Scheiderer Motives and Equivariant Higher Topos Theory
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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. -
Relations: Categories, Monoidal Categories, and Props
Logical Methods in Computer Science Vol. 14(3:14)2018, pp. 1–25 Submitted Oct. 12, 2017 https://lmcs.episciences.org/ Published Sep. 03, 2018 UNIVERSAL CONSTRUCTIONS FOR (CO)RELATIONS: CATEGORIES, MONOIDAL CATEGORIES, AND PROPS BRENDAN FONG AND FABIO ZANASI Massachusetts Institute of Technology, United States of America e-mail address: [email protected] University College London, United Kingdom e-mail address: [email protected] Abstract. Calculi of string diagrams are increasingly used to present the syntax and algebraic structure of various families of circuits, including signal flow graphs, electrical circuits and quantum processes. In many such approaches, the semantic interpretation for diagrams is given in terms of relations or corelations (generalised equivalence relations) of some kind. In this paper we show how semantic categories of both relations and corelations can be characterised as colimits of simpler categories. This modular perspective is important as it simplifies the task of giving a complete axiomatisation for semantic equivalence of string diagrams. Moreover, our general result unifies various theorems that are independently found in literature and are relevant for program semantics, quantum computation and control theory. 1. Introduction Network-style diagrammatic languages appear in diverse fields as a tool to reason about computational models of various kinds, including signal processing circuits, quantum pro- cesses, Bayesian networks and Petri nets, amongst many others. In the last few years, there have been more and more contributions towards a uniform, formal theory of these languages which borrows from the well-established methods of programming language semantics. A significant insight stemming from many such approaches is that a compositional analysis of network diagrams, enabling their reduction to elementary components, is more effective when system behaviour is thought of as a relation instead of a function. -
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. -
PULLBACK and BASE CHANGE Contents
PART II.2. THE !-PULLBACK AND BASE CHANGE Contents Introduction 1 1. Factorizations of morphisms of DG schemes 2 1.1. Colimits of closed embeddings 2 1.2. The closure 4 1.3. Transitivity of closure 5 2. IndCoh as a functor from the category of correspondences 6 2.1. The category of correspondences 6 2.2. Proof of Proposition 2.1.6 7 3. The functor of !-pullback 8 3.1. Definition of the functor 9 3.2. Some properties 10 3.3. h-descent 10 3.4. Extension to prestacks 11 4. The multiplicative structure and duality 13 4.1. IndCoh as a symmetric monoidal functor 13 4.2. Duality 14 5. Convolution monoidal categories and algebras 17 5.1. Convolution monoidal categories 17 5.2. Convolution algebras 18 5.3. The pull-push monad 19 5.4. Action on a map 20 References 21 Introduction Date: September 30, 2013. 1 2 THE !-PULLBACK AND BASE CHANGE 1. Factorizations of morphisms of DG schemes In this section we will study what happens to the notion of the closure of the image of a morphism between schemes in derived algebraic geometry. The upshot is that there is essentially \nothing new" as compared to the classical case. 1.1. Colimits of closed embeddings. In this subsection we will show that colimits exist and are well-behaved in the category of closed subschemes of a given ambient scheme. 1.1.1. Recall that a map X ! Y in Sch is called a closed embedding if the map clX ! clY is a closed embedding of classical schemes. -
DUAL TANGENT STRUCTURES for ∞-TOPOSES In
DUAL TANGENT STRUCTURES FOR 1-TOPOSES MICHAEL CHING Abstract. We describe dual notions of tangent bundle for an 1-topos, each underlying a tangent 1-category in the sense of Bauer, Burke and the author. One of those notions is Lurie's tangent bundle functor for presentable 1-categories, and the other is its adjoint. We calculate that adjoint for injective 1-toposes, where it is given by applying Lurie's tangent bundle on 1-categories of points. In [BBC21], Bauer, Burke, and the author introduce a notion of tangent 1- category which generalizes the Rosick´y[Ros84] and Cockett-Cruttwell [CC14] axiomatization of the tangent bundle functor on the category of smooth man- ifolds and smooth maps. We also constructed there a significant example: diff the Goodwillie tangent structure on the 1-category Cat1 of (differentiable) 1-categories, which is built on Lurie's tangent bundle functor. That tan- gent structure encodes the ideas of Goodwillie's calculus of functors [Goo03] and highlights the analogy between that theory and the ordinary differential calculus of smooth manifolds. The goal of this note is to introduce two further examples of tangent 1- categories: one on the 1-category of 1-toposes and geometric morphisms, op which we denote Topos1, and one on the opposite 1-category Topos1 which, following Anel and Joyal [AJ19], we denote Logos1. These two tangent structures each encode a notion of tangent bundle for 1- toposes, but from dual perspectives. As described in [AJ19, Sec. 4], one can view 1-toposes either from a `geometric' or `algebraic' point of view. -
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. -
Toward a Non-Commutative Gelfand Duality: Boolean Locally Separated Toposes and Monoidal Monotone Complete $ C^{*} $-Categories
Toward a non-commutative Gelfand duality: Boolean locally separated toposes and Monoidal monotone complete C∗-categories Simon Henry February 23, 2018 Abstract ** Draft Version ** To any boolean topos one can associate its category of internal Hilbert spaces, and if the topos is locally separated one can consider a full subcategory of square integrable Hilbert spaces. In both case it is a symmetric monoidal monotone complete C∗-category. We will prove that any boolean locally separated topos can be reconstructed as the classifying topos of “non-degenerate” monoidal normal ∗-representations of both its category of internal Hilbert spaces and its category of square integrable Hilbert spaces. This suggest a possible extension of the usual Gelfand duality between a class of toposes (or more generally localic stacks or localic groupoids) and a class of symmetric monoidal C∗-categories yet to be discovered. Contents 1 Introduction 2 2 General preliminaries 3 3 Monotone complete C∗-categories and booleantoposes 6 4 Locally separated toposes and square integrable Hilbert spaces 7 5 Statementofthemaintheorems 9 arXiv:1501.07045v1 [math.CT] 28 Jan 2015 6 From representations of red togeometricmorphisms 13 6.1 Construction of the geometricH morphism on separating objects . 13 6.2 Functorialityonseparatingobject . 20 6.3 ConstructionoftheGeometricmorphism . 22 6.4 Proofofthe“reduced”theorem5.8 . 25 Keywords. Boolean locally separated toposes, monotone complete C*-categories, recon- struction theorem. 2010 Mathematics Subject Classification. 18B25, 03G30, 46L05, 46L10 . email: [email protected] 1 7 On the category ( ) and its representations 26 H T 7.1 The category ( /X ) ........................ 26 7.2 TensorisationH byT square integrable Hilbert space . -
Monoidal Functors, Equivalence of Monoidal Categories
14 1.4. Monoidal functors, equivalence of monoidal categories. As we have explained, monoidal categories are a categorification of monoids. Now we pass to categorification of morphisms between monoids, namely monoidal functors. 0 0 0 0 0 Definition 1.4.1. Let (C; ⊗; 1; a; ι) and (C ; ⊗ ; 1 ; a ; ι ) be two monoidal 0 categories. A monoidal functor from C to C is a pair (F; J) where 0 0 ∼ F : C ! C is a functor, and J = fJX;Y : F (X) ⊗ F (Y ) −! F (X ⊗ Y )jX; Y 2 Cg is a natural isomorphism, such that F (1) is isomorphic 0 to 1 . and the diagram (1.4.1) a0 (F (X) ⊗0 F (Y )) ⊗0 F (Z) −−F− (X−)−;F− (Y− )−;F− (Z!) F (X) ⊗0 (F (Y ) ⊗0 F (Z)) ? ? J ⊗0Id ? Id ⊗0J ? X;Y F (Z) y F (X) Y;Z y F (X ⊗ Y ) ⊗0 F (Z) F (X) ⊗0 F (Y ⊗ Z) ? ? J ? J ? X⊗Y;Z y X;Y ⊗Z y F (aX;Y;Z ) F ((X ⊗ Y ) ⊗ Z) −−−−−−! F (X ⊗ (Y ⊗ Z)) is commutative for all X; Y; Z 2 C (“the monoidal structure axiom”). A monoidal functor F is said to be an equivalence of monoidal cate gories if it is an equivalence of ordinary categories. Remark 1.4.2. It is important to stress that, as seen from this defini tion, a monoidal functor is not just a functor between monoidal cate gories, but a functor with an additional structure (the isomorphism J) satisfying a certain equation (the monoidal structure axiom). -
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. -
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). -
Combinatorics, Categorification, and Crystals
Combinatorics, Categorification, and Crystals Monica Vazirani UC Davis November 4, 2017 Fall Western Sectional Meeting { UCR Monica Vazirani (UC Davis ) Combinatorics, Categorification & Crystals Nov 4, 2017 1 / 39 I. The What? Why? When? How? of Categorification II. Focus on combinatorics of symmetric group Sn, crystal graphs, quantum groups Uq(g) Monica Vazirani (UC Davis ) Combinatorics, Categorification & Crystals Nov 4, 2017 2 / 39 from searching titles ::: Special Session on Combinatorial Aspects of the Polynomial Ring Special Session on Combinatorial Representation Theory Special Session on Non-Commutative Birational Geometry, Cluster Structures and Canonical Bases Special Session on Rational Cherednik Algebras and Categorification and more ··· Monica Vazirani (UC Davis ) Combinatorics, Categorification & Crystals Nov 4, 2017 3 / 39 Combinatorics ··· ; ··· ··· Monica Vazirani (UC Davis ) Combinatorics, Categorification & Crystals Nov 4, 2017 4 / 39 Representation theory of the symmetric group Branching Rule: Restriction and Induction n n The graph encodes the functors Resn−1 and Indn−1 of induction and restriction arising from Sn−1 ⊆ Sn: Example (multiplicity free) × × ResS4 S = S ⊕ S S3 (3) 4 (3;1) (2;1) 4 (3;1) or dim Hom(S ; Res3 S ) = 1, dim Hom(S ; Res3 S ) = 1 Monica Vazirani (UC Davis ) Combinatorics, Categorification & Crystals Nov 4, 2017 5 / 39 label diagonals ··· ; ··· ··· Monica Vazirani (UC Davis ) Combinatorics, Categorification & Crystals Nov 4, 2017 6 / 39 label diagonals 0 1 2 3 ··· −1 0 1 2 ··· −2 −1 0 1 ··· 0 ··· . Monica