Operads, Algebras and Modules in Model Categories and Motives
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Arxiv:2001.09075V1 [Math.AG] 24 Jan 2020
A topos-theoretic view of difference algebra Ivan Tomašić Ivan Tomašić, School of Mathematical Sciences, Queen Mary Uni- versity of London, London, E1 4NS, United Kingdom E-mail address: [email protected] arXiv:2001.09075v1 [math.AG] 24 Jan 2020 2000 Mathematics Subject Classification. Primary . Secondary . Key words and phrases. difference algebra, topos theory, cohomology, enriched category Contents Introduction iv Part I. E GA 1 1. Category theory essentials 2 2. Topoi 7 3. Enriched category theory 13 4. Internal category theory 25 5. Algebraic structures in enriched categories and topoi 41 6. Topos cohomology 51 7. Enriched homological algebra 56 8. Algebraicgeometryoverabasetopos 64 9. Relative Galois theory 70 10. Cohomologyinrelativealgebraicgeometry 74 11. Group cohomology 76 Part II. σGA 87 12. Difference categories 88 13. The topos of difference sets 96 14. Generalised difference categories 111 15. Enriched difference presheaves 121 16. Difference algebra 126 17. Difference homological algebra 136 18. Difference algebraic geometry 142 19. Difference Galois theory 148 20. Cohomologyofdifferenceschemes 151 21. Cohomologyofdifferencealgebraicgroups 157 22. Comparison to literature 168 Bibliography 171 iii Introduction 0.1. The origins of difference algebra. Difference algebra can be traced back to considerations involving recurrence relations, recursively defined sequences, rudi- mentary dynamical systems, functional equations and the study of associated dif- ference equations. Let k be a commutative ring with identity, and let us write R = kN for the ring (k-algebra) of k-valued sequences, and let σ : R R be the shift endomorphism given by → σ(x0, x1,...) = (x1, x2,...). The first difference operator ∆ : R R is defined as → ∆= σ id, − and, for r N, the r-th difference operator ∆r : R R is the r-th compositional power/iterate∈ of ∆, i.e., → r r ∆r = (σ id)r = ( 1)r−iσi. -
An Introduction to Operad Theory
AN INTRODUCTION TO OPERAD THEORY SAIMA SAMCHUCK-SCHNARCH Abstract. We give an introduction to category theory and operad theory aimed at the undergraduate level. We first explore operads in the category of sets, and then generalize to other familiar categories. Finally, we develop tools to construct operads via generators and relations, and provide several examples of operads in various categories. Throughout, we highlight the ways in which operads can be seen to encode the properties of algebraic structures across different categories. Contents 1. Introduction1 2. Preliminary Definitions2 2.1. Algebraic Structures2 2.2. Category Theory4 3. Operads in the Category of Sets 12 3.1. Basic Definitions 13 3.2. Tree Diagram Visualizations 14 3.3. Morphisms and Algebras over Operads of Sets 17 4. General Operads 22 4.1. Basic Definitions 22 4.2. Morphisms and Algebras over General Operads 27 5. Operads via Generators and Relations 33 5.1. Quotient Operads and Free Operads 33 5.2. More Examples of Operads 38 5.3. Coloured Operads 43 References 44 1. Introduction Sets equipped with operations are ubiquitous in mathematics, and many familiar operati- ons share key properties. For instance, the addition of real numbers, composition of functions, and concatenation of strings are all associative operations with an identity element. In other words, all three are examples of monoids. Rather than working with particular examples of sets and operations directly, it is often more convenient to abstract out their common pro- perties and work with algebraic structures instead. For instance, one can prove that in any monoid, arbitrarily long products x1x2 ··· xn have an unambiguous value, and thus brackets 2010 Mathematics Subject Classification. -
Arxiv:Math/0701299V1 [Math.QA] 10 Jan 2007 .Apiain:Tfs F,S N Oooyagba 11 Algebras Homotopy and Tqfts ST CFT, Tqfts, 9.1
FROM OPERADS AND PROPS TO FEYNMAN PROCESSES LUCIAN M. IONESCU Abstract. Operads and PROPs are presented, together with examples and applications to quantum physics suggesting the structure of Feynman categories/PROPs and the corre- sponding algebras. Contents 1. Introduction 2 2. PROPs 2 2.1. PROs 2 2.2. PROPs 3 2.3. The basic example 4 3. Operads 4 3.1. Restricting a PROP 4 3.2. The basic example 5 3.3. PROP generated by an operad 5 4. Representations of PROPs and operads 5 4.1. Morphisms of PROPs 5 4.2. Representations: algebras over a PROP 6 4.3. Morphisms of operads 6 5. Operations with PROPs and operads 6 5.1. Free operads 7 5.1.1. Trees and forests 7 5.1.2. Colored forests 8 5.2. Ideals and quotients 8 5.3. Duality and cooperads 8 6. Classical examples of operads 8 arXiv:math/0701299v1 [math.QA] 10 Jan 2007 6.1. The operad Assoc 8 6.2. The operad Comm 9 6.3. The operad Lie 9 6.4. The operad Poisson 9 7. Examples of PROPs 9 7.1. Feynman PROPs and Feynman categories 9 7.2. PROPs and “bi-operads” 10 8. Higher operads: homotopy algebras 10 9. Applications: TQFTs, CFT, ST and homotopy algebras 11 9.1. TQFTs 11 1 9.1.1. (1+1)-TQFTs 11 9.1.2. (0+1)-TQFTs 12 9.2. Conformal Field Theory 12 9.3. String Theory and Homotopy Lie Algebras 13 9.3.1. String backgrounds 13 9.3.2. -
Algebra + Homotopy = Operad
Symplectic, Poisson and Noncommutative Geometry MSRI Publications Volume 62, 2014 Algebra + homotopy = operad BRUNO VALLETTE “If I could only understand the beautiful consequences following from the concise proposition d 2 0.” —Henri Cartan D This survey provides an elementary introduction to operads and to their ap- plications in homotopical algebra. The aim is to explain how the notion of an operad was prompted by the necessity to have an algebraic object which encodes higher homotopies. We try to show how universal this theory is by giving many applications in algebra, geometry, topology, and mathematical physics. (This text is accessible to any student knowing what tensor products, chain complexes, and categories are.) Introduction 229 1. When algebra meets homotopy 230 2. Operads 239 3. Operadic syzygies 253 4. Homotopy transfer theorem 272 Conclusion 283 Acknowledgements 284 References 284 Introduction Galois explained to us that operations acting on the solutions of algebraic equa- tions are mathematical objects as well. The notion of an operad was created in order to have a well defined mathematical object which encodes “operations”. Its name is a portemanteau word, coming from the contraction of the words “operations” and “monad”, because an operad can be defined as a monad encoding operations. The introduction of this notion was prompted in the 60’s, by the necessity of working with higher operations made up of higher homotopies appearing in algebraic topology. Algebra is the study of algebraic structures with respect to isomorphisms. Given two isomorphic vector spaces and one algebra structure on one of them, 229 230 BRUNO VALLETTE one can always define, by means of transfer, an algebra structure on the other space such that these two algebra structures become isomorphic. -
Two Constructions in Monoidal Categories Equivariantly Extended Drinfel’D Centers and Partially Dualized Hopf Algebras
Two constructions in monoidal categories Equivariantly extended Drinfel'd Centers and Partially dualized Hopf Algebras Dissertation zur Erlangung des Doktorgrades an der Fakult¨atf¨urMathematik, Informatik und Naturwissenschaften Fachbereich Mathematik der Universit¨atHamburg vorgelegt von Alexander Barvels Hamburg, 2014 Tag der Disputation: 02.07.2014 Folgende Gutachter empfehlen die Annahme der Dissertation: Prof. Dr. Christoph Schweigert und Prof. Dr. Sonia Natale Contents Introduction iii Topological field theories and generalizations . iii Extending braided categories . vii Algebraic structures and monoidal categories . ix Outline . .x 1. Algebra in monoidal categories 1 1.1. Conventions and notations . .1 1.2. Categories of modules . .3 1.3. Bialgebras and Hopf algebras . 12 2. Yetter-Drinfel'd modules 25 2.1. Definitions . 25 2.2. Equivalences of Yetter-Drinfel'd categories . 31 3. Graded categories and group actions 39 3.1. Graded categories and (co)graded bialgebras . 39 3.2. Weak group actions . 41 3.3. Equivariant categories and braidings . 48 4. Equivariant Drinfel'd center 51 4.1. Half-braidings . 51 4.2. The main construction . 55 4.3. The Hopf algebra case . 61 5. Partial dualization of Hopf algebras 71 5.1. Radford biproduct and projection theorem . 71 5.2. The partial dual . 73 5.3. Examples . 75 A. Category theory 89 A.1. Basic notions . 89 A.2. Adjunctions and monads . 91 i ii Contents A.3. Monoidal categories . 92 A.4. Modular categories . 97 References 99 Introduction The fruitful interplay between topology and algebra has a long tradi- tion. On one hand, invariants of topological spaces, such as the homotopy groups, homology groups, etc. -
Categories of Modules with Differentials
JOURNAL OF ALGEBRA 185, 50]73Ž. 1996 ARTICLE NO. 0312 Categories of Modules with Differentials Paul Popescu Department of Mathematics, Uni¨ersity of Craio¨a, 13, A.I. Cuza st., Craio¨a, 1100, Romania Communicated by Walter Feit CORE Metadata, citation and similar papers at core.ac.uk Received August 1, 1994 Provided by Elsevier - Publisher Connector 1. INTRODUCTION The definitions of the module with arrow Ž.module fleche , the infinitesi- mal module, and the Lie pseudoalgebra, as used here, are considered inwx 7 . Moreover, we define the preinfinitesimal module, called inwx 1 ``un pre- espace d'Elie Cartan regulier.'' Inwx 7 the Lie functor is constructed from the category of differentiable groupoids in the category of Lie algebroids, but it is inwx 2 that the first general and abstract treatment of the algebraic properties of Lie alge- broids is made, giving a clear construction of the morphisms of Lie algebroids. We shall use it fully in this paper to define completely the categories M W A Ž.Ž.modules and arrows , P I M preinfinitesimal modules , IMŽ.Ž.infinitesimal modules , and L P A Lie pseudoalgebras ; we call these categories the categories of modules with differentials. We notice that in wx7 and wx 1 the objects of these categories and the subcategories of modules over a fixed algebra are considered. Inwx 3 the category of L P A of Lie pseudoalgebras is defined and some aspects related to the functional covariance or contravariance correspondence with categories constructed with additional structures on vector bundles are studied. In Section 2 we give a brief description of the category MA , of modules over commutative k-algebrasŽ considered also inwx 3. -
Notes on the Lie Operad
Notes on the Lie Operad Todd H. Trimble Department of Mathematics University of Chicago These notes have been reLATEXed by Tim Hosgood, who takes responsibility for any errors. The purpose of these notes is to compare two distinct approaches to the Lie operad. The first approach closely follows work of Joyal [9], which gives a species-theoretic account of the relationship between the Lie operad and homol- ogy of partition lattices. The second approach is rooted in a paper of Ginzburg- Kapranov [7], which generalizes within the framework of operads a fundamental duality between commutative algebras and Lie algebras. Both approaches in- volve a bar resolution for operads, but in different ways: we explain the sense in which Joyal's approach involves a \right" resolution and the Ginzburg-Kapranov approach a \left" resolution. This observation is exploited to yield a precise com- parison in the form of a chain map which induces an isomorphism in homology, between the two approaches. 1 Categorical Generalities Let FB denote the category of finite sets and bijections. FB is equivalent to the permutation category P, whose objects are natural numbers and whose set of P morphisms is the disjoint union Sn of all finite symmetric groups. P (and n>0 therefore also FB) satisfies the following universal property: given a symmetric monoidal category C and an object A of C, there exists a symmetric monoidal functor P !C which sends the object 1 of P to A and this functor is unique up to a unique monoidal isomorphism. (Cf. the corresponding property for the braid category in [11].) Let V be a symmetric monoidal closed category, with monoidal product ⊕ and unit 1. -
The Elliptic Drinfeld Center of a Premodular Category Arxiv
The Elliptic Drinfeld Center of a Premodular Category Ying Hong Tham Abstract Given a tensor category C, one constructs its Drinfeld center Z(C) which is a braided tensor category, having as objects pairs (X; λ), where X ∈ Obj(C) and λ is a el half-braiding. For a premodular category C, we construct a new category Z (C) which 1 2 i we call the Elliptic Drinfeld Center, which has objects (X; λ ; λ ), where the λ 's are half-braidings that satisfy some compatibility conditions. We discuss an SL2(Z)-action el on Z (C) that is related to the anomaly appearing in Reshetikhin-Turaev theory. This construction is motivated from the study of the extended Crane-Yetter TQFT, in particular the category associated to the once punctured torus. 1 Introduction and Preliminaries In [CY1993], Crane and Yetter define a 4d TQFT using a state-sum involving 15j symbols, based on a sketch by Ooguri [Oog1992]. The state-sum begins with a color- ing of the 2- and 3-simplices of a triangulation of the four manifold by integers from 0; 1; : : : ; r. These 15j symbols then arise as the evaluation of a ribbon graph living on the boundary of a 4-simplex. The labels 0; 1; : : : ; r correspond to simple objects of the Verlinde modular category, the semi-simple subquotient of the category of finite dimen- πi r 2 sional representations of the quantum group Uqsl2 at q = e as defined in [AP1995]. Later Crane, Kauffman, and Yetter [CKY1997] extend this~ definition+ to colorings with objects from a premodular category (i.e. -
Stable Model Categories Are Categories of Modules 1
STABLE MODEL CATEGORIES ARE CATEGORIES OF MODULES STEFAN SCHWEDE AND BROOKE SHIPLEY Abstract: A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard’s work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R. 1. Introduction The recent discovery of highly structured categories of spectra has opened the way for a new wholesale use of algebra in stable homotopy theory. In this paper we use this new algebra of spectra to characterize stable model categories, the settings for doing stable homotopy theory, as categories of highly structured modules. This characterization also leads to a Morita theory for equivalences between categories of highly structured modules. -
Arxiv:1909.05807V1 [Math.RA] 12 Sep 2019 Cs T
MODULES OVER TRUSSES VS MODULES OVER RINGS: DIRECT SUMS AND FREE MODULES TOMASZ BRZEZINSKI´ AND BERNARD RYBOLOWICZ Abstract. Categorical constructions on heaps and modules over trusses are con- sidered and contrasted with the corresponding constructions on groups and rings. These include explicit description of free heaps and free Abelian heaps, coprod- ucts or direct sums of Abelian heaps and modules over trusses, and description and analysis of free modules over trusses. It is shown that the direct sum of two non-empty Abelian heaps is always infinite and isomorphic to the heap associated to the direct sums of the group retracts of both heaps and Z. Direct sum is used to extend a given truss to a ring-type truss or a unital truss (or both). Free mod- ules are constructed as direct sums of a truss. It is shown that only free rank-one modules are free as modules over the associated truss. On the other hand, if a (finitely generated) module over a truss associated to a ring is free, then so is the corresponding quotient-by-absorbers module over this ring. 1. Introduction Trusses and skew trusses were defined in [3] in order to capture the nature of the distinctive distributive law that characterises braces and skew braces [12], [6], [9]. A (one-sided) truss is a set with a ternary operation which makes it into a heap or herd (see [10], [11], [1] or [13]) together with an associative binary operation that distributes (on one side or both) over the heap ternary operation. If the specific bi- nary operation admits it, a choice of a particular element could fix a group structure on a heap in a way that turns the truss into a ring or a brace-like system (which becomes a brace provided the binary operation admits inverses). -
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. -
Derived Categories of Modules and Coherent Sheaves
Derived Categories of Modules and Coherent Sheaves Yuriy A. Drozd Abstract We present recent results on derived categories of modules and coherent sheaves, namely, tame{wild dichotomy and semi-continuity theorem for derived categories over finite dimensional algebras, as well as explicit calculations for derived categories of modules over nodal rings and of coherent sheaves over projective configurations of types A and A~. This paper is a survey of some recent results on the structure of derived categories obtained by the author in collaboration with Viktor Bekkert and Igor Burban [6, 11, 12]. The origin of this research was the study of Cohen{ Macaulay modules and vector bundles by Gert-Martin Greuel and myself [27, 28, 29, 30] and some ideas from the work of Huisgen-Zimmermann and Saor´ın [42]. Namely, I understood that the technique of \matrix problems," briefly explained below in subsection 2.3, could be successfully applied to the calculations in derived categories, almost in the same way as it was used in the representation theory of finite-dimensional algebras, in study of Cohen{ Macaulay modules, etc. The first step in this direction was the semi-continuity theorem for derived categories [26] presented in subsection 2.1. Then Bekkert and I proved the tame{wild dichotomy for derived categories over finite di- mensional algebras (see subsection 2.2). At the same time, Burban and I described the indecomposable objects in the derived categories over nodal rings (see Section 3) and projective configurations of types A and A~ (see Sec- tion 4). Note that it follows from [23, 29] that these are the only cases, where such a classification is possible; for all other pure noetherian rings (or projec- tive curves) even the categories of modules (respectively, of vector bundles) are wild.