Categories and Non-Commutative Geometry
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Commutative Hopf Algebras and Cocommutative Hopf Algebras in Positive Characteristic
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 79, 375-392 (1982) Commutative Hopf Algebras and Cocommutative Hopf Algebras in Positive Characteristic MITSUHIRO TAKEUCHI * Department of Mathematics, University of Tsukuba, Ibaraki 305, Japan Communicated by N. Jacobson Received January 1, 1981 Let A be a commutative Hopf algebra over a field k of characteristic p > 0. Let o: C + B be a surjective map of commutative algebras such that xP = 0 for any x in Ker(yl), so that the map Fc: k’@ @ C+ C, 10 a t+ lPaP factors through k’lp @ 9, yielding a map $: k”P @ B -+ C. A map of algebras f: A + B can be lifted to an algebra map f: A -+ C such that f = q 0 f if and only if Ker(F,) c Ker(Fr o (k”” @ f )). In particular, if FA is injective, any algebra map A + B can be lifted to A + C. The dual results will be given for cocommutative Hopf algebras and coalgebra maps. It is known that every reduced algebraic group over a perfect field is smooth [ 1, p. 2391. This means that if A is a finitely generated commutative Hopf algebra over a field k such that k@ A is reduced, then for any surjective map of commutative algebras p: C+ B whose kernel Ker(q) is nilpotent, any algebra mapf: A + B can be lifted to an algebra mapf: A -+ C such that v, of= f [I, Corollary 4.6, p. -
From Topological Field Theory to Deformation Quantization and Reduction
Cattaneo, Alberto (2006). From topological field theory to deformation quantization and reduction. In: International Congress of Mathematicians. Vol. III. 339-365. Postprint available at: http://www.zora.uzh.ch University of Zurich Posted at the Zurich Open Repository and Archive, University of Zurich. Zurich Open Repository and Archive http://www.zora.uzh.ch Originally published at: 2006. International Congress of Mathematicians. Vol. III. 339-365. Winterthurerstr. 190 CH-8057 Zurich http://www.zora.uzh.ch Year: 2006 From topological field theory to deformation quantization and reduction Cattaneo, Alberto Cattaneo, Alberto (2006). From topological field theory to deformation quantization and reduction. In: International Congress of Mathematicians. Vol. III. 339-365. Postprint available at: http://www.zora.uzh.ch Posted at the Zurich Open Repository and Archive, University of Zurich. http://www.zora.uzh.ch Originally published at: 2006. International Congress of Mathematicians. Vol. III. 339-365. From topological field theory to deformation quantization and reduction Alberto S. Cattaneo∗ Abstract. This note describes the functional-integral quantization of two-dimensional topolog- ical field theories together with applications to problems in deformation quantization of Poisson manifolds and reduction of certain submanifolds. A brief introduction to smooth graded mani- folds and to the Batalin–Vilkovisky formalism is included. Mathematics Subject Classification (2000). Primary 81T45; Secondary 51P05, 53D55, 58A50, 81T70. Keywords. Topological quantum field theory, BV formalism, graded manifolds, deformation quantization, formality, Poisson reduction, L∞- and A∞-algebras. 1. Introduction: a 2D TFT 0 1.1. The basic setting. Let be a smooth compact 2-manifold. On M1 := ()⊕ 1() one may define the following very simple action functional: S(ξ,η) := η dξ, ξ ∈ 0(), η ∈ 1(), (1.1) which is invariant under the distribution 0 ⊕ dβ, β ∈ 0(). -
Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and Their Applications
ARCHIVUM MATHEMATICUM (BRNO) Tomus 47 (2011), 415–471 INTRODUCTION TO GRADED GEOMETRY, BATALIN-VILKOVISKY FORMALISM AND THEIR APPLICATIONS Jian Qiu and Maxim Zabzine Abstract. These notes are intended to provide a self-contained introduction to the basic ideas of finite dimensional Batalin-Vilkovisky (BV) formalism and its applications. A brief exposition of super- and graded geometries is also given. The BV–formalism is introduced through an odd Fourier transform and the algebraic aspects of integration theory are stressed. As a main application we consider the perturbation theory for certain finite dimensional integrals within BV-formalism. As an illustration we present a proof of the isomorphism between the graph complex and the Chevalley-Eilenberg complex of formal Hamiltonian vectors fields. We briefly discuss how these ideas can be extended to the infinite dimensional setting. These notes should be accessible toboth physicists and mathematicians. Table of Contents 1. Introduction and motivation 416 2. Supergeometry 417 2.1. Idea 418 2.2. Z2-graded linear algebra 418 2.3. Supermanifolds 420 2.4. Integration theory 421 3. Graded geometry 423 3.1. Z-graded linear algebra 423 3.2. Graded manifold 425 4. Odd Fourier transform and BV-formalism 426 4.1. Standard Fourier transform 426 4.2. Odd Fourier transform 427 4.3. Integration theory 430 4.4. Algebraic view on the integration 433 5. Perturbation theory 437 5.1. Integrals in Rn-Gaussian Integrals and Feynman Diagrams 437 2010 Mathematics Subject Classification: primary 58A50; secondary 16E45, 97K30. Key words and phrases: Batalin-Vilkovisky formalism, graded symplectic geometry, graph homology, perturbation theory. -
The Formal Theory of Hopf Algebras, Part II
The Formal Theory of Hopf Algebras Part II: The case of Hopf algebras Hans{E. Porst Department of Mathematical Sciences, University of Stellenbosch, Stellenbosch, South Africa [email protected] Abstract The category HopfR of Hopf algebras over a commutative unital ring R is analyzed with respect to its categorical properties. The main results are: (1) For every ring R the category HopfR is locally presentable, it is coreflective in the category of bialgebras over R, over every R-algebra there exists a cofree Hopf algebra. (2) If, in addition, R is absoluty flat, then HopfR is reflective in the category of bialgebras as well, and there exists a free Hopf algebra over every R-coalgebra. Similar results are obtained for relevant subcategories of HopfR. Moreover it is shown that, for every commutative unital ring R, the so-called \dual algebra functor" has a left adjoint and that, more generally, universal measuring coalgebras exist. MSC 2000: Primary 16T05, Secondary 18D10 Keywords: Hopf algebras, ?bialgebras, limits, colimits, free Hopf algebras, cofree Hopf algebras, Hopf envelope, universal measuring coalgebra Introduction The first monograph on Hopf algebra theory (Sweedler 1969 [27]) paid considerable attention to categorical properties of those. Somewhat surprisingly more recent successors like e.g. [11] | though using categorical language throughout | hardly touch these questions. The question we here have in mind are, e.g., Does the category of Hopf algebras have products (or, more generally, all limits) and how are they constructed? Does it have colimits? Or Do the naturally occurring functors (e.g., the embedding of the category of Hopf algebras into that of bialgebras) have adjoints? Some scattered results exist, as for example 1. -
A Necessary and Sufficient Condition for Induced Model Structures
A NECESSARY AND SUFFICIENT CONDITION FOR INDUCED MODEL STRUCTURES KATHRYN HESS, MAGDALENA KE¸DZIOREK, EMILY RIEHL, AND BROOKE SHIPLEY Abstract. A common technique for producing a new model category struc- ture is to lift the fibrations and weak equivalences of an existing model struc- ture along a right adjoint. Formally dual but technically much harder is to lift the cofibrations and weak equivalences along a left adjoint. For either technique to define a valid model category, there is a well-known necessary \acyclicity" condition. We show that for a broad class of accessible model structures | a generalization introduced here of the well-known combinato- rial model structures | this necessary condition is also sufficient in both the right-induced and left-induced contexts, and the resulting model category is again accessible. We develop new and old techniques for proving the acyclity condition and apply these observations to construct several new model struc- tures, in particular on categories of differential graded bialgebras, of differential graded comodule algebras, and of comodules over corings in both the differen- tial graded and the spectral setting. We observe moreover that (generalized) Reedy model category structures can also be understood as model categories of \bialgebras" in the sense considered here. Contents 1. Introduction 2 1.1. Acknowledgments 5 2. Accessible model categories, acyclicity, and induced model structures 5 2.1. Induced model structures and acyclicity 5 2.2. Techniques for proving acyclicity 8 2.3. Adjoint squares 11 3. Left and right induced algebraic weak factorization systems 13 3.1. Algebraic weak factorization systems and accessible model categories 14 3.2. -
Cocommutative Coalgebras: Homotopy Theory and Koszul Duality
Homology, Homotopy and Applications, vol. 18(2), 2016, pp.303–336 COCOMMUTATIVE COALGEBRAS: HOMOTOPY THEORY AND KOSZUL DUALITY JOSEPH CHUANG, ANDREY LAZAREV and W. H. MANNAN (communicated by J.P.C. Greenlees) Abstract We extend a construction of Hinich to obtain a closed model category structure on all differential graded cocommutative coalgebras over an algebraically closed field of characteristic zero. We further show that the Koszul duality between commu- tative and Lie algebras extends to a Quillen equivalence between cocommutative coalgebras and formal coproducts of curved Lie algebras. 1. Introduction Differential graded (dg) coalgebras arise naturally as invariants of topological spaces, for example as the normalized singular chains of a space. They also serve as representing objects for formal deformation functors [8, 13, 16] and feature promi- nently in rational homotopy theory [17, 14]. As such it is natural to ask if they may be placed in the framework of a closed model category (CMC), at least in the case of cocommutative dg coalgebras over a field of characteristic zero. The first result of this kind is due to D. Quillen [17] under a rather strong connectivity as- sumption; this assumption was subsequently removed by Hinich in [8]. The crucial difference between these two approaches is that Quillen defined weak equivalences to be quasi-isomorphisms whereas Hinich considered a finer (i.e., harder to satisfy) notion of a filtered quasi-isomorphism. A particularly nice feature of Hinich’s model is the so-called Koszul duality: it turns out to be Quillen equivalent to the category of dg Lie algebras, again without any grading restrictions. -
Cofree Coalgebras and Differential Linear Logic
Cofree coalgebras and differential linear logic James Clift, Daniel Murfet January 9, 2019 Abstract We prove that the semantics of intuitionistic linear logic in vector spaces which uses cofree coalgebras is also a model of differential linear logic, and that the Carte- sian closed category of cofree coalgebras is a model of the simply-typed differential lambda calculus. Contents 1 Introduction 2 2 Sweedler semantics 5 2.1 Cofreecoalgebras ................................ 5 2.2 Group-like elements are points . 13 2.3 Primitive elements are tangent vectors . 13 2.4 Definition of the Sweedler semantics . 14 2.5 Local cohomology and distributions . 17 3 Differential linear logic 18 3.1 Codereliction, cocontraction, coweakening . ... 23 arXiv:1701.01285v2 [cs.LO] 7 Jan 2019 4 Examples 25 4.1 Churchnumerals ................................ 26 4.2 Binaryintegers ................................. 27 4.3 Multiplication . 31 5 Differential lambda calculus 32 5.1 Cartesian differential categories . 36 A Kleisli categories and coalgebras 43 1 In the discrete world of computing, there is no meaningful metric in which “small” changes and “small” effects go hand in hand, and there never will be. E.W.Dijkstra, On the cruelty of really teaching computer science 1 Introduction The idea of taking derivatives of programs is an old one [34, §2] with manifestations in- cluding automatic differentiation of algorithms computing real-valued functions [8] and incremental computation [35]. However, these approaches are limited to restricted classes of computations, and it is only recently with the development of the differential λ-calculus by Ehrhard-Regnier [13] and its refinement by differential linear logic [2, 11] that deriva- tives have been defined for general higher-order programs. -
Graded-And-Microformal-Prop
The University of Manchester Research Graded Geometry, Q-Manifolds, and Microformal Geometry DOI: 10.1002/prop.201910023 Document Version Accepted author manuscript Link to publication record in Manchester Research Explorer Citation for published version (APA): Voronov, T. (2019). Graded Geometry, Q-Manifolds, and Microformal Geometry: LMS/EPSRC Durham Symposium on Higher Structures in M-Theory. Fortschritte der Physik. https://doi.org/10.1002/prop.201910023 Published in: Fortschritte der Physik Citing this paper Please note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscript or Proof version this may differ from the final Published version. If citing, it is advised that you check and use the publisher's definitive version. General rights Copyright and moral rights for the publications made accessible in the Research Explorer are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Takedown policy If you believe that this document breaches copyright please refer to the University of Manchester’s Takedown Procedures [http://man.ac.uk/04Y6Bo] or contact [email protected] providing relevant details, so we can investigate your claim. Download date:06. Oct. 2021 Graded geometry, Q-manifolds, and microformal geometry 1,2 Theodore Th. Voronov ¤ below), to which we give a brief introduction as well. We give an exposition of graded and microformal ge- “Thick morphisms” (defined for ordinary manifolds, ometry, and the language of Q-manifolds. Q-manifolds supermanifolds or graded manifolds) generalize ordinary are supermanifolds endowed with an odd vector field of maps or supermanifold morphisms, but are not maps square zero. -
Torsion Constraints in Supergeometry
Commun. Math. Phys. 133, 563-615(1990) Communications ΪΠ Mathematical Physics ©Springer-Verlagl990 Torsion Constraints in Supergeometry John Lott* I.H.E.S., F-91440 Bures-sur-Yvette, France Received November 9, 1989; in revised form April 2, 1990 Abstract. We derive the torsion constraints for superspace versions of supergravity theories by means of the theory of G-stmctures. We also discuss superconformal geometry and superKahler geometry. I. Introduction Supersymmetry is a now well established topic in quantum field theory [WB, GGRS]. The basic idea is that one can construct actions in ordinary spacetime which involve both even commuting fields and odd anticommuting fields, with a symmetry which mixes the two types of fields. These actions can then be interpreted as arising from actions in a superspace with both even and odd coordinates, upon doing a partial integration over the odd coordinates. A mathematical framework to handle the differential topology of supermanifolds, manifolds with even and odd coordinates, was developed by Berezin, Kostant and others. A very readable account of this theory is given in the book of Manin [Ma]. The right notion of differential geometry for supermanifolds is less clear. Such a geometry is necessary in order to write supergravity theories in superspace. One could construct a supergeometry by ΊL2 grading what one usually does in (pseudo) Riemannian geometry, to have supermetrics, super Levi-Civita connections, etc. The local frame group which would take the place of the orthogonal group in standard geometry would be the orthosymplectic group. However, it turns out that this would be physically undesirable. Such a program would give more fields than one needs for a minimal supergravity theory, i.e. -
Introduction to Graded Geometry
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector European Journal of Mathematics DOI 10.1007/s40879-017-0138-4 RESEARCH/REVIEW ARTICLE Introduction to graded geometry Maxime Fairon1 Received: 7 November 2016 / Revised: 1 March 2017 / Accepted: 8 March 2017 © The Author(s) 2017. This article is an open access publication Abstract This paper aims at setting out the basics of Z-graded manifolds theory. We introduce Z-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and exhibit their graded local basis. The paper also reviews some correspondences between differential Z-graded manifolds and algebraic structures. Keywords Supergeometry · Graded manifold · Differential graded manifold · Q-manifold Mathematics Subject Classification 58A50 · 51-02 1 Introduction In the 70s, supermanifolds were introduced and studied to provide a geometric back- ground to the developing theory of supersymmetry. In the Berezin–Leites [2] and Kostant [11] approach, they are defined as Z2-graded locally ringed spaces. More pre- cisely, this means that a supermanifold is a pair (|N|, ON ) such that |N| is a topological space and ON is a sheaf of Z2-graded algebras which satisfies, for all sufficiently small open subsets U, ∞ m ON (U) CRn (U)⊗ R , B Maxime Fairon [email protected] 1 School of Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom 123 M. Fairon m where R is endowed with its canonical Z2-grading. -
Cofree Coalgebras Over Operads and Representative Functions
Cofree coalgebras over operads and representative functions M. Anel September 16, 2014 Abstract We give a recursive formula to compute the cofree coalgebra P _(C) over any colored operad P in V = Set; CGHaus; (dg)Vect. The construction is closed to that of [Smith] but different. We use a more conceptual approach to simplify the proofs that P _ is the cofree P -coalgebra functor and also the comonad generating P -coalgebras. In a second part, when V = (dg)Vect is the category of vector spaces or chain com- plexes over a field, we generalize to operads the notion of representative functions of [Block-Leroux] and prove that P _(C) is simply the subobject of representative elements in the "completed P -algebra" P ^(C). This says that our recursion (as well as that of [Smith]) stops at the first step. Contents 1 Introduction 2 2 The cofree coalgebra 4 2.1 Analytic functors . .4 2.2 Einstein convention . .5 2.3 Operads and associated functors . .7 2.4 Coalgebras over an operad . 10 2.5 The comonad of coendomorphisms . 12 2.6 Lax comonads and their coalgebras . 13 2.7 The coreflection theorem . 16 3 Operadic representative functions 21 3.1 Representative functions . 23 3.2 Translations and recursive functions . 25 3.3 Cofree coalgebras and representative functions . 27 1 1 Introduction This work has two parts. In a first part we prove that coalgebras over a (colored) operad P are coalgebras over a certain comonad P _ by giving a recursive construction of P _. In a second part we prove that the recursion is unnecessary in the case where the operad is enriched over vector spaces or chain complexes. -
Cofree Compositions of Coalgebras
FPSAC 2011, Reykjavik, Iceland DMTCS proc. (subm.), by the authors, 1–12 Cofree compositions of coalgebras Stefan Forcey1, Aaron Lauve2†, and Frank Sottile3‡ 1Department of Theoretical and Applied Mathematics, The University of Akron, Akron, OH 44325 USA 2Department of Mathematics, Loyola University of Chicago, Chicago, IL 60660 USA 3Department of Mathematics, Texas A&M University, College Station, TX 77843 USA Abstract. We develop the notion of the composition of two coalgebras, which arises naturally in higher category theory and the theory of species. We prove that the composition of two cofree coalgebras is cofree and give conditions which imply that the composition is a one-sided Hopf algebra. These conditions hold when one coalgebra is a graded Hopf operad D and the other is a connected graded coalgebra with coalgebra map to D. We conclude by discussing these structures for compositions with bases the vertices of multiplihedra, composihedra, and hypercubes. Resum´ e.´ Keywords: multiplihedron, cofree coalgebra, Hopf algebra, operad, species 1 Introduction The Hopf algebras of ordered trees (Malvenuto and Reutenauer (1995)) and of planar binary trees (Loday and Ronco (1998)) are cofree coalgebras that are connected by cellular maps from permutahedra to as- sociahedra. Related polytopes include the multiplihedra (Stasheff (1970)) and the composihedra (Forcey (2008b)), and it is natural to study what Hopf structures may be placed on these objects. The map from permutahedra to associahedra factors through the multiplihedra, and in (Forcey et al. (2010)) we used this factorization to place Hopf structures on bi-leveled trees, which correspond to vertices of multiplihedra. Multiplihedra form an operad module over associahedra.