DOCSLIB.ORG

UC Santa Cruz UC Santa Cruz Electronic Theses and Dissertations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
UC Santa Cruz UC Santa Cruz Electronic Theses and Dissertations UC Santa Cruz UC Santa Cruz Electronic Theses and Dissertations Title Fusion systems and biset functors via ghost algebras Permalink https://escholarship.org/uc/item/0ts629vq Author O'Hare, Shawn Michael Publication Date 2013 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA SANTA CRUZ FUSION SYSTEMS AND BISET FUNCTORS VIA GHOST ALGEBRAS A dissertation submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in MATHEMATICS by Shawn Michael O’Hare June 2013 The Dissertation of Shawn Michael O’Hare is approved: Professor Robert Boltje, Chair Professor Geoff Mason Professor Martin Weissman Tyrus Miller Vice Provost and Dean of Graduate Studies Copyright c by Shawn Michael O’Hare 2013 Table of Contents Abstract v Dedication vi Acknowledgments vii Introduction1 Some Notation . .4 1 Background7 1.1 Fusion System Basics . .7 1.2 Bisets . 10 1.3 Double Burnside Groups . 15 1.4 Two Ghost Groups . 17 1.5 Subgroups of Burnside Groups . 21 1.6 Biset Categories . 24 2 Biset Categories 27 2.1 A Special Class of Groups . 28 2.2 Fusion Preserving Isomorphisms . 31 3 Characteristic Idempotents 36 3.1 Calculating the Characteristic Idempotent . 36 3.2 Bideflation of Characteristic Idempotents . 42 4 A Generalized Burnside Functor 47 4.1 Pseudo-rings . 47 Condensation and Decondensation . 49 4.2 Decondensation of the Burnside Functor . 51 4.3 The Action of Elementary Subgroups . 53 4.4 Subfunctors . 57 References 67 iii A Single Burnside Rings 70 A.1 Sets with a Group Action . 70 A.2 Operations on Sets with a Group Action . 76 A.3 The Single Burnside Ring . 78 Notation 86 Index 88 iv Abstract Fusion Systems and Biset Functors via Ghost Algebras by Shawn Michael O’Hare In this work we utilize ghost groups of Burnside groups introduced by Boltje and Danz in order to investigate fusion systems of finite groups, double Burnside modules, and biset functors. We give an expression for the coefficients of the characteristic idempotent !F associated to an arbitrary fusion system F, and demonstrate that bideflation does not generally preserve this idempotent when F is unsaturated. Motivated by the theory of p-completed classifying spaces, we study when two groups are isomorphic in the left-free p-local biset category B¡p , and prove that two groups G and H are isomorphic in this category when there exists an isomorphism between their Frobenius p-fusion systems. Finally, we consider a process mirroring Green’s theory of idempotent condensation and demonstrate that a generalized Burnside functor is the decondensation of the usual Burnside functor, and that this decondensation preserves the subfunctor lattice. v To my family, for their encouragement and support. vi Acknowledgments First and foremost I am indebted to Robert Boltje, without whose insight and patience this thesis would not exist. I have learned a tremendous amount from each of my committee members, and they have helped me develop as a mathematician. Martin Flashman’s enthusiasm for the subject convinced me to pursue it in earnest. David Kinberg has my profound thanks for recognizing and fostering my intellectual curiosity when I was a teenager. I am also grateful for my friends in Santa Cruz, many of them also colleagues in the Mathematics Department. They have enriched my life to an ineffable degree. My early collaborations with Corey Shanbrom instilled me with the requisite tenacity to come as far as I have. Rob Laber and David Deconde have helped me expand my knowledge base and focus the direction of my future studies. I am in debt to Victor Dods and Wyatt Howard for their many hours of stimulating conversation. vii Introduction Double Burnside groups serve as the common core linking together all the objects we study in this thesis. For a finite group G, the set of isomorphism classes of finite G-sets is a commutative monoid under the coproduct of sets, and the Grothendieck group B(G) of this monoid is enriched with a commutative ring structure via the direct product of sets. We call B(G) the single Burnside ring of G. Given another finite group H and commutative ring R, we define the double Burnside R-module RB(G; H) to be R ⊗Z B(G × H). There is a natural correspondence between (G × H)-sets and (G; H)-bisets, which are sets endowed with a left G-action and a right H-action that commute, given by viewing a left (G × H)-set X as a (G; H)-biset via gxh := (g; h−1)x for g 2 G, h 2 H, x 2 X. When G = H, the tensor product of bisets induces a generally non-commutative ring structure on B(G; G), which we then call the double Burnside ring of G. While the single and double Burnside rings of G encode much representation theoretic information about G, their multiplicative structures can be complicated. One successful remedy is to embed the Burnside rings into other rings, called ghost rings, with more tractable multiplicative structures. This is classically done by taking marks, i.e., computing fixed points. Our study of fusion systems and biset functors primarily exploits ghost groups for the double Burnside group introduced by Boltje and Danz in [BD12; BD13]. Fusion systems are of interest to abstract group theorists, modular representa- tion theorists, and topologists. There is hope that the techniques involved in classifying saturated fusion systems could streamline the classification of finite simple groups. As- sociated to every p-block of a finite group is a saturated fusion system, and these fusion systems provide a nice context in which to investigate aspects of modular representation theory. The Martino-Priddy conjecture [MP96], proved by Oliver [BLO03; Oli04; Oli06], asserts that two groups have homotopy equivalent p-completed classifying spaces if and 1 only if their Frobenius p-fusion systems are equivalent. Ragnarsson and Stancu [RS13] create a bijection between saturated fusion systems on a p-group S and the set of characteristic idempotents in the bifree dou- ∆ ble Burnside Z(p)-algebra Z(p)B (S; S). Boltje and Danz [BD12] construct a ghost ring B~∆(S; S) for the bifree double Burnside ring B∆(S; S), a mark homomorphism ρ: B∆(S; S) ! B~∆(S; S), and are able to extend the correspondence in [RS13] to the set of all fusion systems on S and a set of certain explicitly described idempotents in ~∆ ∆ the rational bifree ghost algebra QB (S; S). While the image of Z(p)B (S; S) under ρ ~∆ ~∆ is contained in Z(p)B (S; S), the idempotents in Z(p)B (S; S) do not necessarily corre- spond to saturated fusion systems. One of our initial hopes was to furnish a completely algebraic proof that a fusion system on S is saturated if and only if its corresponding ∆ characteristic idempotent !F is an element of Z(p)B (S; S), which was initially proved by topological means in [RS13]. As of this writing, our goal remains unachieved. The basic theory of saturated fusion systems can be reformulated in terms of characteristic idempotents [RS13, Section 8]. In particular, the characteristic idempo- tents of saturated fusion systems are well-behaved with respect to factoring, in the sense that that if F is a saturated fusion system on S and F=T is a quotient system over a strongly F-closed subgroup T , then the characteristic idempotents of F and F=T are S related by bideflation, i.e., BidefT (!F ) = !F=T . We show in Section 3.2 that this result does not extend generally by constructing an unsaturated fusion system F and quotient S system F=T such that BidefT (!F ) 6= !F=T . A convenient setting for our studies are various subcategories of the R-biset category BR, and categories related to them by means of ghost groups. The category BR has as objects the class of finite groups (or, to get an equivalent category, a transversal for the isomorphism classes of finite groups) and for two objects G and H we define HomBR (G; H) := RB(H; G). Morphism composition is induced from the tensor product of bisets. Both [Rag07, Theorem A] and [RS13, Section 9] relate the stable homotopy of classifying spaces of finite groups to the left-free p-local biset category B¡p , which is a subcategory of the Z(p)-biset category with the same object class, but whose morphisms are generated by transitive bisets corresponding to p-groups. To better understand stable equivalences, it would be useful to determine whether B¡p has the same isomorphism classes as the bifree p-local biset category B∆p , a subcategory of B¡p whose morphisms are generated by bifree bisets. In Chapter2 we use the N-grading of ghost algebras 2 introduced in [BD12, Section 6] to explore when two groups are isomorphic in B¡p . In Section 2.2 we prove that if there is a fusion preserving isomorphism between the Frobenius p-fusion systems of G and H, then G and H are isomorphic in B∆p , further relating fusion systems to stable homotopy of classifying spaces. Many natural operations that occur in the representation theory of finite groups– for example, restriction and induction–arise as functors involving bisets. As these opera- tions appear in a variety of contexts, such as group cohomology, the algebraic K-theory of group rings, and algebraic number theory, the abstract study of these operations is sufficiently motivated. The formalism of operations like restriction and induction is en- coded by biset functors, which are R-bilinear functors from some subcategory of BR to the category of R-modules.
Load more

Recommended publications
  • Math 601: Algebra Problem Set 2 Due: September 20, 2017 Emily Riehl Exercise 1. the Group Z Is the Free Group on One Generator
    Math 601: Algebra Problem Set 2 due: September 20, 2017 Emily Riehl Exercise 1. The group Z is the free group on one generator. What is its universal property?1 Exercise 2. Prove that there are no non-zero group homomorphisms between the Klein four group and Z=7. Exercise 3. • Define an element of order d in Sn for any d < n. • For which n is Sn abelian? Give a proof or supply a counterexample for each n ≥ 1. Exercise 4. Is the product of cyclic groups cyclic? If so, give a proof. If not, find a counterexample. Exercise 5. Prove that if A and B are abelian groups, then A × B satisfies the universal property of the coproduct in the category Ab of abelian groups. Explain why the commutativity hypothesis is necessary. Exercise 6. Prove that the function g 7! g−1 defines a homomorphism G ! G if and only if G is abelian.2 Exercise 7. (i) Fix an element g in a group G. Prove that the conjugation function x 7! −1 gxg defines a homomorphism γg : G ! G. (ii) Prove that the function g 7! γg defines a homomorphism γ : G ! Aut(G). The image of this function is the subgroup of inner automorphisms of G. (iii) Prove that γ is the zero homomorphism if and only if G is abelian. Exercise 8. We have seen that the set of homomorphisms hom(B; A) between two abelian groups is an abelian group with addition defined pointwise in A. In particular End(A) := hom(A; A) is an abelian group under pointwise addition.
    [Show full text]
  • AN INTRODUCTION to CATEGORY THEORY and the YONEDA LEMMA Contents Introduction 1 1. Categories 2 2. Functors 3 3. Natural Transfo
    AN INTRODUCTION TO CATEGORY THEORY AND THE YONEDA LEMMA SHU-NAN JUSTIN CHANG Abstract. We begin this introduction to category theory with definitions of categories, functors, and natural transformations. We provide many examples of each construct and discuss interesting relations between them. We proceed to prove the Yoneda Lemma, a central concept in category theory, and motivate its significance. We conclude with some results and applications of the Yoneda Lemma. Contents Introduction 1 1. Categories 2 2. Functors 3 3. Natural Transformations 6 4. The Yoneda Lemma 9 5. Corollaries and Applications 10 Acknowledgments 12 References 13 Introduction Category theory is an interdisciplinary field of mathematics which takes on a new perspective to understanding mathematical phenomena. Unlike most other branches of mathematics, category theory is rather uninterested in the objects be- ing considered themselves. Instead, it focuses on the relations between objects of the same type and objects of different types. Its abstract and broad nature allows it to reach into and connect several different branches of mathematics: algebra, geometry, topology, analysis, etc. A central theme of category theory is abstraction, understanding objects by gen- eralizing rather than focusing on them individually. Similar to taxonomy, category theory offers a way for mathematical concepts to be abstracted and unified. What makes category theory more than just an organizational system, however, is its abil- ity to generate information about these abstract objects by studying their relations to each other. This ability comes from what Emily Riehl calls \arguably the most important result in category theory"[4], the Yoneda Lemma. The Yoneda Lemma allows us to formally define an object by its relations to other objects, which is central to the relation-oriented perspective taken by category theory.
    [Show full text]
  • Categories, Functors, and Natural Transformations I∗
    Lecture 2: Categories, functors, and natural transformations I∗ Nilay Kumar June 4, 2014 (Meta)categories We begin, for the moment, with rather loose definitions, free from the technicalities of set theory. Definition 1. A metagraph consists of objects a; b; c; : : :, arrows f; g; h; : : :, and two operations, as follows. The first is the domain, which assigns to each arrow f an object a = dom f, and the second is the codomain, which assigns to each arrow f an object b = cod f. This is visually indicated by f : a ! b. Definition 2. A metacategory is a metagraph with two additional operations. The first is the identity, which assigns to each object a an arrow Ida = 1a : a ! a. The second is the composition, which assigns to each pair g; f of arrows with dom g = cod f an arrow g ◦ f called their composition, with g ◦ f : dom f ! cod g. This operation may be pictured as b f g a c g◦f We require further that: composition is associative, k ◦ (g ◦ f) = (k ◦ g) ◦ f; (whenever this composition makese sense) or diagrammatically that the diagram k◦(g◦f)=(k◦g)◦f a d k◦g f k g◦f b g c commutes, and that for all arrows f : a ! b and g : b ! c, we have 1b ◦ f = f and g ◦ 1b = g; or diagrammatically that the diagram f a b f g 1b g b c commutes. ∗This talk follows [1] I.1-4 very closely. 1 Recall that a diagram is commutative when, for each pair of vertices c and c0, any two paths formed from direct edges leading from c to c0 yield, by composition of labels, equal arrows from c to c0.
    [Show full text]
  • Nonablian Cocycles and Their Σ-Model Qfts
    Nonablian cocycles and their σ-model QFTs December 31, 2008 Abstract Nonabelian cohomology can be regarded as a generalization of group cohomology to the case where both the group itself as well as the coefficient object are allowed to be generalized to 1-groupoids or even to general 1-categories. Cocycles in nonabelian cohomology in particular represent higher principal bundles (gerbes) { possibly equivariant, possibly with connection { as well as the corresponding associated higher vector bundles. We propose, expanding on considerations in [13, 34, 5], a systematic 1-functorial formalization of the σ-model quantum field theory associated with a given nonabelian cocycle regarded as the background field for a brane coupled to it. We define propagation in these σ-model QFTs and recover central aspects of groupoidification [1, 2] of linear algebra. In a series of examples we show how this formalization reproduces familiar structures in σ-models with finite target spaces such as Dijkgraaf-Witten theory and the Yetter model. The generalization to σ-models with smooth target spaces is developed elsewhere [24]. 1 Contents 1 Introduction 3 2 1-Categories and Homotopy Theory 4 2.1 Shapes for 1-cells . 5 2.2 !-Categories . 5 2.3 !-Groupoids . 7 2.4 Cosimplicial !-categories . 7 2.5 Monoidal biclosed structure on !Categories ............................ 7 2.6 Model structure on !Categories ................................... 9 3 Nonabelian cohomology and higher vector bundles 9 3.1 Principal !-bundles . 10 3.2 Associated !-bundles . 11 3.3 Sections and homotopies . 12 4 Quantization of !-bundles to σ-models 15 4.1 σ-Models . 16 4.2 Branes and bibranes .
    [Show full text]
  • Ring (Mathematics) 1 Ring (Mathematics)
    Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right.
    [Show full text]
  • THE MCM-APPROXIMATION of the TRIVIAL MODULE OVER a CATEGORY ALGEBRA3 Unfactorizable Morphism
    THE MCM-APPROXIMATION OF THE TRIVIAL MODULE OVER A CATEGORY ALGEBRA REN WANG Abstract. For a finite free EI category, we construct an explicit module over its category algebra. If in addition the category is projective over the ground field, the constructed module is Gorenstein-projective and is a maximal Cohen- Macaulay approximation of the trivial module. We give conditions on when the trivial module is Gorenstein-projective. 1. Introduction Let k be a field and C be a finite EI category. Here, the EI condition means that all endomorphisms in C are isomorphisms. In particular, HomC (x, x) = AutC (x) is a finite group for each object x. Denote by kAutC (x) the group algebra. Recall that a finite EI category C is projective over k if each kAutC (y)-kAutC (x)-bimodule kHomC (x, y) is projective on both sides; see [9, Definition 4.2]. The concept of a finite free EI category is introduced in [6, 7]. Let C be a finite α free EI category. For any morphism x → y in C , set V (α) to be the set of objects ′ ′′ α α w such that there are factorizations x → w → y of α with α′′ a non-isomorphism. ′′ For any w ∈ V (α), we set tw(α) = α ◦ ( g), which is an element in g∈AutPC (w) kHomC (w,y). The freeness of C implies that the element tw(α) is independent of the choice of α′′. Denote by k-mod the category of finite dimensional k-vector spaces. We identify covariant functors from C to k-mod with left modules over the category algebra.
    [Show full text]
  • Pathlike Co/Bialgebras and Their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics
    PATHLIKE CO/BIALGEBRAS AND THEIR ANTIPODES WITH APPLICATIONS TO BI- AND HOPF ALGEBRAS APPEARING IN TOPOLOGY, NUMBER THEORY AND PHYSICS. RALPH M. KAUFMANN AND YANG MO Dedicated to Professor Dirk Kreimer on the occasion of his 60th birthday Abstract. We develop an algebraic theory of flavored and pathlike co-, bi- and Hopf algebras. This is the right framework in which to discuss antipodes for bialgebras naturally appearing in topology, number theory and physics. In particular, we can precisely give conditions the invert- ibility of characters needed for renormalization in the formulation of Connes and Kreimer and in the canonical examples these conditions are met. To construct the antipodes directly, we discuss formal localization constructions and quantum deformations. These allow to construct and naturally explain the appearance Brown style coactions. Using previous results, we can tie all the relevant coalgebras to categorical constructions and the bialgebra structures to Feynman categories. Introduction Although the use of Hopf algebras has a long history, the seminal pa- per [CK98] led to a turbocharged development for their use which has pene- trated deeply into mathematical physics, number theory and also topology, their original realm|see [AFS08] for the early beginnings. The important realization in [CK98, CK00, CK01] was that the renormalization procedure for quantum field theory can be based on a character theory for Hopf al- gebras via a so-called Birkhoff factorization and convolution inverses. The relevant Hopf algebras are those of trees with a coproduct given by a sum over so{called admissible cuts |with the factors of the coproduct being the arXiv:2104.08895v1 [math.QA] 18 Apr 2021 left over stump and the collection of cut off branches| and Hopf algebras of graphs in which the factors of the summands of the coproduct are given by a subgraph and the quotient graph.
    [Show full text]
  • Quiver Generalized Weyl Algebras, Skew Category Algebras and Diskew Polynomial Rings
    Math.Comput.Sci. (2017) 11:253–268 DOI 10.1007/s11786-017-0313-5 Mathematics in Computer Science Quiver Generalized Weyl Algebras, Skew Category Algebras and Diskew Polynomial Rings V. V. B av u l a Received: 7 December 2016 / Revised: 3 March 2017 / Accepted: 11 March 2017 / Published online: 28 April 2017 © The Author(s) 2017. This article is an open access publication Abstract The aim of the paper is to introduce new large classes of algebras—quiver generalized Weyl algebras, skew category algebras, diskew polynomial rings and skew semi-Laurent polynomial rings. Keywords Skew category algebra · Quiver generalized Weyl algebra · Diskew polynomial ring · Generalized Weyl algebra · Skew double quiver algebra · Double quiver groupoid Mathematics Subject Classification 16D30 · 16P40 · 16D25 · 16P50 · 16S85 1 Introduction In this paper, K is a commutative ring with 1, algebra means a K -algebra. In general, it is not assumed that a K -algebra has an identity element. Module means a left module. Missing definitions can be found in [11]. The aim of the paper is to introduce new large classes of algebras—quiver generalized Weyl algebras, skew category algebras, diskew polynomial rings, skew semi-Laurent polynomial rings and the simplex generalized Weyl algebras. 2 Skew Category Algebras The aim of this section is to introduce skew category algebras; to consider new interesting examples (skew tree rings, skew semi-Laurent polynomial rings, etc); to give criteria for a skew category algebra to be a left/right Noetherian algebra (Theorem 2.2, Proposition 2.3). In Sect. 3, skew category algebras are used to define the quiver generalized Weyl algebras.
    [Show full text]
  • Monoids in Representation Theory, Markov Chains, Combinatorics and Operator Algebras
    Monoids in Representation Theory, Markov Chains, Combinatorics and Operator Algebras Benjamin Steinberg, City College of New York Australian Mathematical Society Meeting December 2019 Group representation theory and Markov chains • Group representation theory is sometimes a valuable tool for analyzing Markov chains. Group representation theory and Markov chains • Group representation theory is sometimes a valuable tool for analyzing Markov chains. • The method (at least for nonabelian groups) was perhaps first popularized in a paper of Diaconis and Shahshahani from 1981. Group representation theory and Markov chains • Group representation theory is sometimes a valuable tool for analyzing Markov chains. • The method (at least for nonabelian groups) was perhaps first popularized in a paper of Diaconis and Shahshahani from 1981. • They analyzed shuffling a deck of cards by randomly transposing cards. Group representation theory and Markov chains • Group representation theory is sometimes a valuable tool for analyzing Markov chains. • The method (at least for nonabelian groups) was perhaps first popularized in a paper of Diaconis and Shahshahani from 1981. • They analyzed shuffling a deck of cards by randomly transposing cards. • Because the transpositions form a conjugacy class, the eigenvalues and convergence time of the walk can be understood via characters of the symmetric group. Group representation theory and Markov chains • Group representation theory is sometimes a valuable tool for analyzing Markov chains. • The method (at least for nonabelian groups) was perhaps first popularized in a paper of Diaconis and Shahshahani from 1981. • They analyzed shuffling a deck of cards by randomly transposing cards. • Because the transpositions form a conjugacy class, the eigenvalues and convergence time of the walk can be understood via characters of the symmetric group.
    [Show full text]
  • Categories, Functors, Natural Transformations
    CHAPTERCHAPTER 1 1 Categories, Functors, Natural Transformations Frequently in modern mathematics there occur phenomena of “naturality”. Samuel Eilenberg and Saunders Mac Lane, “Natural isomorphisms in group theory” [EM42b] A group extension of an abelian group H by an abelian group G consists of a group E together with an inclusion of G E as a normal subgroup and a surjective homomorphism → E H that displays H as the quotient group E/G. This data is typically displayed in a diagram of group homomorphisms: 0 G E H 0.1 → → → → A pair of group extensions E and E of G and H are considered to be equivalent whenever there is an isomorphism E E that commutes with the inclusions of G and quotient maps to H, in a sense that is made precise in §1.6. The set of equivalence classes of abelian group extensions E of H by G defines an abelian group Ext(H, G). In 1941, Saunders Mac Lane gave a lecture at the University of Michigan in which 1 he computed for a prime p that Ext(Z[ p ]/Z, Z) Zp, the group of p-adic integers, where 1 Z[ p ]/Z is the Prüfer p-group. When he explained this result to Samuel Eilenberg, who had missed the lecture, Eilenberg recognized the calculation as the homology of the 3-sphere complement of the p-adic solenoid, a space formed as the infinite intersection of a sequence of solid tori, each wound around p times inside the preceding torus. In teasing apart this connection, the pair of them discovered what is now known as the universal coefficient theorem in algebraic topology, which relates the homology H and cohomology groups H∗ ∗ associated to a space X via a group extension [ML05]: n (1.0.1) 0 Ext(Hn 1(X), G) H (X, G) Hom(Hn(X), G) 0 .
    [Show full text]
  • Adjoint Associativity: an Invitation to Algebra in -Categories 1 JOSEPHLIPMAN
    Commutative Algebra and Noncommutative Algebraic Geometry, II MSRI Publications Volume 68, 2015 Adjoint associativity: an invitation to algebra in -categories 1 JOSEPH LIPMAN There appeared not long ago a reduction formula for derived Hochschild cohomology, that has been useful, for example, in the study of Gorenstein maps and of rigidity with respect to semidualizing complexes. The formula involves the relative dualizing complex of a ring homomorphism, so brings out a connection between Hochschild homology and Grothendieck duality. The proof, somewhat ad hoc, uses homotopical considerations via a number of noncanonical projective and injective resolutions of differential graded objects. Recent efforts aim at more intrinsic approaches, hopefully upgradable to “higher” contexts — like bimodules over algebras in -categories. This 1 would lead to wider applicability, for example to ring spectra; and the methods might be globalizable, revealing some homotopical generalizations of aspects of Grothendieck duality. (The original formula has a geometric version, proved by completely different methods coming from duality theory.) A first step is to extend Hom-Tensor adjunction — adjoint associativity — to the -category 1 setting. Introduction 266 1. Motivation: reduction of Hochschild (co)homology 266 2. Enter homotopy 268 3. Adjoint associativity 269 4. -categories 270 1 5. The homotopy category of an -category 273 1 6. Mapping spaces; equivalences 276 7. Colimits 279 8. Adjoint functors 280 9. Algebra objects in monoidal -categories 282 1 This expository article is an elaboration of a colloquium talk given April 17, 2013 at the Mathemat- ical Sciences Research Institute in Berkeley, under the Commutative Algebra program supported by the National Science Foundation.
    [Show full text]
  • Operator Algebras in Rigid C*-Tensor Categories Arxiv:1611.04620V2
    Operator algebras in rigid C*-tensor categories Corey Jones and David Penneys December 5, 2016 Abstract In this article, we define operator algebras internal to a rigid C*-tensor category C. A C*/W*-algebra object in C is an algebra object A in ind-C whose category of free modules FreeModC(A) is a C-module C*/W*-category respectively. When C = Hilbf:d:, the category of finite dimensional Hilbert spaces, we recover the usual notions of operator algebras. We generalize basic representation theoretic results, such as the Gelfand-Naimark and von Neumann bicommutant theorems, along with the GNS construction. We define the notion of completely positive maps between C*-algebra objects in C and prove the analog of the Stinespring dilation theorem. As an application, we discuss approximation and rigidity properties, including amenability, the Haagerup property, and property (T) for a connected W*-algebra M in C. Our definitions simultaneously unify the definitions of analytic properties for discrete quantum groups and rigid C*-tensor categories. Contents 1 Introduction2 2 Background7 2.1 Linear, dagger, and C*-categories . .8 2.2 Involutions on tensor categories . .9 2.3 Rigid C*-tensor categories . 10 arXiv:1611.04620v2 [math.OA] 1 Dec 2016 2.4 The category Vec(C)............................... 13 2.5 Graphical calculus for the Yoneda embedding . 15 2.6 The category Hilb(C)............................... 17 3 ∗-Algebras in Vec(C) 20 3.1 Algebras and module categories . 20 3.2 Equivalence between algebras and cyclic module categories . 22 3.3 ∗-Algebra objects and dagger module categories . 25 3.4 C* and W*-algebra objects .
    [Show full text]