DOCSLIB.ORG

Drinfeld Centers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Drinfeld Centers U.U.D.M. Project Report 2018:3 Drinfeld centers Markus Thuresson Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev April 2018 Department of Mathematics Uppsala University Drinfeld centers Markus Thuresson Abstract In the first part of this paper, we describe the structure of the center Z(R-Mod) of the category of left R-modules. Its natural structure as a ring is shown to be isomorphic to the subring Z(R). In the sections that follow, we present the basics of monoidal categories by regarding them as single-object bicategories. The Drinfeld center Z(C ) of a monoidal category C is defined and its basic properties presented. The second half of the paper is devoted to describing the structure of the Drinfeld center of the monoidal categories VectC and Z2-mod. Contents 1 Basics of cateogries . 3 2 Categories of modules . 6 2.1 The center of R-mod . 6 2.2 Tensor product of modules . 11 2.2.1 Tensor product of bimodules . 17 3 2-categories . 21 4 Bicategories . 24 4.1 Basics . 24 4.2 Coherence . 32 5 Monoidal categories . 33 5.1 The Drinfeld center . 34 6 The Drinfeld center of VectC .............................. 37 6.1 Equivalence . 45 7 Categories of group representations . 45 7.1 Drinfeld centers . 50 8 The Drinfeld center of Z2-mod ............................. 55 Markus Thuresson 3 1 Basics of cateogries In order to make this paper as self-contained as possible, we present the basics of category theory. This secion, and those similar to it, are of course highly skippable. Definition 1.1. A category C consists of the following: - a class Ob (C) of objects. - for every pair of objects X; Y 2 Ob (C), a class of morphisms or arrows, denoted by Hom (X; Y ). In particular, for the pair (X; X), we require the existence of an identity morphism idX : X ! X. - for every three objects X; Y; Z 2 Ob (C) and morphisms ' 2 Hom (X; Y ) and 2 Hom (Y; Z), a binary operation ◦ : Hom (X; Y ) × Hom (X; Z) ! Hom (Y; Z) with ('; ) 7! ◦ ', called the composition. We require that the composition satisfies the following axioms: i) if ' 2 Hom (X; Y ) ; 2 Hom (Y; Z) and ξ 2 Hom (Z; W ) then ξ ◦ ( ◦ ') = (ξ ◦ ) ◦ '. ii) if ' 2 Hom (X; Y ) and 2 Hom (Y; Z) then ' ◦ idX = ' and idZ ◦ = . 0 Remark 1.2. The identity morphism idX is unique for every object X 2 Ob (C). If idX were another identity morphism, we would have 0 0 idX = idX ◦ idX = idX : Definition 1.3. A category C is said to be small if both the class of objects and the morphism classes are sets, and not proper classes. Definition 1.4. The terminal category, denoted by 11, is the category having a single object and a single morphism(the identity). Definition 1.5. Let C be a category. A morphism ' 2 Hom (X; Y ) is called an isomorphism if if there exists 2 Hom (Y; X) such that ◦ ' = idX and ' ◦ = idY . Definition 1.6. Let C be a category. Two objects X; Y 2 Ob (C) are said to be isomorphic if there exists ' 2 Hom (X; Y ) which is an isomorphism. Definition 1.7. Let C and D be categories. Then a functor F : C!D consists of the following: - a map F : Ob (C) ! Ob (D). - for every pair of objects X; Y 2 Ob (C), a map F : HomC (X; Y ) ! HomD (F (X);F (Y )) such that i) for every X 2 Ob (C), we have F (idX ) = idF (X). ii) for composable morphisms ' and we have F ( ◦ ') = F ( ) ◦ F ('). Drinfeld centers 4 Definition 1.8. Let C; D be categories and let F : C!D be a functor. Then F is full if each map FX;Y : HomC (X; Y ) ! HomD (F (X);F (Y )) is surjective. Definition 1.9. Let C; A be categories and let F : C!D be a functor. Then F is faithful if each map FX;Y : HomC (X; Y ) ! HomD (F (X);F (Y )) is injective. Definition 1.10. Let C; D be categories and let F : C!D be a functor. Then F is dense if each Y 2 Ob (D) is isomorphic to an object F (X) for some X 2 Ob (C). Definition 1.11. Let C and D be categories and let F; G : C!D be functors. A natural transformation η : F ) G is a map Ob (C) ! HomD (F (X);G(X)) such that the diagram F (') F (X) / F (Y ) ηX ηY G(X) / G(Y ) G(') commutes for every morphism ' 2 Hom (X; Y ). Definition 1.12. A natural transformation η such that each component ηX is an isomorphism is called a natural isomorphism. Definition 1.13. Let C; D be categories. An equivalence of C and D is a functor F : C!D such that there exists another functor G : D!C and two natural isomorphisms η : F ◦G ! idD and µ : G ◦ F ! idC. Theorem 1.14. Let C, D be categories and F : C!D a functor. Then F is an equivalence if and only if F is full, faithful and dense. Example 1.15. Let the categories C and D be defined as follows: Fix a positive integer n. The category C has a single object and its morphisms are n × n complex matrices, with composition given by matrix multiplication. The category D has as objects complex vector spaces of dimension n, and its morphisms are linear maps. Composition is given by the usual composition of maps. n Fixing the standard basis of the vector space C , we define a functor F : C!D by mapping n the object of C to the vector space C and mapping each matrix to the linear map which in the standard basis is given by that matrix. Then it is clear by linear algebra that F is a functor and that F is full and faithful. More- over, F is dense since all objects of D, being vector spaces of the same finite dimension, are isomorphic. Since the functor F is full, faithful and dense it is an equivalence. Markus Thuresson 5 Definition 1.16. Let C, D be categories. Then the product category C × D consists of the following: - pairs (X; Y ) of objects, where X 2 Ob (C) and Y 2 Ob (D). - pairs ('; ):(X; Y ) ! (Z; W ) of morphisms, where for X; Z 2 Ob (C) and Y; W 2 Ob (D), we have ' 2 HomC (X; Z) and 2 HomD (Y; W ). - componentwise composition ('2; 2) ◦C×D ('1; 1) = ('2 ◦C '1; 2 ◦D 1). - identity morphisms of the form id(X;Y ) = (idX ; idY ). Proposition 1.17. Let C be a small category and 1 the terminal category. Then we have C × 1 =∼ C (as objects in the category of small categories). Proof. Consider the map F : C ! C × 1 given by X 7! (X; 1) f 7! (f; id): Then we have F (idX ) = (idX ; id) = id(X;1) = idF (X) and F (g ◦ f) = (g ◦ f; id) = (g; id) ◦ (f; id) = F (g) ◦ F (f) so F is a functor. It is obvious that we can construct the inverse functor F −1 : C × 1 !C by (X; 1) 7! X (f; id) 7! f: Definition 1.18. Let C be a category. We define the center of C, denoted by Z(C) to be the class of natural transformations from the identity functor idC to itself. Example 1.19. We recall that a monoid is a set with a binary associative operation with identity. Any monoid M may be regarded as a category. Define the category M as follows: - M has one object, •. - Morphisms are all elements of M. - Composition is given by the multiplication of M. The identity of M then acts as identity for the composition, and composition is associative since the multiplication of M is associative. Drinfeld centers 6 It is clear that given such a category we can reconstruct our monoid, so the above is an equivalent definition. Now consider the center Z(M) of M. Since M has only one object and the morphisms are elements of M, a natural transformation from the identity functor to itself consists of an element z 2 M such that the diagram M x / M z z / M x M commutes for all x 2 M. Since the composition is just multiplication in M, this is equivalent to zx = xz for all x 2 M. Now it is clear that Z(M) = Z(M) = fz 2 M : zx = xz 8x 2 Mg. 2 Categories of modules In the following section, let R be a unital ring. Recall that the center of a ring R is the subring Z(R) = fz 2 R : zr = rz 8r 2 Rg. Definition 2.1. A left R-module is an abelian group (M; +) together with a binary operation · : R × M ! M such that i) r · (x + y) = r · x + r · y ii)(r + s) · x = r · x + s · x iii)(rs) · x = r · (s · x) iv) 1R · x = x for all x; y 2 M and r; s 2 R. When necessary, we refer to the action of R on M as ·M . Definition 2.2. Let M; N be left R-modules. A homomorphism of R-modules is a map ' : M ! N such that '(r ·M x + s ·M y) = r ·N '(x) + s ·N '(y) for all x; y 2 M and r; s 2 R.
Load more

Recommended publications
  • Frobenius and the Derived Centers of Algebraic Theories
    Frobenius and the derived centers of algebraic theories William G. Dwyer and Markus Szymik October 2014 We show that the derived center of the category of simplicial algebras over every algebraic theory is homotopically discrete, with the abelian monoid of components isomorphic to the center of the category of discrete algebras. For example, in the case of commutative algebras in characteristic p, this center is freely generated by Frobenius. Our proof involves the calculation of homotopy coherent centers of categories of simplicial presheaves as well as of Bousfield localizations. Numerous other classes of examples are dis- cussed. 1 Introduction Algebra in prime characteristic p comes with a surprise: For each commutative ring A such that p = 0 in A, the p-th power map a 7! ap is not only multiplica- tive, but also additive. This defines Frobenius FA : A ! A on commutative rings of characteristic p, and apart from the well-known fact that Frobenius freely gen- erates the Galois group of the prime field Fp, it has many other applications in 1 algebra, arithmetic and even geometry. Even beyond fields, Frobenius is natural in all rings A: Every map g: A ! B between commutative rings of characteris- tic p (i.e. commutative Fp-algebras) commutes with Frobenius in the sense that the equation g ◦ FA = FB ◦ g holds. In categorical terms, the Frobenius lies in the center of the category of commutative Fp-algebras. Furthermore, Frobenius freely generates the center: If (PA : A ! AjA) is a family of ring maps such that g ◦ PA = PB ◦ g (?) holds for all g as above, then there exists an integer n > 0 such that the equa- n tion PA = (FA) holds for all A.
    [Show full text]
  • Types Are Internal Infinity-Groupoids Antoine Allioux, Eric Finster, Matthieu Sozeau
    Types are internal infinity-groupoids Antoine Allioux, Eric Finster, Matthieu Sozeau To cite this version: Antoine Allioux, Eric Finster, Matthieu Sozeau. Types are internal infinity-groupoids. 2021. hal- 03133144 HAL Id: hal-03133144 https://hal.inria.fr/hal-03133144 Preprint submitted on 5 Feb 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Types are Internal 1-groupoids Antoine Allioux∗, Eric Finstery, Matthieu Sozeauz ∗Inria & University of Paris, France [email protected] yUniversity of Birmingham, United Kingdom ericfi[email protected] zInria, France [email protected] Abstract—By extending type theory with a universe of defini- attempts to import these ideas into plain homotopy type theory tionally associative and unital polynomial monads, we show how have, so far, failed. This appears to be a result of a kind of to arrive at a definition of opetopic type which is able to encode circularity: all of the known classical techniques at some point a number of fully coherent algebraic structures. In particular, our approach leads to a definition of 1-groupoid internal to rely on set-level algebraic structures themselves (presheaves, type theory and we prove that the type of such 1-groupoids is operads, or something similar) as a means of presenting or equivalent to the universe of types.
    [Show full text]
  • Product Category Rules
    Product Category Rules PRé’s own Vee Subramanian worked with a team of experts to publish a thought provoking article on product category rules (PCRs) in the International Journal of Life Cycle Assessment in April 2012. The full article can be viewed at http://www.springerlink.com/content/qp4g0x8t71432351/. Here we present the main body of the text that explores the importance of global alignment of programs that manage PCRs. Comparing Product Category Rules from Different Programs: Learned Outcomes Towards Global Alignment Vee Subramanian • Wesley Ingwersen • Connie Hensler • Heather Collie Abstract Purpose Product category rules (PCRs) provide category-specific guidance for estimating and reporting product life cycle environmental impacts, typically in the form of environmental product declarations and product carbon footprints. Lack of global harmonization between PCRs or sector guidance documents has led to the development of duplicate PCRs for same products. Differences in the general requirements (e.g., product category definition, reporting format) and LCA methodology (e.g., system boundaries, inventory analysis, allocation rules, etc.) diminish the comparability of product claims. Methods A comparison template was developed to compare PCRs from different global program operators. The goal was to identify the differences between duplicate PCRs from a broad selection of product categories and propose a path towards alignment. We looked at five different product categories: Milk/dairy (2 PCRs), Horticultural products (3 PCRs), Wood-particle board (2 PCRs), and Laundry detergents (4 PCRs). Results & discussion Disparity between PCRs ranged from broad differences in scope, system boundaries and impacts addressed (e.g. multi-impact vs. carbon footprint only) to specific differences of technical elements.
    [Show full text]
  • An Introduction to Category Theory and Categorical Logic
    An Introduction to Category Theory and Categorical Logic Wolfgang Jeltsch Category theory An Introduction to Category Theory basics Products, coproducts, and and Categorical Logic exponentials Categorical logic Functors and Wolfgang Jeltsch natural transformations Monoidal TTU¨ K¨uberneetika Instituut categories and monoidal functors Monads and Teooriaseminar comonads April 19 and 26, 2012 References An Introduction to Category Theory and Categorical Logic Category theory basics Wolfgang Jeltsch Category theory Products, coproducts, and exponentials basics Products, coproducts, and Categorical logic exponentials Categorical logic Functors and Functors and natural transformations natural transformations Monoidal categories and Monoidal categories and monoidal functors monoidal functors Monads and comonads Monads and comonads References References An Introduction to Category Theory and Categorical Logic Category theory basics Wolfgang Jeltsch Category theory Products, coproducts, and exponentials basics Products, coproducts, and Categorical logic exponentials Categorical logic Functors and Functors and natural transformations natural transformations Monoidal categories and Monoidal categories and monoidal functors monoidal functors Monads and Monads and comonads comonads References References An Introduction to From set theory to universal algebra Category Theory and Categorical Logic Wolfgang Jeltsch I classical set theory (for example, Zermelo{Fraenkel): I sets Category theory basics I functions from sets to sets Products, I composition
    [Show full text]
  • Toy Industry Product Categories
    Definitions Document Toy Industry Product Categories Action Figures Action Figures, Playsets and Accessories Includes licensed and theme figures that have an action-based play pattern. Also includes clothing, vehicles, tools, weapons or play sets to be used with the action figure. Role Play (non-costume) Includes role play accessory items that are both action themed and generically themed. This category does not include dress-up or costume items, which have their own category. Arts and Crafts Chalk, Crayons, Markers Paints and Pencils Includes singles and sets of these items. (e.g., box of crayons, bucket of chalk). Reusable Compounds (e.g., Clay, Dough, Sand, etc.) and Kits Includes any reusable compound, or items that can be manipulated into creating an object. Some examples include dough, sand and clay. Also includes kits that are intended for use with reusable compounds. Design Kits and Supplies – Reusable Includes toys used for designing that have a reusable feature or extra accessories (e.g., extra paper). Examples include Etch-A-Sketch, Aquadoodle, Lite Brite, magnetic design boards, and electronic or digital design units. Includes items created on the toy themselves or toys that connect to a computer or tablet for designing / viewing. Design Kits and Supplies – Single Use Includes items used by a child to create art and sculpture projects. These items are all-inclusive kits and may contain supplies that are needed to create the project (e.g., crayons, paint, yarn). This category includes refills that are sold separately to coincide directly with the kits. Also includes children’s easels and paint-by-number sets.
    [Show full text]
  • 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.
    [Show full text]
  • Arxiv:1204.3607V6 [Math.KT] 6 Nov 2015 At1 Ar N Waldhausen and Pairs 1
    ON THE ALGEBRAIC K-THEORY OF HIGHER CATEGORIES CLARK BARWICK In memoriam Daniel Quillen, 1940–2011, with profound admiration. Abstract. We prove that Waldhausen K-theory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, K-theory spaces admit canonical (connective) deloopings, and the K-theory functor enjoys a simple universal property. Using this, we give new, higher categorical proofs of the Approximation, Additivity, and Fibration Theorems of Waldhausen in this context. As applications of this technology, we study the algebraic K-theory of associative rings in a wide range of homotopical contexts and of spectral Deligne–Mumford stacks. Contents 0. Introduction 3 Relation to other work 6 A word on higher categories 7 Acknowledgments 7 Part 1. Pairs and Waldhausen ∞-categories 8 1. Pairs of ∞-categories 8 Set theoretic considerations 9 Simplicial nerves and relative nerves 10 The ∞-category of ∞-categories 11 Subcategories of ∞-categories 12 Pairs of ∞-categories 12 The ∞-category of pairs 13 Pair structures 14 arXiv:1204.3607v6 [math.KT] 6 Nov 2015 The ∞-categories of pairs as a relative nerve 15 The dual picture 16 2. Waldhausen ∞-categories 17 Limits and colimits in ∞-categories 17 Waldhausen ∞-categories 18 Some examples 20 The ∞-category of Waldhausen ∞-categories 21 Equivalences between maximal Waldhausen ∞-categories 21 The dual picture 22 3. Waldhausen fibrations 22 Cocartesian fibrations 23 Pair cartesian and cocartesian fibrations 27 The ∞-categoriesofpair(co)cartesianfibrations 28 A pair version of 3.7 31 1 2 CLARK BARWICK Waldhausencartesianandcocartesianfibrations 32 4. The derived ∞-category of Waldhausen ∞-categories 34 Limits and colimits of pairs of ∞-categories 35 Limits and filtered colimits of Waldhausen ∞-categories 36 Direct sums of Waldhausen ∞-categories 37 Accessibility of Wald∞ 38 Virtual Waldhausen ∞-categories 40 Realizations of Waldhausen cocartesian fibrations 42 Part 2.
    [Show full text]
  • RESEARCH STATEMENT 1. Introduction My Interests Lie
    RESEARCH STATEMENT BEN ELIAS 1. Introduction My interests lie primarily in geometric representation theory, and more specifically in diagrammatic categorification. Geometric representation theory attempts to answer questions about representation theory by studying certain algebraic varieties and their categories of sheaves. Diagrammatics provide an efficient way of encoding the morphisms between sheaves and doing calculations with them, and have also been fruitful in their own right. I have been applying diagrammatic methods to the study of the Iwahori-Hecke algebra and its categorification in terms of Soergel bimodules, as well as the categorifications of quantum groups; I plan on continuing to research in these two areas. In addition, I would like to learn more about other areas in geometric representation theory, and see how understanding the morphisms between sheaves or the natural transformations between functors can help shed light on the theory. After giving a general overview of the field, I will discuss several specific projects. In the most naive sense, a categorification of an algebra (or category) A is an additive monoidal category (or 2-category) A whose Grothendieck ring is A. Relations in A such as xy = z + w will be replaced by isomorphisms X⊗Y =∼ Z⊕W . What makes A a richer structure than A is that these isomorphisms themselves will have relations; that between objects we now have morphism spaces equipped with composition maps. 2-category). In their groundbreaking paper [CR], Chuang and Rouquier made the key observation that some categorifications are better than others, and those with the \correct" morphisms will have more interesting properties. Independently, Rouquier [Ro2] and Khovanov and Lauda (see [KL1, La, KL2]) proceeded to categorify quantum groups themselves, and effectively demonstrated that categorifying an algebra will give a precise notion of just what morphisms should exist within interesting categorifications of its modules.
    [Show full text]
  • The Monoidal Center Construction and Bimodules
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 158 (2001) 325–346 www.elsevier.com/locate/jpaa The monoidal center construction and bimodules Peter Schauenburg Mathematisches Institut der Universitat Munchen, Theresienstr. 39, 80333 Munchen, Germany Received 29 January 1999; received in revised form 4 October 1999 Communicated by C. Kassel Abstract Let C be a cocomplete monoidal category such that the tensor product in C preserves colimits in each argument. Let A be an algebra in C. We show (under some assumptions including “faithful 2atness” of A) that the center of the monoidal category (ACA; ⊗A)ofA–A-bimodules ∼ is equivalent to the center of C (hence in a sense trivial): Z(ACA) = Z(C). Assuming A to be a commutative algebra in the center Z(C), we compute the center Z(CA) of the category of right A-modules (considered as a subcategory of ACA using the structure of A ∈ Z(C). We ÿnd ∼ Z(CA) = dys Z(C)A, the category of dyslectic right A-modules in the braided category Z(C). c 2001 Elsevier Science B.V. All rights reserved. MSC: 18D10; 16W30 1. Introduction Braided monoidal categories are situated at the intersection of quantum group theory and low-dimensional topology (more precisely invariants of knots and three-manifolds, see [12,4] for general references). They also provide a framework for generalizations of commutative algebra based on a 2ip of tensor factors more general than the ordinary 2ip (as proposed by Manin [6] for symmetric monoidal categories and well-established for the special case of Z=2Z-graded vector spaces) or generalizations of Hopf algebra theory [5].
    [Show full text]
  • Product Category Rule for Environmental Product Declarations
    Product Category Rule for Environmental Product Declarations BIFMA PCR for Seating: UNCPC 3811 Version 3 Program Operator NSF International National Center for Sustainability Standards Valid through September 30, 2019 Extended per PCRext 2019-107 valid through September 30, 2020 [email protected] Product Category Rule for Environmental Product Declarations BIFMA PCR for Seating: UNCPC 3811 – Version 3 PRODUCT CATEGORY RULES REVIEW PANEL Thomas P. Gloria, Ph. D. Industrial Ecology Consultants 35 Bracebridge Rd. Newton, MA 02459-1728 [email protected] Jack Geibig, P.E. Ecoform 2624 Abelia Way Knoxville, TN 37931 [email protected] Dr. Michael Overcash Environmental Clarity 2908 Chipmunk Lane Raleigh, NC 27607-3117 [email protected] PCR review panel comments may be obtained by contacting NSF International’s National Center for Sustainability Standards at [email protected]. No participation fees were charged by NSF to interested parties. NSF International ensured that reasonable representation among the members of the PCR committee was achieved and potential conflicts of interest were resolved prior to commencing this PCR development. NSF International National Center for Sustainability Standards Valid through September 30, 2019 Extended per PCRext 2019-107 valid through September 30, 2020 Page 2 Product Category Rule for Environmental Product Declarations BIFMA PCR for Seating: UNCPC 3811 – Version 3 TABLE OF CONTENTS BIFMA PRODUCT CATEGORY RULES....................................................................................................................4
    [Show full text]
  • Centers and Homotopy Centers in Enriched Monoidal Categories
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Available online at www.sciencedirect.com Advances in Mathematics 230 (2012) 1811–1858 www.elsevier.com/locate/aim Centers and homotopy centers in enriched monoidal categories Michael Batanina,∗, Martin Marklb a Macquarie University, NSW 2109, Australia b Mathematical Institute of the Academy, Zitnˇ a´ 25, 115 67 Prague 1, Czech Republic Received 20 September 2011; accepted 12 April 2012 Available online 16 May 2012 Communicated by Ross Street Abstract We consider a theory of centers and homotopy centers of monoids in monoidal categories which themselves are enriched in duoidal categories. The duoidal categories (introduced by Aguiar and Mahajan under the name 2-monoidal categories) are categories with two monoidal structures which are related by some, not necessary invertible, coherence morphisms. Centers of monoids in this sense include many examples which are not ‘classical.’ In particular, the 2-category of categories is an example of a center in our sense. Examples of homotopy center (analogue of the classical Hochschild complex) include the Gray- category Gray of 2-categories, 2-functors and pseudonatural transformations and Tamarkin’s homotopy 2-category of dg-categories, dg-functors and coherent dg-transformations. ⃝c 2012 Elsevier Inc. All rights reserved. MSC: primary 18D10; 18D20; 18D50; secondary 55U40; 55P48 Keywords: Monoidal categories; Center; Hochschild complex; Operads; Deligne’s conjecture Contents 1. Introduction.................................................................................................................. 1812 2. Monoidal V -categories and duoidal V -categories.............................................................. 1814 ∗ Corresponding author. E-mail addresses: [email protected], [email protected] (M. Batanin), [email protected] (M.
    [Show full text]
  • Mathematics and the Brain: a Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness
    entropy Review Mathematics and the Brain: A Category Theoretical Approach to Go Beyond the Neural Correlates of Consciousness 1,2,3, , 4,5,6,7, 8, Georg Northoff * y, Naotsugu Tsuchiya y and Hayato Saigo y 1 Mental Health Centre, Zhejiang University School of Medicine, Hangzhou 310058, China 2 Institute of Mental Health Research, University of Ottawa, Ottawa, ON K1Z 7K4 Canada 3 Centre for Cognition and Brain Disorders, Hangzhou Normal University, Hangzhou 310036, China 4 School of Psychological Sciences, Faculty of Medicine, Nursing and Health Sciences, Monash University, Melbourne, Victoria 3800, Australia; [email protected] 5 Turner Institute for Brain and Mental Health, Monash University, Melbourne, Victoria 3800, Australia 6 Advanced Telecommunication Research, Soraku-gun, Kyoto 619-0288, Japan 7 Center for Information and Neural Networks (CiNet), National Institute of Information and Communications Technology (NICT), Suita, Osaka 565-0871, Japan 8 Nagahama Institute of Bio-Science and Technology, Nagahama 526-0829, Japan; [email protected] * Correspondence: georg.northoff@theroyal.ca All authors contributed equally to the paper as it was a conjoint and equally distributed work between all y three authors. Received: 18 July 2019; Accepted: 9 October 2019; Published: 17 December 2019 Abstract: Consciousness is a central issue in neuroscience, however, we still lack a formal framework that can address the nature of the relationship between consciousness and its physical substrates. In this review, we provide a novel mathematical framework of category theory (CT), in which we can define and study the sameness between different domains of phenomena such as consciousness and its neural substrates. CT was designed and developed to deal with the relationships between various domains of phenomena.
    [Show full text]