RESEARCH STATEMENT 1. Introduction My Interests Lie
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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. -
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. -
Elements of -Category Theory Joint with Dominic Verity
Emily Riehl Johns Hopkins University Elements of ∞-Category Theory joint with Dominic Verity MATRIX seminar ∞-categories in the wild A recent phenomenon in certain areas of mathematics is the use of ∞-categories to state and prove theorems: • 푛-jets correspond to 푛-excisive functors in the Goodwillie tangent structure on the ∞-category of differentiable ∞-categories — Bauer–Burke–Ching, “Tangent ∞-categories and Goodwillie calculus” • 푆1-equivariant quasicoherent sheaves on the loop space of a smooth scheme correspond to sheaves with a flat connection as an equivalence of ∞-categories — Ben-Zvi–Nadler, “Loop spaces and connections” • the factorization homology of an 푛-cobordism with coefficients in an 푛-disk algebra is linearly dual to the factorization homology valued in the formal moduli functor as a natural equivalence between functors between ∞-categories — Ayala–Francis, “Poincaré/Koszul duality” Here “∞-category” is a nickname for (∞, 1)-category, a special case of an (∞, 푛)-category, a weak infinite dimensional category in which all morphisms above dimension 푛 are invertible (for fixed 0 ≤ 푛 ≤ ∞). What are ∞-categories and what are they for? It frames a possible template for any mathematical theory: the theory should have nouns and verbs, i.e., objects, and morphisms, and there should be an explicit notion of composition related to the morphisms; the theory should, in brief, be packaged by a category. —Barry Mazur, “When is one thing equal to some other thing?” An ∞-category frames a template with nouns, verbs, adjectives, adverbs, pronouns, prepositions, conjunctions, interjections,… which has: • objects • and 1-morphisms between them • • • • composition witnessed by invertible 2-morphisms 푓 푔 훼≃ ℎ∘푔∘푓 • • • • 푔∘푓 푓 ℎ∘푔 훾≃ witnessed by • associativity • ≃ 푔∘푓 훼≃ 훽 ℎ invertible 3-morphisms 푔 • with these witnesses coherent up to invertible morphisms all the way up. -
Diagrammatics in Categorification and Compositionality
Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 ABSTRACT Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 Copyright c 2019 by Dmitry Vagner All rights reserved Abstract In the present work, I explore the theme of diagrammatics and their capacity to shed insight on two trends|categorification and compositionality|in and around contemporary category theory. The work begins with an introduction of these meta- phenomena in the context of elementary sets and maps. Towards generalizing their study to more complicated domains, we provide a self-contained treatment|from a pedagogically novel perspective that introduces almost all concepts via diagrammatic language|of the categorical machinery with which we may express the broader no- tions found in the sequel. The work then branches into two seemingly unrelated disciplines: dynamical systems and knot theory. In particular, the former research defines what it means to compose dynamical systems in a manner analogous to how one composes simple maps. The latter work concerns the categorification of the slN link invariant. In particular, we use a virtual filtration to give a more diagrammatic reconstruction of Khovanov-Rozansky homology via a smooth TQFT. -
The Hecke Bicategory
Axioms 2012, 1, 291-323; doi:10.3390/axioms1030291 OPEN ACCESS axioms ISSN 2075-1680 www.mdpi.com/journal/axioms Communication The Hecke Bicategory Alexander E. Hoffnung Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, PA 19122, USA; E-Mail: [email protected]; Tel.: +215-204-7841; Fax: +215-204-6433 Received: 19 July 2012; in revised form: 4 September 2012 / Accepted: 5 September 2012 / Published: 9 October 2012 Abstract: We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid—the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest an explicit connection to new higher isomorphisms arising from incidence geometries, which are solutions of the Zamolodchikov tetrahedron equation. This paper is expository in style and is meant as a companion to Higher Dimensional Algebra VII: Groupoidification and an exploration of structures arising in the work in progress, Higher Dimensional Algebra VIII: The Hecke Bicategory, which introduces the Hecke bicategory in detail. Keywords: Hecke algebras; categorification; groupoidification; Yang–Baxter equations; Zamalodchikov tetrahedron equations; spans; enriched bicategories; buildings; incidence geometries 1. Introduction Categorification is, in part, the attempt to shed new light on familiar mathematical notions by replacing a set-theoretic interpretation with a category-theoretic analogue. Loosely speaking, categorification replaces sets, or more generally n-categories, with categories, or more generally (n + 1)-categories, and functions with functors. By replacing interesting equations by isomorphisms, or more generally equivalences, this process often brings to light a new layer of structure previously hidden from view. -
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. -
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. -
Why Category Theory?
The beginning of topology Categorification of the concepts Category theory as a research field Grothendieck's n-categories Why category theory? Daniel Tubbenhauer - 02.04.2012 Georg-August-Universit¨at G¨ottingen Gr G ACT Cat C Hom(-,O) GRP Free - / [-,-] Forget (-) ι (-) Δ SET n=1 -x- Forget Free K(-,n) CxC n>1 AB π (-) n TOP Hom(-,-) * Drop (-) Forget * Hn Free op TOP Join AB * K-VS V - Hom(-,V) The beginning of topology Categorification of the concepts Category theory as a research field Grothendieck's n-categories A generalisation of well-known notions Closed Total Associative Unit Inverses Group Yes Yes Yes Yes Yes Monoid Yes Yes Yes Yes No Semigroup Yes Yes Yes No No Magma Yes Yes No No No Groupoid Yes No Yes Yes Yes Category No No Yes Yes No Semicategory No No Yes No No Precategory No No No No No The beginning of topology Categorification of the concepts Category theory as a research field Grothendieck's n-categories Leonard Euler and convex polyhedron Euler's polyhedron theorem Polyhedron theorem (1736) 3 Let P ⊂ R be a convex polyhedron with V vertices, E edges and F faces. Then: χ = V − E + F = 2: Here χ denotes the Euler Leonard Euler (15.04.1707-18.09.1783) characteristic. The beginning of topology Categorification of the concepts Category theory as a research field Grothendieck's n-categories Leonard Euler and convex polyhedron Euler's polyhedron theorem Polyhedron theorem (1736) 3 Polyhedron E K F χ Let P ⊂ R be a convex polyhedron with V vertices, E Tetrahedron 4 6 4 2 edges and F faces. -
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]. -
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. -
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. -
CAREER: Categorical Representation Theory of Hecke Algebras
CAREER: Categorical representation theory of Hecke algebras The primary goal of this proposal is to study the categorical representation theory (or 2- representation theory) of the Iwahori-Hecke algebra H(W ) attached to a Coxeter group W . Such 2-representations (and related structures) are ubiquitous in classical representation theory, includ- ing: • the category O0(g) of representations of a complex semisimple lie algebra g, introduced by Bernstein, Gelfand, and Gelfand [9], or equivalently, the category Perv(F l) of perverse sheaves on flag varieties [75], • the category Rep(g) of finite dimensional representations of g, via the geometric Satake equivalence [44, 67], • the category Rat(G) of rational representations an algebraic group G (see [1, 78]), • and most significantly, integral forms and/or quantum deformations of the above, which can be specialized to finite characteristic or to roots of unity. Categorical representation theory is a worthy topic of study in its own right, but can be viewed as a fantastic new tool with which to approach these classical categories of interest. In §1 we define 2-representations of Hecke algebras. One of the major questions in the field is the computation of local intersection forms, and we mention some related projects. In §2 we discuss a project, joint with Geordie Williamson and Daniel Juteau, to compute local intersection forms in the antispherical module of an affine Weyl group, with applications to rational representation theory and quantum groups at roots of unity. In §3 we describe a related project, partially joint with Benjamin Young, to describe Soergel bimodules for the complex reflection groups G(m,m,n).