Diffeology, Groupoids & Morita Equivalence
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Hodge Decomposition for Higher Order Hochschild Homology
ANNALES SCIENTIFIQUES DE L’É.N.S. TEIMURAZ PIRASHVILI Hodge decomposition for higher order Hochschild homology Annales scientifiques de l’É.N.S. 4e série, tome 33, no 2 (2000), p. 151-179 <http://www.numdam.org/item?id=ASENS_2000_4_33_2_151_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 2000, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Sclent. EC. Norm. Sup., 4° serie, t 33, 2000, p. 151 a 179. HODGE DECOMPOSITION FOR HIGHER ORDER HOCHSCHILD HOMOLOGY TEIMURAZ PIRASHVILI ABSTRACT. - Let r be the category of finite pointed sets and F be a functor from r to the category of vector spaces over a characteristic zero field. Loday proved that one has the natural decomposition 7TnF(S1) ^ ©n^ Q^^CF), n ^ 0. We show that for any d ^ 1, there exists a similar decomposition for 7^nF(Sd). Here 3d is a simplicial model of the d-dimensional sphere. The striking point is, that the knowledge of the decomposition for 7VnF(S1) (respectively 7TnF(S2)) completely determines the decomposition of 7^nF(Sd) for any odd (respectively even) d. These results can be applied to the cohomology of the mapping space Xs , where X is a c^-connected space. -
Morita Equivalence and Generalized Kähler Geometry by Francis
Morita Equivalence and Generalized Kahler¨ Geometry by Francis Bischoff A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mathematics University of Toronto c Copyright 2019 by Francis Bischoff Abstract Morita Equivalence and Generalized K¨ahlerGeometry Francis Bischoff Doctor of Philosophy Graduate Department of Mathematics University of Toronto 2019 Generalized K¨ahler(GK) geometry is a generalization of K¨ahlergeometry, which arises in the study of super-symmetric sigma models in physics. In this thesis, we solve the problem of determining the underlying degrees of freedom for the class of GK structures of symplectic type. This is achieved by giving a reformulation of the geometry whereby it is represented by a pair of holomorphic Poisson structures, a holomorphic symplectic Morita equivalence relating them, and a Lagrangian brane inside of the Morita equivalence. We apply this reformulation to solve the longstanding problem of representing the metric of a GK structure in terms of a real-valued potential function. This generalizes the situation in K¨ahlergeometry, where the metric can be expressed in terms of the partial derivatives of a function. This result relies on the fact that the metric of a GK structure corresponds to a Lagrangian brane, which can be represented via the method of generating functions. We then apply this result to give new constructions of GK structures, including examples on toric surfaces. Next, we study the Picard group of a holomorphic Poisson structure, and explore its relationship to GK geometry. We then apply our results to the deformation theory of GK structures, and explain how a GK metric can be deformed by flowing the Lagrangian brane along a Hamiltonian vector field. -
Morita Equivalence and Continuous-Trace C*-Algebras, 1998 59 Paul Howard and Jean E
http://dx.doi.org/10.1090/surv/060 Selected Titles in This Series 60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras, 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya> and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert, Integer-valued polynomials, 1997 47 A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May (with an appendix by M. Cole), Rings, modules, and algebras in stable homotopy theory, 1997 46 Stephen Lipscomb, Symmetric inverse semigroups, 1996 45 George M. Bergman and Adam O. Hausknecht, Cogroups and co-rings in categories of associative rings, 1996 44 J. Amoros, M. Burger, K. Corlette, D. -
AN INTRODUCTION to DIFFEOLOGY I Since Its Creation, In
AN INTRODUCTION TO DIFFEOLOGY PATRICK IGLESIAS-ZEMMOUR Abstract. This text presents the basics of Difeology and the main domains: Homotopy, Fiber Bundles, Quotients, Singularities, Cartan-de Rham Calculus — which form the core of diferential geometry — from the point of view of this theory. We show what makes difeology special and relevant in regard to these traditional subjects. Introduction Since its creation, in the early 1980s, Dieology has become an alternative, or rather a natural extension of traditional diferential geometry. With its developments in higher homotopy theory, ber bundles, modeling spaces, Cartan-de Rham calcu- lus, moment map and symplectic1 program, for examples, difeology now covers a large spectrum of traditional elds and deploys them from singular quotients to innite-dimensional spaces — and mixes the two — treating mathematical objects that are or are not strictly speaking manifolds, and other constructions, on an equal footing in a common framework. We shall see some of its achievements through a series of examples, chosen because they are not covered by the geom- etry of manifolds, because they involve innite-dimensional spaces or singular quotients, or both. The growing interest in difeology comes from the conjunction of two strong properties of the theory: 1. Mainly, the category {Difeology} is stable under all set-theoretic construc- tions: sums, products, subsets, and quotients. One says that it is a complete and co-complete category. It is also Cartesian closed, the space of smooth maps in difeology has itself a natural functional dieology. Date: August 25, 2017 – Revision May 15, 2020. Text of the talk given at the Conference “New Spaces in Mathematics & Physics”, September 28 – October 2, 2015 Institut Henri Poincaré Paris, France. -
Derived Smooth Manifolds
DERIVED SMOOTH MANIFOLDS DAVID I. SPIVAK Abstract. We define a simplicial category called the category of derived man- ifolds. It contains the category of smooth manifolds as a full discrete subcat- egory, and it is closed under taking arbitrary intersections in a manifold. A derived manifold is a space together with a sheaf of local C1-rings that is obtained by patching together homotopy zero-sets of smooth functions on Eu- clidean spaces. We show that derived manifolds come equipped with a stable normal bun- dle and can be imbedded into Euclidean space. We define a cohomology theory called derived cobordism, and use a Pontrjagin-Thom argument to show that the derived cobordism theory is isomorphic to the classical cobordism theory. This allows us to define fundamental classes in cobordism for all derived man- ifolds. In particular, the intersection A \ B of submanifolds A; B ⊂ X exists on the categorical level in our theory, and a cup product formula [A] ^ [B] = [A \ B] holds, even if the submanifolds are not transverse. One can thus consider the theory of derived manifolds as a categorification of intersection theory. Contents 1. Introduction 1 2. The axioms 8 3. Main results 14 4. Layout for the construction of dMan 21 5. C1-rings 23 6. Local C1-ringed spaces and derived manifolds 26 7. Cotangent Complexes 32 8. Proofs of technical results 39 9. Derived manifolds are good for doing intersection theory 50 10. Relationship to similar work 52 References 55 1. Introduction Let Ω denote the unoriented cobordism ring (though what we will say applies to other cobordism theories as well, e.g. -
Arxiv:Math/0310146V1 [Math.AT] 10 Oct 2003 Usinis: Question Fr Most Aut the the of Algebra”
MORITA THEORY IN ABELIAN, DERIVED AND STABLE MODEL CATEGORIES STEFAN SCHWEDE These notes are based on lectures given at the Workshop on Structured ring spectra and their applications. This workshop took place January 21-25, 2002, at the University of Glasgow and was organized by Andy Baker and Birgit Richter. Contents 1. Introduction 1 2. Morita theory in abelian categories 2 3. Morita theory in derived categories 6 3.1. The derived category 6 3.2. Derived equivalences after Rickard and Keller 14 3.3. Examples 19 4. Morita theory in stable model categories 21 4.1. Stable model categories 22 4.2. Symmetric ring and module spectra 25 4.3. Characterizing module categories over ring spectra 32 4.4. Morita context for ring spectra 35 4.5. Examples 38 References 42 1. Introduction The paper [Mo58] by Kiiti Morita seems to be the first systematic study of equivalences between module categories. Morita treats both contravariant equivalences (which he calls arXiv:math/0310146v1 [math.AT] 10 Oct 2003 dualities of module categories) and covariant equivalences (which he calls isomorphisms of module categories) and shows that they always arise from suitable bimodules, either via contravariant hom functors (for ‘dualities’) or via covariant hom functors and tensor products (for ‘isomorphisms’). The term ‘Morita theory’ is now used for results concerning equivalences of various kinds of module categories. The authors of the obituary article [AGH] consider Morita’s theorem “probably one of the most frequently used single results in modern algebra”. In this survey article, we focus on the covariant form of Morita theory, so our basic question is: When do two ‘rings’ have ‘equivalent’ module categories ? We discuss this question in different contexts: • (Classical) When are the module categories of two rings equivalent as categories ? Date: February 1, 2008. -
Arxiv:Math/0301347V1
Contemporary Mathematics Morita Contexts, Idempotents, and Hochschild Cohomology — with Applications to Invariant Rings — Ragnar-Olaf Buchweitz Dedicated to the memory of Peter Slodowy Abstract. We investigate how to compare Hochschild cohomology of algebras related by a Morita context. Interpreting a Morita context as a ring with distinguished idempotent, the key ingredient for such a comparison is shown to be the grade of the Morita defect, the quotient of the ring modulo the ideal generated by the idempotent. Along the way, we show that the grade of the stable endomorphism ring as a module over the endomorphism ring controls vanishing of higher groups of selfextensions, and explain the relation to various forms of the Generalized Nakayama Conjecture for Noetherian algebras. As applications of our approach we explore to what extent Hochschild cohomology of an invariant ring coincides with the invariants of the Hochschild cohomology. Contents Introduction 1 1. Morita Contexts 3 2. The Grade of the Stable Endomorphism Ring 6 3. Equivalences to the Generalized Nakayama Conjecture 12 4. The Comparison Homomorphism for Hochschild Cohomology 14 5. Hochschild Cohomology of Auslander Contexts 17 6. Invariant Rings: The General Case 20 7. Invariant Rings: The Commutative Case 23 References 27 arXiv:math/0301347v1 [math.AC] 29 Jan 2003 Introduction One of the basic features of Hochschild (co-)homology is its invariance under derived equivalence, see [25], in particular, it is invariant under Morita equivalence, a result originally established by Dennis-Igusa [18], see also [26]. This raises the question how Hochschild cohomology compares in general for algebras that are just ingredients of a Morita context. -
Math 395: Category Theory Northwestern University, Lecture Notes
Math 395: Category Theory Northwestern University, Lecture Notes Written by Santiago Can˜ez These are lecture notes for an undergraduate seminar covering Category Theory, taught by the author at Northwestern University. The book we roughly follow is “Category Theory in Context” by Emily Riehl. These notes outline the specific approach we’re taking in terms the order in which topics are presented and what from the book we actually emphasize. We also include things we look at in class which aren’t in the book, but otherwise various standard definitions and examples are left to the book. Watch out for typos! Comments and suggestions are welcome. Contents Introduction to Categories 1 Special Morphisms, Products 3 Coproducts, Opposite Categories 7 Functors, Fullness and Faithfulness 9 Coproduct Examples, Concreteness 12 Natural Isomorphisms, Representability 14 More Representable Examples 17 Equivalences between Categories 19 Yoneda Lemma, Functors as Objects 21 Equalizers and Coequalizers 25 Some Functor Properties, An Equivalence Example 28 Segal’s Category, Coequalizer Examples 29 Limits and Colimits 29 More on Limits/Colimits 29 More Limit/Colimit Examples 30 Continuous Functors, Adjoints 30 Limits as Equalizers, Sheaves 30 Fun with Squares, Pullback Examples 30 More Adjoint Examples 30 Stone-Cech 30 Group and Monoid Objects 30 Monads 30 Algebras 30 Ultrafilters 30 Introduction to Categories Category theory provides a framework through which we can relate a construction/fact in one area of mathematics to a construction/fact in another. The goal is an ultimate form of abstraction, where we can truly single out what about a given problem is specific to that problem, and what is a reflection of a more general phenomenom which appears elsewhere. -
Infinite Dimensional Lie Theory from the Point of View of Functional
Infinite dimensional Lie Theory from the point of view of Functional Analysis Josef Teichmann F¨urmeine Eltern, Michael, Michaela und Hannah Zsofia Hazel. Abstract. Convenient analysis is enlarged by a powerful theory of Hille-Yosida type. More precisely asymptotic spectral properties of bounded operators on a convenient vector space are related to the existence of smooth semigroups in a necessary and sufficient way. An approximation theorem of Trotter-type is proved, too. This approximation theorem is in fact an existence theorem for smooth right evolutions of non-autonomous differential equations on convenient locally convex spaces and crucial for the following applications. To enlighten the generically "unsolved" (even though H. Omori et al. gave interesting and concise conditions for regularity) question of the existence of product integrals on convenient Lie groups, we provide by the given approximation formula some simple criteria. On the one hand linearization is used, on the other hand remarkable families of right invariant distance functions, which exist on all up to now known Lie groups, are the ingredients: Assuming some natural global conditions regularity can be proved on convenient Lie groups. The existence of product integrals is an essential basis for Lie theory in the convenient setting, since generically differential equations cannot be solved on non-normable locally convex spaces. The relationship between infinite dimensional Lie algebras and Lie groups, which is well under- stood in the regular case, is also reviewed from the point of view of local Lie groups: Namely the question under which conditions the existence of a local Lie group for a given convenient Lie algebra implies the existence of a global Lie group is treated by cohomological methods. -
MORITA EQUIVALENCE of C*-CORRESPONDENCES PASSES to the RELATED OPERATOR ALGEBRAS 1. Introduction Introduced by Rieffel in the 19
MORITA EQUIVALENCE OF C*-CORRESPONDENCES PASSES TO THE RELATED OPERATOR ALGEBRAS GEORGE K. ELEFTHERAKIS, EVGENIOS T.A. KAKARIADIS, AND ELIAS G. KATSOULIS Abstract. We revisit a central result of Muhly and Solel on operator algebras of C*-correspondences. We prove that (possibly non-injective) strongly Morita equivalent C*-correspondences have strongly Morita equivalent relative Cuntz-Pimsner C*-algebras. The same holds for strong Morita equivalence (in the sense of Blecher, Muhly and Paulsen) and strong ∆-equivalence (in the sense of Eleftherakis) for the related tensor algebras. In particular, we obtain stable isomorphism of the operator algebras when the equivalence is given by a σ-TRO. As an ap- plication we show that strong Morita equivalence coincides with strong ∆-equivalence for tensor algebras of aperiodic C*-correspondences. 1. Introduction Introduced by Rieffel in the 1970's [45, 46], Morita theory provides an important equivalence relation between C*-algebras. In the past 25 years there have been fruitful extensions to cover more general (possibly nonselfad- joint) spaces of operators. These directions cover (dual) operator algebras and (dual) operator spaces, e.g. [3, 4, 6, 7, 16{22, 34]. There are two main streams in this endeavour. Blecher, Muhly and Paulsen [6] introduced a SME strong Morita equivalence s , along with a Morita Theorem I, where the operator algebras A and B are symmetrically decomposed by two bimodules M and N, i.e. A' M ⊗B N and B' N ⊗A M: SME Morita Theorems II and III for s were provided by Blecher [3]. On the ∆ other hand Eleftherakis [16] introduced a strong ∆-equivalence s that is given by a generalized similarity under a TRO M, i.e. -
Morita Equivalence and Infinite Matrix Rings X
proceedings of the american mathematical society Volume 90, Number 2. February 1984 MORITA EQUIVALENCEAND INFINITE MATRIX RINGS VICTOR CAMILLO Abstract. This paper contains the proof of a theorem conjectured by William Stephenson in his thesis [3], Two rings A and S are Morita equivalent if their categories of right A modules are isomorphic. A standard example of two Morita equivalent rings are A and A„, the n X n matrices over A. It is therefore true that if Rn and Sm are isomorphic for integers n and m then A and S are Morita equivalent. An easy example (below) shows that there are Morita equivalent rings A and S with A„ and Sm not isomorphic for any integers m and n. This means that category isomorphism cannot be reduced to a ring isomorphism in this way. Regard A„ as End A*"1, where A'"1 is a direct sum of n copies of A. Let R(N) be a direct sum of countably many copies of A. It follows from a result of Eilenberg (below) that if A and S are Morita equivalent, then End A(A° * End S(N). We prove the following Theorem. A and S are Morita equivalent if and only if End A(/v> « End S(N), where N is a countably infinite set. Note End R(N) — RN is just the ring of column finite matrices over A. The following example was suggested by Kent Fuller. I would also like to thank him for several helpful conversations. Example. Two Morita equivalent rings A and S with Rn not isomorphic to S„, for any integers n and m. -
On the Smoothness of Functors
Iranian Journal of Mathematical Sciences and Informatics Vol. 5, No. 1 (2010), pp. 27-39 On the Smoothness of Functors A. Bajravania and A. Rastegarb,∗ aDepartment of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P. O. Box 14115-134, Tehran, Iran b Faculty of Mathematics, Sharif University, P. O. Box 11155, Tehran, Iran E-mail: [email protected] E-mail: [email protected] Abstract. In this paper we will try to introduce a good smoothness no- tion for a functor. We consider properties and conditions from geometry and algebraic geometry which we expect a smooth functor should have. Keywords: Abelian Category, First Order Deformations, Multicategory, Tangent Category, Topologizing Subcategory. 2000 Mathematics subject classification: 14A20, 14A15, 14A22. 1. Introduction Nowadays noncommutative algebraic geometry is in the focus of many basic topics in mathematics and mathematical physics. In these fields, any under consideration space is an abelian category and a morphism between noncom- mutative spaces is a functor between abelian categories. So one may ask to gen- eralize some aspects of morphisms between commutative spaces to morphisms between noncommutative ones. One of the important aspects in commuta- tive case is the notion of smoothness of a morphism which is stated in some languages, for example: by lifting property as a universal language, by projec- tivity of relative cotangent sheaves as an algebraic language and by inducing a surjective morphism on tangent spaces as a geometric language. ∗Corresponding Author Received 25 December 2009; Accepted 10 April 2010 c 2010 Academic Center for Education, Culture and Research TMU 27 28 A.