DOCSLIB.ORG

LIE GROUPOIDS and LIE ALGEBROIDS LECTURE NOTES, FALL 2017 Contents 1. Lie Groupoids 4 1.1. Definitions 4 1.2. Examples 6 1.3. Ex

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
LIE GROUPOIDS and LIE ALGEBROIDS LECTURE NOTES, FALL 2017 Contents 1. Lie Groupoids 4 1.1. Definitions 4 1.2. Examples 6 1.3. Ex LIE GROUPOIDS AND LIE ALGEBROIDS LECTURE NOTES, FALL 2017 ECKHARD MEINRENKEN Abstract. These notes are under construction. They contain errors and omissions, and the references are very incomplete. Apologies! Contents 1. Lie groupoids 4 1.1. Definitions 4 1.2. Examples 6 1.3. Exercises 9 2. Foliation groupoids 10 2.1. Definition, examples 10 2.2. Monodromy and holonomy 12 2.3. The monodromy and holonomy groupoids 12 2.4. Appendix: Haefliger’s approach 14 3. Properties of Lie groupoids 14 3.1. Orbits and isotropy groups 14 3.2. Bisections 15 3.3. Local bisections 17 3.4. Transitive Lie groupoids 18 4. More constructions with groupoids 19 4.1. Vector bundles in terms of scalar multiplication 20 4.2. Relations 20 4.3. Groupoid structures as relations 22 4.4. Tangent groupoid, cotangent groupoid 22 4.5. Prolongations of groupoids 24 4.6. Pull-backs and restrictions of groupoids 24 4.7. A result on subgroupoids 25 4.8. Clean intersection of submanifolds and maps 26 4.9. Intersections of Lie subgroupoids, fiber products 28 4.10. The universal covering groupoid 29 5. Groupoid actions, groupoid representations 30 5.1. Actions of Lie groupoids 30 5.2. Principal actions 31 5.3. Representations of Lie groupoids 33 6. Lie algebroids 33 1 2 ECKHARD MEINRENKEN 6.1. Definitions 33 6.2. Examples 34 6.3. Lie subalgebroids 36 6.4. Intersections of Lie subalgebroids 37 6.5. Direct products of Lie algebroids 38 7. Morphisms of Lie algebroids 38 7.1. Definition of morphisms 38 7.2. Morphisms and sections 41 7.3. Fibered products, pre-images 42 7.4. Pull-backs 42 7.5. Further Constructions 43 7.6. Lie algebroid actions, representations of Lie algebroids 45 8. The generalized foliation of a Lie algebroid 46 8.1. Integral submanifolds 46 8.2. Normal bundles and tubular neighborhoods 47 8.3. Euler-like vector fields 49 8.4. Some applications of Theorem 8.9 51 8.5. The splitting theorem for Lie algebroids 53 8.6. The generalized foliation 56 9. The Lie functor 56 9.1. The Lie algebra of a Lie group 56 9.2. The Lie algebroid of a Lie groupoid 58 9.3. Left-and right-invariant vector fields 58 9.4. The Lie functor from Lie groupoids to Lie algebroids 61 9.5. Examples 62 9.6. Groupoid multiplication via σL; σR 62 10. Integrability of Lie algebroids: The transitive case 64 10.1. The Almeida-Molino counter-example 64 10.2. Transitive Lie algebroids 65 10.3. Splittings 66 10.4. Gauge transformations of transitive Lie algebroids 67 10.5. Classification of Lie algebroids over 2-spheres 68 10.6. The monodromy groups 70 10.7. Construction of an integration 71 11. Integrability of non-transitive Lie algebroids 74 12. Lie algebroid cohomology, Lie groupoid cohomology 76 12.1. The de Rham complex of a Lie algebroid 76 12.2. The Lie algebroid structure 78 12.3. Examples 80 12.4. The Lie groupoid complex 81 12.5. Weinstein-Xu's van Est map 83 12.6. The Crainic double complex 84 12.7. Perturbation lemma 86 12.8. Construction of the van Est map 87 Appendix A. Deformation to the normal cone 90 LIE GROUPOIDS AND LIE ALGEBROIDS 3 A.1. Basic properties 90 A.2. Charts on D(M; N) 92 A.3. Euler-like vector fields 93 A.4. Vector bundles 94 A.5. Lie groupoids 94 A.6. Lie algebroids 96 References 97 4 ECKHARD MEINRENKEN 1. Lie groupoids Symmetries in mathematics, as well as in nature, are often defined to be invariance properties under actions of groups. Lie groupoids are given by a manifold M of `objects' together with a type of symmetry of M that is more general than those provided by group actions. For example, a foliation of M provides an example of such a generalized symmetry, but foliations need not be obtained from group actions in any obvious way. 1.1. Definitions. The groupoid will assign to any two objects m0; m1 2 M a collection (possibly empty) of arrows from m1 to m0. These arrows are thought of as `symmetries', but in contrast to Lie group actions this symmetry need not be defined for all m 2 M { only pointwise. On the other hand, we require that the collection of all such arrows (with arbitrary end points) fit together smoothly to define a manifold, and that arrows can be composed provided the end point (target) of one arrow is the starting point (source) of the next. The formal definition of a Lie groupoid G ⇒ M involves a manifold G of arrows, a submanifold i: M,!G of units (or objects), and two surjective submersions s; t: G! M called source and target such that t ◦ i = s ◦ i = idM : One thinks of g as an arrow from its source s(g) to its target t(g), with M embedded as trivial arrows. g (1) t(g) s(g) Using that s; t are submersion, one finds (cf. Exercise 1.1 below) that for all k = 1; 2;::: the set of k-arrows (k) k G = f(g1; : : : ; gk) 2 G j s(gi) = t(gi+1)g g gk g1 g2 g3 k−1 (2) m0 m1 m2 ··· ··· mk−1 mk k (k) is a smooth submanifold of G , and the two maps G ! M taking (g1; : : : ; gk) to s(gk), (0) respectively to t(g1), are submersions. For k = 0 one puts G = M. The definition of a Lie groupoid also involves a smooth multiplication map, defined on composable arrows (i.e., 2-arrows) (2) MultG : G !G; (g1; g2) 7! g1 ◦ g2; such that s(g1 ◦ g2) = s(g2); t(g1 ◦ g2) = t(g1). It is thought of as a concatenation of arrows. Note that when picturing this composition rule, it is best to draw arrows from the right to the left. g1◦g2 g1 g2 (3) m0 m1 m2 m0 m2 LIE GROUPOIDS AND LIE ALGEBROIDS 5 Definition 1.1. The above data define a Lie groupoid G ⇒ M if the following axioms are satisfied: (3) 1. Associativity: (g1 ◦ g2) ◦ g3 = g1 ◦ (g2 ◦ g3) for all (g1; g2; g3) 2 G . 2. Units: t(g) ◦ g = g = g ◦ s(g) for all g 2 G. 3. Inverses: For all g 2 G there exists h 2 G such that s(h) = t(g); t(h) = s(g), and such that g ◦ h; h ◦ g are units. The inverse of an element is necessarily unique (cf. Exercise). Denoting this element by g−1, we have that g ◦ g−1 = t(g); g−1 ◦ g = s(g). Inversion is pictured as reversing the direction of arrows. From now on, when we write g = g1 ◦ g2 we implicitly assume that g1; g2 are composable, i.e. s(g1) = t(g2). Let 3 Gr(MultG) = f(g; g1; g2) 2 G j g = g1 ◦ g2g: be the graph of the multiplication map; we will think of MultG as a smooth relation from G ×G to G. Remark 1.2. A groupoid structure on a manifold G is completely determined by Gr(MultG), i.e. by declaring when g = g1 ◦ g2. Indeed, the units are the elements m 2 G such that m = m ◦ m. Given g 2 G, the source s(g) and target t(g) are the unique units for which g = g◦s(g) = t(g)◦g. The inverse of g is the unique element g−1 such that g ◦ g−1 is a unit. Remark 1.3. In the definition above, our manifolds are always assumed to satisfy the Hausdorff separation axiom. For a (possibly) non-Hausdorff Lie groupoid, we allow the space G to be a non-Hausdorff manifold, but still require that the fibers of the source and target maps, as well as the units M, are Hausdorff. 1 Non-Hausdorff Lie groupoids are very common in the theory of foliations; see below. Remark 1.4. One may similarly consider `set-theoretic' groupoids G ⇒ M, by taking s; t, and MultG to be set maps (with s; t surjective). Such a set-theoretic groupoid is the same as a category for which the objects M and arrows G are sets, and with the property that every arrow is invertible. −1 Remark 1.5. Let InvG : G!G; g 7! g be the inversion map. As an application of the implicit function theorem, it is automatic that InvG is a diffeomorphism. Definition 1.6. A morphism of Lie groupoids F : H!G is a smooth map such that F (h1 ◦ h2) = F (h1) ◦ F (h2) (2) for all (h1; h2) 2 H . If F is an inclusion as a submanifold, we say that H is a Lie subgroupoid of G. 1One of the consequences of the Hausdorff property is the uniqueness of flows of vector fields. But in the theory to be developed below, the vector fields that we integrate are all tangent to source fibers, the target fibers, or the units. 6 ECKHARD MEINRENKEN By Remark 1.2, it is automatic that such a morphism takes units of H to units of G, and that it intertwines the source, target, and inversion maps. We will often present Lie groupoid homomorphisms by diagrams, as follows: G // M H / N If G ⇒ M is a Lie groupoid, and m 2 M, the intersection of the source and target fibers −1 −1 Gm = t (m) \ s (m) is a Lie group, with group structure induced by the groupoid multiplication. (We will prove later that it is a submanifold.) It is called the isotropy group of G at m.
Load more

Recommended publications
  • Generalized Connections and Higgs Fields on Lie Algebroids Nijmegen, April 5, 2016
    Generalized connections and Higgs fields on Lie algebroids Nijmegen, April 5, 2016 Thierry Masson Centre de Physique Théorique Campus de Luminy, Marseille Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy Motivations Discovery of Higgs particle in 2012. ➙ need for a mathematical validation of the Higgs sector in the SM. No clue from “traditional” schemes and tools. NCG: Higgs field is part of a “generalized connection”. Dubois-Violette, M., Kerner, R., and Madore, J. (1990). Noncommutative Differential Geometry and New Models of Gauge Theory. J. Math. Phys. 31, p. 323 Connes, A. and Lott, J. (1991). Particle models and noncommutative geometry. Nucl. Phys. B Proc. Suppl. 18.2, pp. 29–47 Models in NCG can reproduce the Standard Model up to the excitement connected to the diphoton resonance at 750 GeV “seen” by ATLAS and CMS! Mathematical structures difficult to master by particle physicists. Transitive Lie algebroids: ➙ generalized connections, gauge symmetries, Yang-Mills-Higgs models… ➙ Direct filiation from Dubois-Violette, Kerner, and Madore (1990). Mathematics close to “usual” mathematics of Yang-Mills theories. No realistic theory yet. 2 Generalized connections and Higgs fields on Lie algebroids, Nijmegen, April 5, 2016 Thierry Masson, CPT-Luminy How to construct a gauge field theory? The basic ingredients are: 1 A space of local symmetries (space-time dependence): ➙ a gauge group. 2 An implementation of the symmetry on matter fields: ➙ a representation theory. 3 A notion of derivation: ➙ some differential structures. 4 A (gauge compatible) replacement of ordinary derivations: ➙ a covariant derivative. 5 A way to write a gauge invariant Lagrangian density: ➙ action functional.
    [Show full text]
  • Introduction to Dynamical Triangulations
    Introduction to Dynamical Triangulations Andrzej G¨orlich Niels Bohr Institute, University of Copenhagen Naxos, September 12th, 2011 Andrzej G¨orlich Causal Dynamical Triangulation Outline 1 Path integral for quantum gravity 2 Causal Dynamical Triangulations 3 Numerical setup 4 Phase diagram 5 Background geometry 6 Quantum fluctuations Andrzej G¨orlich Causal Dynamical Triangulation Path integral formulation of quantum mechanics A classical particle follows a unique trajectory. Quantum mechanics can be described by Path Integrals: All possible trajectories contribute to the transition amplitude. To define the functional integral, we discretize the time coordinate and approximate each path by linear pieces. space classical trajectory t1 time t2 Andrzej G¨orlich Causal Dynamical Triangulation Path integral formulation of quantum mechanics A classical particle follows a unique trajectory. Quantum mechanics can be described by Path Integrals: All possible trajectories contribute to the transition amplitude. To define the functional integral, we discretize the time coordinate and approximate each path by linear pieces. quantum trajectory space classical trajectory t1 time t2 Andrzej G¨orlich Causal Dynamical Triangulation Path integral formulation of quantum mechanics A classical particle follows a unique trajectory. Quantum mechanics can be described by Path Integrals: All possible trajectories contribute to the transition amplitude. To define the functional integral, we discretize the time coordinate and approximate each path by linear pieces. quantum trajectory space classical trajectory t1 time t2 Andrzej G¨orlich Causal Dynamical Triangulation Path integral formulation of quantum gravity General Relativity: gravity is encoded in space-time geometry. The role of a trajectory plays now the geometry of four-dimensional space-time. All space-time histories contribute to the transition amplitude.
    [Show full text]
  • Topics in Equivariant Cohomology
    Topics in Equivariant Cohomology Luke Keating Hughes Thesis submitted for the degree of Master of Philosophy in Pure Mathematics at The University of Adelaide Faculty of Mathematical and Computer Sciences School of Mathematical Sciences February 1, 2017 Contents Abstract v Signed Statement vii Acknowledgements ix 1 Introduction 1 2 Classical Equivariant Cohomology 7 2.1 Topological Equivariant Cohomology . 7 2.1.1 Group Actions . 7 2.1.2 The Borel Construction . 9 2.1.3 Principal Bundles and the Classifying Space . 11 2.2 TheGeometryofPrincipalBundles. 20 2.2.1 The Action of a Lie Algebra . 20 2.2.2 Connections and Curvature . 21 2.2.3 Basic Di↵erentialForms ............................ 26 2.3 Equivariant de Rham Theory . 28 2.3.1 TheWeilAlgebra................................ 28 2.3.2 TheWeilModel ................................ 34 2.3.3 The Chern-Weil Homomorphism . 35 2.3.4 The Mathai-Quillen Isomorphism . 36 2.3.5 The Cartan Model . 37 3 Simplicial Methods 39 3.1 SimplicialandCosimplicialObjects. 39 3.1.1 The Simplicial Category . 39 3.1.2 CosimplicialObjects .............................. 41 3.1.3 SimplicialObjects ............................... 43 3.1.4 The Nerve of a Category . 47 3.1.5 Geometric Realisation . 49 iii 3.2 A Simplicial Construction of the Universal Bundle . 53 3.2.1 Basic Properties of NG ........................... 53 | •| 3.2.2 Principal Bundles and Local Trivialisations . 56 3.2.3 The Homotopy Extension Property and NDR Pairs . 57 3.2.4 Constructing Local Sections . 61 4 Simplicial Equivariant de Rham Theory 65 4.1 Dupont’s Simplicial de Rham Theorem . 65 4.1.1 The Double Complex of a Simplicial Space .
    [Show full text]
  • Vertex-Unfoldings of Simplicial Manifolds Erik D
    Masthead Logo Smith ScholarWorks Computer Science: Faculty Publications Computer Science 2002 Vertex-Unfoldings of Simplicial Manifolds Erik D. Demaine Massachusetts nI stitute of Technology David Eppstein University of California, Irvine Jeff rE ickson University of Illinois at Urbana-Champaign George W. Hart State University of New York at Stony Brook Joseph O'Rourke Smith College, [email protected] Follow this and additional works at: https://scholarworks.smith.edu/csc_facpubs Part of the Computer Sciences Commons, and the Geometry and Topology Commons Recommended Citation Demaine, Erik D.; Eppstein, David; Erickson, Jeff; Hart, George W.; and O'Rourke, Joseph, "Vertex-Unfoldings of Simplicial Manifolds" (2002). Computer Science: Faculty Publications, Smith College, Northampton, MA. https://scholarworks.smith.edu/csc_facpubs/60 This Article has been accepted for inclusion in Computer Science: Faculty Publications by an authorized administrator of Smith ScholarWorks. For more information, please contact [email protected] Vertex-Unfoldings of Simplicial Manifolds Erik D. Demaine∗ David Eppstein† Jeff Erickson‡ George W. Hart§ Joseph O’Rourke¶ Abstract We present an algorithm to unfold any triangulated 2-manifold (in particular, any simplicial polyhedron) into a non-overlapping, connected planar layout in linear time. The manifold is cut only along its edges. The resulting layout is connected, but it may have a disconnected interior; the triangles are connected at vertices, but not necessarily joined along edges. We extend our algorithm to establish a similar result for simplicial manifolds of arbitrary dimension. 1 Introduction It is a long-standing open problem to determine whether every convex polyhe- dron can be cut along its edges and unfolded flat in one piece without overlap, that is, into a simple polygon.
    [Show full text]
  • Monodromy Representations Associated to a Continuous Map
    Monodromy representations associated to a continuous map Salvador Borrós Cullell Supervisor: Dr. Pietro Polesello ALGANT Master Program Universität Duisburg-Essen; Università degli studi di Padova This dissertation is submitted for the degree of Master in Science July 2017 Acknowledgements First and foremost, I would like to thank my advisor Dr. Polesello for his infinite patience and his guidance while writing the thesis. His emphasis on writing mathematics in a precise and concise manner will forever remain a valuable lesson. I would also like to thank my parents for their constant support, specially during these two years I have been out of home. Despite the distance, they have kept me going when things weren’t going my way. A very special mention must be made of Prof. Marc Levine who introduced me to many of the notions that now appear in this thesis and showed unparalleled patience and kindness while helping me understand them in depth. Finally, I would like to thank my friends Shehzad Hathi and Jan Willem Frederik van Beck for taking up the role of a supportive family while living abroad. Abstract The aim of this thesis is to give a geometrical meaning to the induced monodromy rep- resentation. More precisely, given f : X ! Y a continuous map, the associated functor ind f f : P1(X) ! P1(Y) induces a functor Repk(P1(X)) ! Repk(P1(Y)) of the corresponding LCSH categories of representations. We will define a functor f∗ : LCSH(kX ) ! LCSH(kY ) from the category of locally constant sheaves on X to that of locally constant sheaves on Y in a way that the monodromy representation m LCSH is given by ind (m ), where m denotes f∗ F f F F the monodromy representation of a locally constant sheaf F on X.
    [Show full text]
  • Arxiv:Math/0507171V1 [Math.AG] 8 Jul 2005 Monodromy
    Monodromy Wolfgang Ebeling Dedicated to Gert-Martin Greuel on the occasion of his 60th birthday. Abstract Let (X,x) be an isolated complete intersection singularity and let f : (X,x) → (C, 0) be the germ of an analytic function with an isolated singularity at x. An important topological invariant in this situation is the Picard-Lefschetz monodromy operator associated to f. We give a survey on what is known about this operator. In particular, we re- view methods of computation of the monodromy and its eigenvalues (zeta function), results on the Jordan normal form of it, definition and properties of the spectrum, and the relation between the monodromy and the topology of the singularity. Introduction The word ’monodromy’ comes from the greek word µoνo − δρoµψ and means something like ’uniformly running’ or ’uniquely running’. According to [99, 3.4.4], it was first used by B. Riemann [135]. It arose in keeping track of the solutions of the hypergeometric differential equation going once around arXiv:math/0507171v1 [math.AG] 8 Jul 2005 a singular point on a closed path (cf. [30]). The group of linear substitutions which the solutions are subject to after this process is called the monodromy group. Since then, monodromy groups have played a substantial rˆole in many areas of mathematics. As is indicated on the webside ’www.monodromy.com’ of N. M. Katz, there are several incarnations, classical and l-adic, local and global, arithmetic and geometric. Here we concentrate on the classical lo- cal geometric monodromy in singularity theory. More precisely we focus on the monodromy operator of an isolated hypersurface or complete intersection singularity.
    [Show full text]
  • Arxiv:2010.15657V1 [Math.NT] 29 Oct 2020
    LOW-DEGREE PERMUTATION RATIONAL FUNCTIONS OVER FINITE FIELDS ZHIGUO DING AND MICHAEL E. ZIEVE Abstract. We determine all degree-4 rational functions f(X) 2 Fq(X) 1 which permute P (Fq), and answer two questions of Ferraguti and Micheli about the number of such functions and the number of equivalence classes of such functions up to composing with degree-one rational func- tions. We also determine all degree-8 rational functions f(X) 2 Fq(X) 1 which permute P (Fq) in case q is sufficiently large, and do the same for degree 32 in case either q is odd or f(X) is a nonsquare. Further, for several other positive integers n < 4096, for each sufficiently large q we determine all degree-n rational functions f(X) 2 Fq(X) which permute 1 P (Fq) but which are not compositions of lower-degree rational func- tions in Fq(X). Some of these results are proved by using a new Galois- theoretic characterization of additive (linearized) polynomials among all rational functions, which is of independent interest. 1. Introduction Let q be a power of a prime p.A permutation polynomial is a polyno- mial f(X) 2 Fq[X] for which the map α 7! f(α) is a permutation of Fq. Such polynomials have been studied both for their own sake and for use in various applications. Much less work has been done on permutation ra- tional functions, namely rational functions f(X) 2 Fq(X) which permute 1 P (Fq) := Fq [ f1g. However, the topic of permutation rational functions seems worthy of study, both because permutation rational functions have the same applications as permutation polynomials, and because of the con- struction in [24] which shows how to use permutation rational functions over Fq to produce permutation polynomials over Fq2 .
    [Show full text]
  • Mechanics, Control and Lie Algebroids, Geometry Meeting 2009
    Mechanics, Control and Lie algebroids Geometry Meeting 2009 Eduardo Martínez University of Zaragoza [email protected] Escola Universitaria Politécnica, Ferrol, October 29–30, 2009 Abstract The most relevant ideas and results about mechanical systems defined on Lie algebroids are presented. This was a program originally proposed by Alan Weinstein (1996) and developed by many authors. 1 Several reasons for formulating Mechanics on Lie algebroids The inclusive nature of the Lie algebroid framework: under the same formalism one can consider standard mechanical systems, systems on Lie algebras, systems on semidirect products, systems with symme- tries. The reduction of a mechanical system on a Lie algebroid is a mechan- ical system on a Lie algebroid, and this reduction procedure is done via morphisms of Lie algebroids. Well adapted: the geometry of the underlying Lie algebroid deter- mines some dynamical properties as well as the geometric structures associated to it (e.g. Symplectic structure). Provides a natural way to use quasi-velocities in Mechanics. 2 Introduction Lagrangian systems Given a Lagrangian L 2 C1(TQ), the Euler-Lagrange equations define a dynamical system q_i = vi d @L @L = : dt @vi @qi Variational Calculus. Symplectic formalism 4 Geodesics of left invariant metrics For instance, if k is a Riemannian metric on a manifold Q , the Euler- 1 Lagrange equations for L(v) = 2 k(v; v) are the equation for the geodesics rvv = 0 If Q = G is a Lie group and k is left invariant, then L defines a function 1 l on the Lie algebra g, by restriction l(ξ) = 2 k(ξ; ξ).
    [Show full text]
  • Hausdorff Morita Equivalence of Singular Foliations
    Hausdorff Morita Equivalence of singular foliations Alfonso Garmendia∗ Marco Zambony Abstract We introduce a notion of equivalence for singular foliations - understood as suitable families of vector fields - that preserves their transverse geometry. Associated to every singular foliation there is a holonomy groupoid, by the work of Androulidakis-Skandalis. We show that our notion of equivalence is compatible with this assignment, and as a consequence we obtain several invariants. Further, we show that it unifies some of the notions of transverse equivalence for regular foliations that appeared in the 1980's. Contents Introduction 2 1 Background on singular foliations and pullbacks 4 1.1 Singular foliations and their pullbacks . .4 1.2 Relation with pullbacks of Lie groupoids and Lie algebroids . .6 2 Hausdorff Morita equivalence of singular foliations 7 2.1 Definition of Hausdorff Morita equivalence . .7 2.2 First invariants . .9 2.3 Elementary examples . 11 2.4 Examples obtained by pushing forward foliations . 12 2.5 Examples obtained from Morita equivalence of related objects . 15 3 Morita equivalent holonomy groupoids 17 3.1 Holonomy groupoids . 17 arXiv:1803.00896v1 [math.DG] 2 Mar 2018 3.2 Morita equivalent holonomy groupoids: the case of projective foliations . 21 3.3 Pullbacks of foliations and their holonomy groupoids . 21 3.4 Morita equivalence for open topological groupoids . 27 3.5 Holonomy transformations . 29 3.6 Further invariants . 30 3.7 A second look at Hausdorff Morita equivalence of singular foliations . 31 4 Further developments 33 4.1 An extended equivalence for singular foliations . 33 ∗KU Leuven, Department of Mathematics, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium.
    [Show full text]
  • Groupoids and Local Cartesian Closure
    Groupoids and local cartesian closure Erik Palmgren∗ Department of Mathematics, Uppsala University July 2, 2003 Abstract The 2-category of groupoids, functors and natural isomorphisms is shown to be locally cartesian closed in a weak sense of a pseudo- adjunction. Key words: groupoids, 2-categories, local cartesian closure. Mathematics Subject Classification 2000: 18B40, 18D05, 18A40. 1 Introduction A groupoid is a category in which each morphism is invertible. The notion may be considered as a common generalisation of the notions of group and equivalence relation. A group is a one-object groupoid. Every equivalence relation on a set, viewed as a directed graph, is a groupoid. A recent indepen- dence proof in logic used groupoids. Hofmann and Streicher [8] showed that groupoids arise from a construction in Martin-Löf type theory. This is the so-called identity type construction, which applied to a type gives a groupoid turning the type into a projective object, in the category of types with equiv- alence relations. As a consequence there are plenty of choice objects in this category: every object has a projective cover [12] — a property which seems essential for doing constructive mathematics according to Bishop [1] inter- nally to the category. For some time the groupoids associated with types were believed to be discrete. However, in [8] a model of type theory was given using fibrations over groupoids, showing that this need not be the case. The purpose of this article is to draw some further conclusions for group- oids from the idea of Hofmann and Streicher. Since type theory has a Π- construction, one could expect that some kind of (weak) local cartesian clo- sure should hold for groupoids.
    [Show full text]
  • Arxiv:1705.00466V2 [Math.DG] 1 Nov 2017 Bet,I Atclrqoins Si Hywr Moh Hsi Bec Is This Smooth
    ORBISPACES AS DIFFERENTIABLE STRATIFIED SPACES MARIUS CRAINIC AND JOAO˜ NUNO MESTRE Abstract. We present some features of the smooth structure, and of the canonical stratification on the orbit space of a proper Lie groupoid. One of the main features is that of Morita invariance of these structures - it allows us to talk about the canonical structure of differentiable stratified space on the orbispace (an object analogous to a separated stack in algebraic geometry) presented by the proper Lie groupoid. The canonical smooth structure on an orbispace is studied mainly via Spallek’s framework of differentiable spaces, and two alternative frameworks are then presented. For the canonical stratification on an orbispace, we ex- tend the similar theory coming from proper Lie group actions. We make no claim to originality. The goal of these notes is simply to give a complemen- tary exposition to those available, and to clarify some subtle points where the literature can sometimes be confusing, even in the classical case of proper Lie group actions. Contents 1. Introduction 1 2. Background 5 3. Orbispaces as differentiable spaces 13 4. Orbispaces as stratified spaces 27 5. Orbispaces as differentiable stratified spaces 43 References 47 1. Introduction Lie groupoids are geometric objects that generalize Lie groups and smooth man- arXiv:1705.00466v2 [math.DG] 1 Nov 2017 ifolds, and permit a unified approach to the study of several objects of interest in differential geometry, such as Lie group actions, foliations and principal bundles (see for example [9, 37, 44] and the references therein). They have found wide use in Poisson and Dirac geometry (e.g.
    [Show full text]
  • Regular Jacobi Structures and Generalized Contact Bundles
    Regular Jacobi Structures and Generalized Contact Bundles Jonas Schnitzer∗ Dipartimento di Matematica Università degli Studi di Salerno Via Giovanni Paolo II, 132 84084 Fisciano (SA) Italy June 14, 2021 Abstract A Jacobi structure J on a line bundle L → M is weakly regular if the sharp map J ♯ : J 1L → DL has constant rank. A generalized contact bundle with regular Jacobi structure possess a transverse complex structure. Paralleling the work of Bailey in generalized complex geometry, we find condition on a pair consisting of a regular Jacobi structure and an transverse complex structure to come from a generalized contact structure. In this way we are able to construct interesting examples of generalized contact bundles. As applications: 1) we prove that every 5- dimensional nilmanifold is equipped with an invariant generalized contact structure, 2) we show that, unlike the generalized complex case, all contact bundles over a complex manifold possess a compatible generalized contact structure. Finally we provide a counterexample presenting a locally conformal symplectic bundle over a generalized contact manifold of complex type that do not possess a compatible generalized contact structure. Contents 1 Introduction 2 2 Preliminaries and Notation 2 3 Tranversally Complex Jacobi Structures and Generalized Contact Bundles 8 arXiv:1806.10489v1 [math.DG] 27 Jun 2018 4 The Splitting of the Spectral Sequence of a Transversally Complex Subbundle 10 5 Examples I: Five dimensional nilmanifolds 15 6 Examples II: Contact Fiber Bundles 19 7 A Counterexample 21 References 23 ∗[email protected] 1 1 Introduction Generalized geometry was set up in the early 2000’s by Hitchin [9] and further developed by Gualtieri [8], and the literature about them is now rather wide.
    [Show full text]