g \in G</math> with <math>s(h)=t(h)=s(g)</math>, one has <math>ghg^{-1} \in H</math>. The isotropy groups of <math>H</math> are therefore [[normal subgroup]]s of the isotropy groups of <math>G</math>. A '''Lie groupoid morphism''' between two Lie groupoids <math>G \rightrightarrows M</math> and <math>H \rightrightarrows N</math> is a groupoid morphism <math>F: G \to H, f: M \to N</math> (i.e. a [[functor]] between the categories <math>G</math> and <math>H</math>), where both <math>F</math> and <math>f</math> are smooth. The [[Kernel (algebra)|kernel]] <math>\ker(F):= \{ g \in G \mid F(g) = 1_{s(g)} \}</math> of a morphism between Lie groupoids over the same base manifold is automatically a normal Lie subgroupoid. The [[quotient]] <math>G/\ker(F) \rightrightarrows M</math> has a natural groupoid structure such that the projection <math>G \to G/\ker(F)</math> is a groupoid morphism; however, unlike [[Quotient group#Quotients of Lie groups|quotients of Lie groups]], <math>G/\ker(F)</math> 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.<ref name=":1">{{Cite book |last=Mackenzie |first=K. |url=https://www.cambridge.org/core/books/lie-groupoids-and-lie-algebroids-in-differential-geometry/6B8B1D00E8B7672ABA80A0C950FD4979 |title=Lie Groupoids and Lie Algebroids in Differential Geometry |date=1987 |publisher=Cambridge University Press |isbn=978-0-521-34882-9 |series=London Mathematical Society Lecture Note Series |location=Cambridge |doi=10.1017/cbo9780511661839}}</ref> A Lie groupoid is called '''abelian''' if its isotropy Lie groups are [[Abelian group|abelian]]. For similar reasons as above, while the definition of [[Abelianization|abelianisation]] of a group extends to set-theoretical groupoids, in the Lie case the analogue of the quotient <math>G^{ab} = G/(IG, IG)</math> may not exist or be smooth.<ref>{{Cite journal |last1=Contreras |first1=Ivan |last2=Fernandes |first2=Rui Loja |author-link2=Rui Loja Fernandes |date=2021-06-28 |title=Genus Integration, Abelianization, and Extended Monodromy |journal=International Mathematics Research Notices |volume=2021 |issue=14 |pages=10798–10840 |doi=10.1093/imrn/rnz133 |issn=1073-7928|doi-access=free |arxiv=1805.12043 }}</ref> === Bisections === A '''bisection''' of a Lie groupoid <math>G \rightrightarrows M</math> is a smooth map <math>b: M \to G</math> such that <math>s \circ b = id_M</math> and <math>t \circ b</math> is a diffeomorphism of <math>M</math>. In order to overcome the lack of symmetry between the source and the target, a bisection can be equivalently defined as a submanifold <math>B \subseteq G</math> such that <math>s_{\mid B}: B \to M</math> and <math>t_{\mid B}: B \to M</math> are diffeomorphisms; the relation between the two definitions is given by <math>B = b(M)</math>.<ref name=":3">{{Cite journal |last1=Albert |first1=Claude |last2=Dazord |first2=Pierre |last3=Weinstein |first3=Alan |author-link3=Alan Weinstein |date=1987 |title=Groupoïdes Symplectiques |trans-title=Symplectic Groupoids |url=http://www.numdam.org/item/PDML_1987___2A_1_0/ |journal=Pub. Dept. Math. Lyon |issue=2A |language=fr |pages=1–62 |via={{Interlanguage link|Numérisation de documents anciens mathématiques|lt=NUMDAM|fr}}}}</ref> The set of bisections forms a [[Group (mathematics)|group]], with the multiplication <math>b_1 \cdot b_2</math> defined as<math display="block">(b_1 \cdot b_2) (x) := b_1 (b_2 (x)) b_2(x).</math>and inversion defined as<math display="block">b_1^{-1} (x) := i \circ b_1 \left( (t\circ b_2)^{-1} (x) \right)</math>Note that the definition is given in such a way that, if <math>t \circ b_1 = \phi_1</math> and <math>t \circ b_2 = \phi_2</math>, then <math>t \circ (b_1 \cdot b_2) = \phi_1 \circ \phi_2</math> and <math>t \circ b_1^{-1} = \phi_1^{-1}</math>. 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''' <math>b: U \subseteq M \to G</math> is defined analogously, but the multiplication between local bisections is of course only partially defined. ==Examples== === Trivial and extreme cases === *Lie groupoids <math>G \rightrightarrows {*}</math> with one object are the same thing as Lie groups. *Given any manifold <math>M</math>, there is a Lie groupoid <math>M \times M \rightrightarrows M</math> 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''' <math>M \times G \times M \rightrightarrows M</math>, with structure maps <math>s(x,g,y)= y</math>, <math>t(x,g,y)=x</math>, <math>m ((x,g,y),(y,h,z))=(x,gh,z)</math>, <math>u(x) = (x,1,x)</math> and <math>i(x,g,y)= (y,g^{-1},x)</math>. *Given any manifold <math>M</math>, there is a Lie groupoid <math>u(M) \rightrightarrows M</math> 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 <math>s=t</math> 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 <math>G \rightrightarrows M</math> and a surjective submersion <math>\mu: N \to M</math>, there is a Lie groupoid <math>\mu^*G \rightrightarrows N</math>, called its '''pullback groupoid''' or '''induced groupoid''', where <math>\mu^*G \subseteq N \times G \times N</math> contains triples <math>(x,g,y)</math> such that <math>s(g)=\mu(y)</math> and <math>t(g) = \mu(x)</math>, and the multiplication is defined using the multiplication of <math>G</math>. For instance, the pullback of the pair groupoid of <math>M</math> is the pair groupoid of <math>N</math>. *Given any two Lie groupoids <math>G_1 \rightrightarrows M_1</math> and <math>G_2 \rightrightarrows M_2</math>, there is a Lie groupoid <math>G_1 \times G_2 \rightrightarrows M_1 \times M_2</math>, called their '''[[direct product]]''', such that the groupoid morphisms <math>G_1 \times G_2 \to \mathrm{pr}_{M_1}^*G_1</math> and <math>G_1 \times G_2 \to \mathrm{pr}_{M_2}^*G_2</math> are surjective submersions. *Given any Lie groupoid <math>G \rightrightarrows M</math>, there is a Lie groupoid <math>TG \rightrightarrows TM</math>, called its '''tangent groupoid''', obtained by considering the [[tangent bundle]] of <math>G</math> and <math>M</math> and the [[Pushforward (differential)|differential]] of the structure maps. *Given any Lie groupoid <math>G \rightrightarrows M</math>, there is a Lie groupoid <math>T^*G \rightrightarrows A^*</math>, called its '''cotangent groupoid''' obtained by considering the [[cotangent bundle]] of <math>G</math>, the [[Dual bundle|dual]] of the Lie algebroid <math>A</math> (see below), and suitable structure maps involving the differentials of the left and right translations. *Given any Lie groupoid <math>G \rightrightarrows M</math>, there is a Lie groupoid <math>J^k G \rightrightarrows M</math>, called its '''jet groupoid''', obtained by considering the [[Jet (mathematics)|k-jets]] of the local bisections of <math>G</math> (with smooth structure inherited from the [[jet bundle]] of <math>s: G \to M</math>) and setting <math>s(j^k_x b) = x</math>, <math>t(j^k_x b) = t(b(x))</math>, <math>m(j^k_{t(b(x))} b_1, j^k_x b_2) = j^k_x (b_1 \cdot b_2)</math>, <math>u(x) = j^k_x u</math> and <math>i(j^k_x b) = j^k_{t(b(x))} b^{-1}</math>. === Examples from differential geometry === *Given a submersion <math>\mu: M \to N</math>, there is a Lie groupoid <math>M \times_\mu M := \{ (x,y) \in M \times M \mid \mu(x)=\mu(y) \} \rightrightarrows M</math>, called the '''submersion groupoid''' or '''fibred pair groupoid''', whose structure maps are induced from the pair groupoid <math>M \times M \rightrightarrows M</math> (the condition that <math>\mu</math> is a submersion ensures the smoothness of <math>M \times_\mu M</math>). If <math>N</math> is a point, one recovers the pair groupoid. *Given a Lie group <math>G</math> [[Lie group action|acting]] on a manifold <math>M</math>, there is a Lie groupoid <math>G \times M \rightrightarrows M</math>, called the '''action groupoid''' or '''translation groupoid''', with one morphism for each triple <math>g \in G, x,y \in M</math> with <math>gx = y</math>. *Given any [[vector bundle]] <math>E\to M</math>, there is a Lie groupoid <math>GL(E) \rightrightarrows M</math>, called the '''general linear groupoid''', with morphisms between <math>x,y \in M</math> being linear isomorphisms between the fibres <math>E_x</math> and <math>E_y</math>. For instance, if <math>E = M \times \mathbb{R}^n</math> is the trivial vector bundle of rank <math>k</math>, then <math>GL(E) \rightrightarrows M</math> is the action groupoid. *Any [[principal bundle]] <math>P\to M</math> with structure group ''<math>G</math>'' defines a Lie groupoid <math>(P\times P)/G \rightrightarrows M</math>, where ''<math>G</math>'' acts on the pairs <math>(p,q) \in P \times P</math> componentwise, called the '''gauge groupoid'''. The multiplication is defined via compatible representatives as in the pair groupoid. *Any [[foliation]] <math>\mathcal{F}</math> on a manifold <math>M</math> defines two Lie groupoids, <math>\mathrm{Mon}(\mathcal{F}) \rightrightarrows M</math> (or <math>\Pi_1(\mathcal{F}) \rightrightarrows M</math>) and <math>\mathrm{Hol}(\mathcal{F}) \rightrightarrows M</math>, called respectively the '''monodromy/homotopy/fundamental groupoid''' and the '''holonomy groupoid''' of <math>\mathcal{F}</math>, whose morphisms consist of the [[homotopy]], respectively [[Foliation#Holonomy|holonomy]], equivalence classes of paths entirely lying in a leaf of <math>\mathcal{F}</math>. For instance, when <math>\mathcal{F}</math> is the trivial foliation with only one leaf, one recovers, respectively, the fundamental groupoid and the pair groupoid of <math>M</math>. On the other hand, when <math>\mathcal{F}</math> is a simple foliation, i.e. the foliation by (connected) fibres of a submersion <math>\mu: M \to N</math>, its holonomy groupoid is precisely the submersion groupoid <math>M \times_\mu M</math> but its monodromy groupoid may even fail to be Hausdorff, due to a general criterion in terms of vanishing cycles.<ref>{{Cite journal |last1=Cuesta |first1=F. Alcalde |last2=Hector |first2=G. |date=1997-09-01 |title=Feuilletages en surfaces, cycles évanouissants et variétés de Poisson |trans-title=Foliations on surfaces, vanishing cycles and Poisson manifolds |url=https://doi.org/10.1007/BF01298244 |journal=Monatshefte für Mathematik |language=fr |volume=124 |issue=3 |pages=191–213 |doi=10.1007/BF01298244 |s2cid=119369484 |issn=1436-5081|url-access=subscription }}</ref> In general, many elementary foliations give rise to monodromy and holonomy groupoids which are not Hausdorff. *Given any [[pseudogroup]] <math>\Gamma \subseteq \mathrm{Diff}_{loc}(M)</math>, there is a Lie groupoid <math>G = \mathrm{Germ}(\Gamma) \rightrightarrows M</math>, 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 groupoid]]s. === 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 <math>(s,t): G \to M \times M</math> (also known as the '''anchor''' of <math>G \rightrightarrows M</math>) 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 <math>G_x</math>-bundle <math>t: s^{-1}(x) \to M</math>, for any point <math>x \in M</math>. For instance: * the trivial Lie groupoid <math>M \times G \times M \rightrightarrows M</math> is transitive and arise from the trivial principal <math>G</math>-bundle <math>G \times M \to M</math>. As particular cases, Lie groups <math>G \rightrightarrows {*}</math> and pair groupoids <math>M \times M \rightrightarrows M</math> are trivially transitive, and arise, respectively, from the principal <math>G</math>-bundle <math>G \to {*}</math>, and from the principal <math>\{e\}</math>-bundle <math>M \to M</math>; * an action groupoid <math>G \times M \rightrightarrows M</math> is transitive if and only if the group action is [[Transitive action|transitive]], and in such case it arises from the principal bundle <math>G \to M</math> with structure group the isotropy group (at an arbitrary point); * the general linear groupoid of <math>E</math> is transitive, and arises from the [[frame bundle]] <math>Fr(E) \to M</math>; *pullback groupoids, jet groupoids and tangent groupoids of <math>G \rightrightarrows M</math> are transitive if and only if <math>G \rightrightarrows M</math> is transitive. As a less trivial instance of the correspondence between transitive Lie groupoids and principal bundles, consider the [[fundamental groupoid]] <math>\Pi_1(M)</math> of a (connected) smooth manifold <math>M</math>. This is naturally a topological groupoid, which is moreover transitive; one can see that <math>\Pi_1(M)</math> is isomorphic to the gauge groupoid of the [[Universal Cover|universal cover]] of <math>M</math>. Accordingly, <math>\Pi_1(M)</math> inherits a smooth structure which makes it into a Lie groupoid. Submersions groupoids <math>M \times_\mu M \rightrightarrows M</math> are an example of non-transitive Lie groupoids, whose orbits are precisely the fibres of <math>\mu</math>. A stronger notion of transitivity requires the anchor <math>(s,t): G \to M \times M</math> to be a surjective submersion. Such condition is also called '''local triviality''', because <math>G</math> becomes locally isomorphic (as Lie groupoid) to a trivial groupoid over any open <math>U \subseteq M</math> (as a consequence of the local triviality of principal bundles).<ref name=":1" /> When the space <math>G</math> 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 <math>\Gamma</math> is a transitive pseudogroup, its germ groupoid <math>\mathrm{Germ}(\Gamma)</math> is transitive but not locally trivial. === Proper groupoids === A Lie groupoid is called '''proper''' if <math>(s,t): G \to M \times M</math> is a [[Proper map|proper]] map. As a consequence * all isotropy groups of <math>G</math> are [[Compact space|compact]]; * all orbits of <math>G</math> are closed submanifolds; * the orbit space <math>M/G</math> is [[Hausdorff space|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 group action|proper]]; *the fundamental groupoid is proper if and only if the fundamental groups are [[Finite topological space|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 <math>s: G \to M</math> is proper (then <math>G \rightrightarrows M</math> is called '''s-proper'''), or that the entire space <math>G</math> is compact (then <math>G \rightrightarrows M</math> is called '''compact'''), but these requirements turns out to be too strict for many examples and applications.<ref>{{Cite journal |last1=Crainic |first1=Marius |author-link=Marius Crainic |last2=Loja Fernandes |first2=Rui |author-link2=Rui Loja Fernandes |last3=Martínez Torres |first3=David |date=2019-11-01 |title=Poisson manifolds of compact types (PMCT 1) |url=https://www.degruyter.com/document/doi/10.1515/crelle-2017-0006/html |journal=Journal für die reine und angewandte Mathematik (Crelle's Journal) |language=en |volume=2019 |issue=756 |pages=101–149 |doi=10.1515/crelle-2017-0006 |issn=1435-5345|arxiv=1510.07108 |s2cid=7668127 }}</ref> === Étale groupoids === A Lie groupoid is called '''étale''' if it satisfies one of the following equivalent conditions: * the dimensions of <math>G</math> and <math>M</math> are equal; * <math>s</math> is a [[local diffeomorphism]]; * all the <math>s</math>-fibres are [[Discrete space|discrete]] As a consequence, also the <math>t</math>-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 <math>G</math> is discrete; * germ groupoids of pseudogroups are always étale. === Effective groupoids === An étale groupoid is called '''effective''' if, for any two local bisections <math>b_1, b_2</math>, the condition <math>t \circ b_1 = t \circ b_2</math> implies <math>b_1 = b_2</math>. For instance: * Lie groups are effective if and only if are trivial; *unit groupoids are always effective; *an action groupoid is effective if the <math>G</math>-action is [[Free action|free]] and <math>G</math> is discrete. In general, any effective étale groupoid arise as the germ groupoid of some pseudogroup.<ref>{{Cite journal |last=Haefliger |first=André |author-link=André Haefliger |date=1958-12-01 |title=Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes |trans-title=Foliated structures and cohomology taking values in a sheaf of groupoids |url=https://doi.org/10.1007/BF02564582 |journal=[[Commentarii Mathematici Helvetici]] |language=fr |volume=32 |issue=1 |pages=248–329 |doi=10.1007/BF02564582 |s2cid=121138118 |issn=1420-8946|url-access=subscription }}</ref> 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 '''<math>s</math>-connected''' if all its <math>s</math>-fibres are [[Connected space|connected]]. Similarly, one talks about '''<math>s</math>-simply connected''' groupoids (when the <math>s</math>-fibres are [[Simply connected space|simply connected]]) or '''source-k-connected''' groupoids (when the <math>s</math>-fibres are [[k-connected]], i.e. the first <math>k</math> [[homotopy group]]s are trivial). Note that the entire space of arrows <math>G</math> is not asked to satisfy any connectedness hypothesis. However, if <math>G</math> is a source-<math>k</math>-connected Lie groupoid over a <math>k</math>-connected manifold, then <math>G</math> itself is automatically <math>k</math>-connected. For instanceː * Lie groups are source <math>k</math>-connected if and only if they are <math>k</math>-connected; * a pair groupoid is source <math>k</math>-connected if and only if <math>M</math> is <math>k</math>-connected; * unit groupoids are always source <math>k</math>-connected; * action groupoids are source <math>k</math>-connected if and only if <math>G</math> is <math>k</math>-connected; * monodromy groupoids (hence also fundamental groupoids) are source simply connected; * a gauge groupoid associated to a principal bundle <math>P\to M</math> is source <math>k</math>-connected if and only if the total space <math>P</math> is. == Further related concepts == === Actions and principal bundles === Recall that an action of a groupoid <math>G \rightrightarrows M</math> on a set <math>P</math> along a function <math>\mu: P \rightrightarrows M</math> is defined via a collection of maps <math>\mu^{-1}(x) \to \mu^{-1}(y), \quad p \mapsto g \cdot p</math> for each morphism <math>g \in G</math> between <math>x,y \in M</math>. Accordingly, an '''action of a Lie groupoid''' <math>G \rightrightarrows M</math> on a manifold <math>P</math> along a smooth map <math>\mu: P \rightrightarrows M</math> consists of a groupoid action where the maps <math>\mu^{-1}(x) \to \mu^{-1}(y)</math> are smooth. Of course, for every <math>x \in M</math> there is an induced smooth action of the isotropy group <math>G_x</math> on the fibre <math>\mu^{-1}(x)</math>. Given a Lie groupoid <math>G \rightrightarrows M</math>, a '''principal <math>G</math>-bundle''' consists of a <math>G</math>-space <math>P</math> and a <math>G</math>-invariant surjective submersion <math>\pi: P \to N</math> such that<math display="block">P \times_N G \to P \times_\pi P, \quad (p,g) \mapsto (p,p \cdot g)</math>is a diffeomorphism. Equivalent (but more involved) definitions can be given using <math>G</math>-valued cocycles or local trivialisations. When <math>G</math> is a Lie groupoid over a point, one recovers, respectively, standard [[Lie group action]]s and [[principal bundle]]s. === Representations === A '''representation''' of a Lie groupoid <math>G \rightrightarrows M</math> consists of a Lie groupoid action on a vector bundle <math>\pi: E \to M</math>, such that the action is fibrewise linear, i.e. each bijection <math>\pi^{-1}(x) \to \pi^{-1}(y)</math> is a linear isomorphism. Equivalently, a representation of <math>G</math> on <math>E</math> can be described as a Lie groupoid morphism from <math>G</math> to the general linear groupoid <math>GL(E)</math>. Of course, any fibre <math>E_x</math> becomes a representation of the isotropy group <math>G_x</math>. More generally, representations of transitive Lie groupoids are uniquely determined by representations of their isotropy groups, via the construction of the [[Frame bundle#Associated vector bundles|associated vector bundle]]. Examples of Lie groupoids representations include the following: * representations of Lie groups <math>G \rightrightarrows {*}</math> recover standard [[representation of a Lie group|Lie group representations]] * representations of pair groupoids <math>M \times M \rightrightarrows M</math> are trivial vector bundles * representations of unit groupoids <math>M \rightrightarrows M</math> are vector bundles * representations of action groupoid <math>G \times M \rightrightarrows M</math> are <math>G</math>-[[equivariant vector bundle]]s * representations of fundamental groupoids <math>\Pi_1(M)</math> are vector bundles endowed with [[Connection (vector bundle)|flat connections]] The set <math>\mathrm{Rep}(G)</math> of isomorphism classes of representations of a Lie groupoid <math>G \rightrightarrows M</math> has a natural structure of [[semiring]], 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 set|simplicial]] structure of the [[Nerve (category theory)|nerve]] <math>N(G)_n = G^{(n)}</math> of <math>G \rightrightarrows M</math>, viewed as a category. More precisely, recall that the space <math>G^{(n)}</math> consists of strings of <math>n</math> composable morphisms, i.e. <math>G^{(n)}:= \{ (g_1,\ldots,g_n) \in G \times \ldots \times G \mid s(g_i)=t(g_{i+1}) \quad \forall i=1,\ldots,n-1 \},</math> and consider the map <math>t^{(n)} = t \circ \mathrm{pr}_1: G^{(n)}\to M, (g_1,\ldots,g_n) \mapsto t(g_1)</math>. A '''differentiable ''<math>n</math>''-cochain''' of <math>G \rightrightarrows M</math> with coefficients in some representation <math>E \to M</math> is a smooth section of the pullback vector bundle <math>(t^{(n)})^*E \to G^{(n)}</math>. One denotes by <math>C^n(G,E)</math> the space of such ''<math>n</math>''-cochains, and considers the differential <math>d_n: C^n(G,E) \to C^{n+1}(G,E)</math>, defined as <math>d_n(c)(g_1,\ldots,g_{n+1}):= g_1 \cdot c(g_2,\ldots,g_{n+1}) +\sum_{i=1}^n (-1)^i c (g_1,\ldots, g_i g_{i+1}, \ldots,g_{n+1}) + (-1)^{n+1} c(g_1,\ldots,g_n).</math> Then <math>(C^n (G, E), d^n)</math> becomes a [[cochain complex]] and its cohomology, denoted by <math>H^n_d (G, E)</math>, is called the '''differentiable cohomology''' of <math>G \rightrightarrows M</math> with coefficients in <math>E \to M</math>. Note that, since the differential at degree zero is <math>d_0(c)(g) = g \cdot c(s(g)) - c(t(g))</math>, one has always <math>H^0_d (G, E) = \ker(d_0) = \Gamma(E)^G</math>. Of course, the differentiable cohomology of <math>G \rightrightarrows {*}</math> as a Lie groupoid coincides with the standard differentiable cohomology of <math>G</math> as a Lie group (in particular, for [[discrete group]]s one recovers the usual [[group cohomology]]). On the other hand, for any ''proper'' Lie groupoid <math>G \rightrightarrows M</math>, one can prove that <math>H^n_d (G, E) = 0</math> for every <math>n > 0</math>.<ref name=":0" /> === The Lie algebroid of a Lie groupoid === {{See also|Lie algebroid#Lie groupoid-Lie algebroid correspondence}} Any Lie groupoid <math>G \rightrightarrows M</math> has an associated [[Lie algebroid]] <math>A \to M</math>, obtained with a construction similar to the one which associates a [[Lie algebra]] to any Lie groupː * the vector bundle <math>A \to M</math> is the vertical bundle with respect to the source map, restricted to the elements tangent to the identities, i.e. <math>A:= \ker (ds)_{\mid M}</math>; * the Lie bracket is obtained by identifying <math>\Gamma(A)</math> with the left-invariant vector fields on <math>G</math>, and by transporting their Lie bracket to <math>A</math>; * the anchor map <math>A \to TM</math> is the differential of the target map <math>t: G \to M</math> restricted to <math>A</math>. The [[Lie group–Lie algebra correspondence]] 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 <math>G</math> there is a unique (up to isomorphism) s-simply connected Lie groupoid <math>\tilde{G}</math> with the same Lie algebroid of <math>G</math>, and a local diffeomorphism <math>\tilde{G} \to G</math> which is a groupoid morphism. For instance, * given any connected manifold <math>M</math> its pair groupoid <math>M \times M</math> is s-connected but not s-simply connected, while its fundamental groupoid <math>\Pi_1(M)</math> is. They both have the same Lie algebroid, namely the tangent bundle <math>TM \to M</math>, and the local diffeomorphism <math>\Pi_1 (M) \to M \times M</math> is given by <math>[\gamma] \mapsto (\gamma(0),\gamma(1))</math>. * given any foliation <math>\mathcal{F}</math> on <math>M</math>, its holonomy groupoid <math>\mathrm{Hol}(\mathcal{F})</math> is s-connected but not s-simply connected, while its monodromy groupoid <math>\mathrm{Mon}(\mathcal{F})</math> is. They both have the same Lie algebroid, namely the foliation algebroid <math>\mathcal{F} \to M</math>, and the local diffeomorphism <math>\mathrm{Mon}(\mathcal{F}) \to \mathrm{Hol}(\mathcal{F})</math> is given by <math>[\gamma] \mapsto [\gamma]</math> (since the homotopy classes are smaller than the holonomy ones). However, there is no analogue of [[Lie's third theorem]]ː while several classes of Lie algebroids are integrable, there are examples of Lie algebroids, for instance related to [[Foliation|foliation theory]], which do not admit an integrating Lie groupoid.<ref>{{Cite journal |last1=Almeida |first1=Rui |last2=Molino |first2=Pierre |date=1985 |title=Suites d'Atiyah et feuilletages transversalement complets |trans-title=Atiyah sequences and transversely complete foliations |url=https://gallica.bnf.fr/ark:/12148/bpt6k5496040t/f29.item |journal=Comptes Rendus de l'Académie des Sciences, Série I |language=fr |volume=300 |pages=13–15 |via=[[Gallica]]}}</ref> The general obstructions to the existence of such integration depend on the topology of <math>G</math>.<ref>{{Cite journal |last1=Crainic |first1=Marius |author-link=Marius Crainic |last2=Fernandes |first2=Rui |author-link2=Rui Loja Fernandes |date=2003-03-01 |title=Integrability of Lie brackets |journal=Annals of Mathematics |volume=157 |issue=2 |pages=575–620 |doi=10.4007/annals.2003.157.575 |issn=0003-486X |doi-access=free|arxiv=math/0105033 }}</ref> ==Morita equivalence== As discussed above, the standard notion of (iso)morphism of groupoids (viewed as [[Functor|functors between categories]]) restricts naturally to Lie groupoids. However, there is a more coarse notion of equivalence, called Morita equivalence, which is more flexible and useful in applications. First, a '''Morita map''' (also known as a weak equivalence or essential equivalence) between two Lie groupoids <math>G_1 \rightrightarrows G_0</math> and <math>H_1\rightrightarrows H_0</math> consists of a Lie groupoid morphism from G to H which is moreover [[Fully faithful functor|fully faithful]] and [[Essentially surjective functor|essentially surjective]] (adapting these categorical notions to the smooth context). We say that two Lie groupoids <math>G_1\rightrightarrows G_0</math> and <math>H_1\rightrightarrows H_0</math> are '''Morita equivalent''' if and only if there exists a third Lie groupoid <math>K_1\rightrightarrows K_0</math> together with two Morita maps from ''G'' to ''K'' and from ''H'' to ''K''. A more explicit description of Morita equivalence (e.g. useful to check that it is an [[equivalence relation]]) requires the existence of two surjective submersions <math>P \to G_0</math> and <math>P \to H_0</math> together with a left <math>G</math>-action and a right <math>H</math>-action, commuting with each other and making <math>P</math> into a principal bi-bundle.<ref>{{Cite journal|last=del Hoyo|first=Matias|date=2013|title=Lie groupoids and their orbispaces|url=http://www.ems-ph.org/doi/10.4171/PM/1930|journal=Portugaliae Mathematica|language=en|volume=70|issue=2|pages=161–209|doi=10.4171/PM/1930|issn=0032-5155|arxiv=1212.6714}}</ref> === Morita invariance === Many properties of Lie groupoids, e.g. being proper, being Hausdorff or being transitive, are Morita invariant. On the other hand, being étale is not Morita invariant. In addition, a Morita equivalence between <math>G_1\rightrightarrows G_0</math> and <math>H_1\rightrightarrows H_0</math> preserves their ''transverse geometry'', i.e. it induces: * a homeomorphism between the orbit spaces <math>G_0/G_1</math> and <math>H_0/H_1</math>; * an isomorphism <math>G_x\cong H_y</math> between the isotropy groups at corresponding points <math>x\in G_0</math> and <math>y\in H_0</math>; * an isomorphism <math>\mathcal{N}_x\cong \mathcal{N}_y</math> between the normal representations of the isotropy groups at corresponding points <math>x\in G_0</math> and <math>y\in H_0</math>. Last, the differentiable cohomologies of two Morita equivalent Lie groupoids are isomorphic.<ref name=":0">{{Cite journal |last=Crainic |first=Marius |author-link=Marius Crainic |date=2003-12-31 |title=Differentiable and algebroid cohomology, Van Est isomorphisms, and characteristic classes |url=https://www.ems-ph.org/journals/show_abstract.php?issn=0010-2571&vol=78&iss=4&rank=2 |journal=[[Commentarii Mathematici Helvetici]] |volume=78 |issue=4 |pages=681–721 |doi=10.1007/s00014-001-0766-9 |issn=0010-2571 |doi-access=free|arxiv=math/0008064 }}</ref> === Examples === * Isomorphic Lie groupoids are trivially Morita equivalent. * Two Lie groups are Morita equivalent if and only if they are isomorphic as Lie groups. *Two unit groupoids are Morita equivalent if and only if the base manifolds are diffeomorphic. * Any transitive Lie groupoid is Morita equivalent to its isotropy groups. * Given a Lie groupoid <math>G\rightrightarrows M</math> and a surjective submersion <math>\mu: N\to M</math>, the pullback groupoid <math>\mu^*G \rightrightarrows N</math> is Morita equivalent to <math>G\rightrightarrows M</math>. * Given a free and proper Lie group action of <math>G</math> on <math>M</math> (therefore the quotient <math>M/G</math> is a manifold), the action groupoid <math>G \times M \rightrightarrows M</math> is Morita equivalent to the unit groupoid <math>u(M/G) \rightrightarrows M/G</math>. * A Lie groupoid <math>G</math> is Morita equivalent to an étale groupoid if and only if all isotropy groups of <math>G</math> are discrete.<ref>{{Cite journal | date=2001-02-10 | title=Foliation Groupoids and Their Cyclic Homology | first1=Marius | last1=Crainic | author-link1=Marius Crainic | first2=Ieke | last2=Moerdijk | author-link2=Ieke Moerdijk | journal=[[Advances in Mathematics]] | language=en | volume=157 | issue=2 | pages=177–197 | doi=10.1006/aima.2000.1944 | doi-access=free | issn=0001-8708| arxiv=math/0003119 }}</ref> A concrete instance of the last example goes as follows. Let ''M'' be a smooth manifold and <math>\{U_\alpha\}</math> an open cover of ''<math>M</math>''. Its '''Čech groupoid''' <math>G_1\rightrightarrows G_0</math> is defined by the disjoint unions <math>G_0:=\bigsqcup_\alpha U_\alpha</math> and <math>G_1:=\bigsqcup_{\alpha,\beta}U_{\alpha\beta}</math>, where <math>U_{\alpha\beta}=U_\alpha \cap U_\beta\subset M</math>. The source and target map are defined as the embeddings <math>s:U_{\alpha\beta}\to U_\alpha</math> and <math>t:U_{\alpha\beta}\to U_\beta</math>, and the multiplication is the obvious one if we read the <math>U_{\alpha\beta}</math> as subsets of ''M'' (compatible points in <math>U_{\alpha\beta}</math> and <math>U_{\beta\gamma}</math> actually are the same in ''<math>M</math>'' and also lie in <math>U_{\alpha\gamma}</math>). The Čech groupoid is in fact the pullback groupoid, under the obvious submersion <math>p:G_0\to M</math>'','' of the unit groupoid <math>M\rightrightarrows M</math>. As such, Čech groupoids associated to different open covers of ''<math>M</math>'' are Morita equivalent. === Smooth stacks === {{See also|differentiable stack}}Investigating the structure of the orbit space of a Lie groupoid leads to the notion of a smooth stack. For instance, the orbit space is a smooth manifold if the isotropy groups are trivial (as in the example of the Čech groupoid), but it is not smooth in general. The solution is to revert the problem and to define a '''smooth stack''' as a Morita-equivalence class of Lie groupoids. The natural geometric objects living on the stack are the geometric objects on Lie groupoids invariant under Morita-equivalence: an example is the Lie groupoid cohomology. Since the notion of smooth stack is quite general, obviously all smooth manifolds are smooth stacks. Other classes of examples include [[orbifold]]s, which are (equivalence classes of) proper étale Lie groupoids, and orbit spaces of foliations. ==References== {{Reflist}} ==Books== * {{Cite journal |last=Weinstein |first=A. |author-link=Alan Weinstein |year=1996 |title=Groupoids: unifying internal and external symmetry |url=https://www.ams.org/notices/199607/weinstein.pdf |journal=Notices of the American Mathematical Society |volume=43 |pages=744–752 |arxiv=math/9602220 |bibcode=<!----> |doi=}} * {{cite book |last1=MacKenzie |first1=K. |year=1987 |title=Lie Groupoids and Lie Algebroids in Differential Geometry |publisher=Cambridge University Press |doi=10.1017/CBO9780511661839 |isbn=9780521348829}} * {{cite book|last1=MacKenzie |first1=K. C. H. |year=2005 |title=General Theory of Lie Groupoids and Lie Algebroids|publisher=Cambridge University Press |doi=10.1017/CBO9781107325883|isbn=9781107325883}} * {{cite journal |last1=Crainic |first1=M. |last2=Fernandes |first2=R. L. |year=2011 |title=Lectures on Integrability of Lie Brackets |url=https://msp.org/gtm/2011/17/gtm-2011-17-001s.pdf |journal=Geometry & Topology Monographs |volume=17 |pages=1–107 |arxiv=math/0611259}} * {{cite book |last1=Moerdijk |first1=I. |last2=Mrcun |first2=J. |year=2003 |title=Introduction to Foliations and Lie Groupoids |publisher=Cambridge University Press |doi=10.1017/CBO9780511615450 |isbn=9780521831970}} {{Authority control}} [[Category:Differential geometry]] [[Category:Lie groups]] [[Category:Lie groupoids| ]] [[Category:Manifolds]] [[Category:Symmetry]]
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