The Structure of Stable Homotopy Theories
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The Homotopy Category Is a Homotopy Category
Vol. XXIII, 19~2 435 The homotopy category is a homotopy category By A~NE S~o~ In [4] Quillen defines the concept of a category o/models /or a homotopy theory (a model category for short). A model category is a category K together with three distingxtished classes of morphisms in K: F ("fibrations"), C ("cofibrations"), and W ("weak equivalences"). These classes are required to satisfy axioms M0--M5 of [4]. A closed model category is a model category satisfying the extra axiom M6 (see [4] for the statement of the axioms M0--M6). To each model category K one can associate a category Ho K called the homotopy category of K. Essentially, HoK is obtained by turning the morphisms in W into isomorphisms. It is shown in [4] that the category of topological spaces is a closed model category ff one puts F---- (Serre fibrations) and IV = (weak homotopy equivalences}, and takes C to be the class of all maps having a certain lifting property. From an aesthetical point of view, however, it would be nicer to work with ordinary (Hurewicz) fibrations, cofibrations and homotopy equivalences. The corresponding homotopy category would then be the ordinary homotopy category of topological spaces, i.e. the objects would be all topological spaces and the morphisms would be all homotopy classes of continuous maps. In the first section of this paper we prove that this is indeed feasible, and in the last section we consider the case of spaces with base points. 1. The model category structure of Top. Let Top be the category of topolo~cal spaces and continuous maps. -
Thick Subcategories of the Stable Module Category
FUNDAMENTA MATHEMATICAE 153 (1997) Thick subcategories of the stable module category by D. J. B e n s o n (Athens, Ga.), Jon F. C a r l s o n (Athens, Ga.) and Jeremy R i c k a r d (Bristol) Abstract. We study the thick subcategories of the stable category of finitely generated modules for the principal block of the group algebra of a finite group G over a field of characteristic p. In case G is a p-group we obtain a complete classification of the thick subcategories. The same classification works whenever the nucleus of the cohomology variety is zero. In case the nucleus is nonzero, we describe some examples which lead us to believe that there are always infinitely many thick subcategories concentrated on each nonzero closed homogeneous subvariety of the nucleus. 1. Introduction. A subcategory of a triangulated category is said to be thick, or ´epaisse, if it is a triangulated subcategory and it is closed under taking direct summands. One product of the deep work of Devinatz, Hop- kins and Smith [8] on stable homotopy theory was a classification of the thick subcategories of the stable homotopy category. This is described in Hopkins’ survey [11], where he also states a corresponding classification of thick subcategories of the homotopy category of bounded chain complexes of finitely generated projective R-modules for a commutative ring R. In fact, this classification requires R to be Noetherian, as Neeman pointed out in [13], where he also gave a complete proof. Recent work in the modular representation theory of finite groups [3, 4, 5, 18] has exploited the analogy between the stable homotopy category of algebraic topology and the stable module category of a finite group, which is also a triangulated category. -
Poincar\'E Duality for $ K $-Theory of Equivariant Complex Projective
POINCARE´ DUALITY FOR K-THEORY OF EQUIVARIANT COMPLEX PROJECTIVE SPACES J.P.C. GREENLEES AND G.R. WILLIAMS Abstract. We make explicit Poincar´eduality for the equivariant K-theory of equivariant complex projective spaces. The case of the trivial group provides a new approach to the K-theory orientation [3]. 1. Introduction In 1 well behaved cases one expects the cohomology of a finite com- plex to be a contravariant functor of its homology. However, orientable manifolds have the special property that the cohomology is covariantly isomorphic to the homology, and hence in particular the cohomology ring is self-dual. More precisely, Poincar´eduality states that taking the cap product with a fundamental class gives an isomorphism between homology and cohomology of a manifold. Classically, an n-manifold M is a topological space locally modelled n on R , and the fundamental class of M is a homology class in Hn(M). Equivariantly, it is much less clear how things should work. If we pick a point x of a smooth G-manifold, the tangent space Vx is a represen- tation of the isotropy group Gx, and its G-orbit is locally modelled on G ×Gx Vx; both Gx and Vx depend on the point x. It may happen that we have a W -manifold, in the sense that there is a single representation arXiv:0711.0346v1 [math.AT] 2 Nov 2007 W so that Vx is the restriction of W to Gx for all x, but this is very restrictive. Even if there are fixed points x, the representations Vx at different points need not be equivalent. -
Faithful Linear-Categorical Mapping Class Group Action 3
A FAITHFUL LINEAR-CATEGORICAL ACTION OF THE MAPPING CLASS GROUP OF A SURFACE WITH BOUNDARY ROBERT LIPSHITZ, PETER S. OZSVÁTH, AND DYLAN P. THURSTON Abstract. We show that the action of the mapping class group on bordered Floer homol- ogy in the second to extremal spinc-structure is faithful. This paper is designed partly as an introduction to the subject, and much of it should be readable without a background in Floer homology. Contents 1. Introduction1 1.1. Acknowledgements.3 2. The algebras4 2.1. Arc diagrams4 2.2. The algebra B(Z) 4 2.3. The algebra C(Z) 7 3. The bimodules 11 3.1. Diagrams for elements of the mapping class group 11 3.2. Type D modules 13 3.3. Type A modules 16 3.4. Practical computations 18 3.5. Equivalence of the two actions 22 4. Faithfulness of the action 22 5. Finite generation 24 6. Further questions 26 References 26 1. Introduction arXiv:1012.1032v2 [math.GT] 11 Feb 2014 Two long-standing, and apparently unrelated, questions in low-dimensional topology are whether the mapping class group of a surface is linear and whether the Jones polynomial detects the unknot. In 2010, Kronheimer-Mrowka gave an affirmative answer to a categorified version of the second question: they showed that Khovanov homology, a categorification of the Jones polynomial, does detect the unknot [KM11]. (Previously, Grigsby and Wehrli had shown that any nontrivially-colored Khovanov homology detects the unknot [GW10].) 2000 Mathematics Subject Classification. Primary: 57M60; Secondary: 57R58. Key words and phrases. Mapping class group, Heegaard Floer homology, categorical group actions. -
An Introduction to Spectra
An introduction to spectra Aaron Mazel-Gee In this talk I’ll introduce spectra and show how to reframe a good deal of classical algebraic topology in their language (homology and cohomology, long exact sequences, the integration pairing, cohomology operations, stable homotopy groups). I’ll continue on to say a bit about extraordinary cohomology theories too. Once the right machinery is in place, constructing all sorts of products in (co)homology you may never have even known existed (cup product, cap product, cross product (?!), slant products (??!?)) is as easy as falling off a log! 0 Introduction n Here is a table of some homotopy groups of spheres πn+k(S ): n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 n πn(S ) Z Z Z Z Z Z Z Z Z Z n πn+1(S ) 0 Z Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 n πn+2(S ) 0 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 Z2 n πn+3(S ) 0 Z2 Z12 Z ⊕ Z12 Z24 Z24 Z24 Z24 Z24 Z24 n πn+4(S ) 0 Z12 Z2 Z2 ⊕ Z2 Z2 0 0 0 0 0 n πn+5(S ) 0 Z2 Z2 Z2 ⊕ Z2 Z2 Z 0 0 0 0 n πn+6(S ) 0 Z2 Z3 Z24 ⊕ Z3 Z2 Z2 Z2 Z2 Z2 Z2 n πn+7(S ) 0 Z3 Z15 Z15 Z30 Z60 Z120 Z ⊕ Z120 Z240 Z240 k There are many patterns here. The most important one for us is that the values πn+k(S ) eventually stabilize. -
Homotopical Categories: from Model Categories to ( ,)-Categories ∞
HOMOTOPICAL CATEGORIES: FROM MODEL CATEGORIES TO ( ;1)-CATEGORIES 1 EMILY RIEHL Abstract. This chapter, written for Stable categories and structured ring spectra, edited by Andrew J. Blumberg, Teena Gerhardt, and Michael A. Hill, surveys the history of homotopical categories, from Gabriel and Zisman’s categories of frac- tions to Quillen’s model categories, through Dwyer and Kan’s simplicial localiza- tions and culminating in ( ;1)-categories, first introduced through concrete mod- 1 els and later re-conceptualized in a model-independent framework. This reader is not presumed to have prior acquaintance with any of these concepts. Suggested exercises are included to fertilize intuitions and copious references point to exter- nal sources with more details. A running theme of homotopy limits and colimits is included to explain the kinds of problems homotopical categories are designed to solve as well as technical approaches to these problems. Contents 1. The history of homotopical categories 2 2. Categories of fractions and localization 5 2.1. The Gabriel–Zisman category of fractions 5 3. Model category presentations of homotopical categories 7 3.1. Model category structures via weak factorization systems 8 3.2. On functoriality of factorizations 12 3.3. The homotopy relation on arrows 13 3.4. The homotopy category of a model category 17 3.5. Quillen’s model structure on simplicial sets 19 4. Derived functors between model categories 20 4.1. Derived functors and equivalence of homotopy theories 21 4.2. Quillen functors 24 4.3. Derived composites and derived adjunctions 25 4.4. Monoidal and enriched model categories 27 4.5. -
Separable Commutative Rings in the Stable Module Category of Cyclic Groups
SEPARABLE COMMUTATIVE RINGS IN THE STABLE MODULE CATEGORY OF CYCLIC GROUPS PAUL BALMER AND JON F. CARLSON Abstract. We prove that the only separable commutative ring-objects in the stable module category of a finite cyclic p-group G are the ones corresponding to subgroups of G. We also describe the tensor-closure of the Kelly radical of the module category and of the stable module category of any finite group. Contents Introduction1 1. Separable ring-objects4 2. The Kelly radical and the tensor6 3. The case of the group of prime order 14 4. The case of the general cyclic group 16 References 18 Introduction Since 1960 and the work of Auslander and Goldman [AG60], an algebra A over op a commutative ring R is called separable if A is projective as an A ⊗R A -module. This notion turns out to be remarkably important in many other contexts, where the module category C = R- Mod and its tensor ⊗ = ⊗R are replaced by an arbitrary tensor category (C; ⊗). A ring-object A in such a category C is separable if multiplication µ : A⊗A ! A admits a section σ : A ! A⊗A as an A-A-bimodule in C. See details in Section1. Our main result (Theorem 4.1) concerns itself with modular representation theory of finite groups: Main Theorem. Let | be a separably closed field of characteristic p > 0 and let G be a cyclic p-group. Let A be a commutative and separable ring-object in the stable category |G- stmod of finitely generated |G-modules modulo projectives. -
L22 Adams Spectral Sequence
Lecture 22: Adams Spectral Sequence for HZ=p 4/10/15-4/17/15 1 A brief recall on Ext We review the definition of Ext from homological algebra. Ext is the derived functor of Hom. Let R be an (ordinary) ring. There is a model structure on the category of chain complexes of modules over R where the weak equivalences are maps of chain complexes which are isomorphisms on homology. Given a short exact sequence 0 ! M1 ! M2 ! M3 ! 0 of R-modules, applying Hom(P; −) produces a left exact sequence 0 ! Hom(P; M1) ! Hom(P; M2) ! Hom(P; M3): A module P is projective if this sequence is always exact. Equivalently, P is projective if and only if if it is a retract of a free module. Let M∗ be a chain complex of R-modules. A projective resolution is a chain complex P∗ of projec- tive modules and a map P∗ ! M∗ which is an isomorphism on homology. This is a cofibrant replacement, i.e. a factorization of 0 ! M∗ as a cofibration fol- lowed by a trivial fibration is 0 ! P∗ ! M∗. There is a functor Hom that takes two chain complexes M and N and produces a chain complex Hom(M∗;N∗) de- Q fined as follows. The degree n module, are the ∗ Hom(M∗;N∗+n). To give the differential, we should fix conventions about degrees. Say that our differentials have degree −1. Then there are two maps Y Y Hom(M∗;N∗+n) ! Hom(M∗;N∗+n−1): ∗ ∗ Q Q The first takes fn to dN fn. -
Generalized Cohomology Theories
Lecture 4: Generalized cohomology theories 1/12/14 We've now defined spectra and the stable homotopy category. They arise naturally when considering cohomology. Proposition 1.1. For X a finite CW-complex, there is a natural isomorphism 1 ∼ r [Σ X; HZ]−r = H (X; Z). The assumption that X is a finite CW-complex is not necessary, but here is a proof in this case. We use the following Lemma. Lemma 1.2. ([A, III Prop 2.8]) Let F be any spectrum. For X a finite CW- 1 n+r complex there is a natural identification [Σ X; F ]r = colimn!1[Σ X; Fn] n+r On the right hand side the colimit is taken over maps [Σ X; Fn] ! n+r+1 n+r [Σ X; Fn+1] which are the composition of the suspension [Σ X; Fn] ! n+r+1 n+r+1 n+r+1 [Σ X; ΣFn] with the map [Σ X; ΣFn] ! [Σ X; Fn+1] induced by the structure map of F ΣFn ! Fn+1. n+r Proof. For a map fn+r :Σ X ! Fn, there is a pmap of degree r of spectra Σ1X ! F defined on the cofinal subspectrum whose mth space is ΣmX for m−n−r m ≥ n+r and ∗ for m < n+r. This pmap is given by Σ fn+r for m ≥ n+r 0 n+r and is the unique map from ∗ for m < n+r. Moreover, if fn+r; fn+r :Σ X ! 1 Fn are homotopic, we may likewise construct a pmap Cyl(Σ X) ! F of degree n+r 1 r. -
Rational Homotopy Theory: a Brief Introduction
Contemporary Mathematics Rational Homotopy Theory: A Brief Introduction Kathryn Hess Abstract. These notes contain a brief introduction to rational homotopy theory: its model category foundations, the Sullivan model and interactions with the theory of local commutative rings. Introduction This overview of rational homotopy theory consists of an extended version of lecture notes from a minicourse based primarily on the encyclopedic text [18] of F´elix, Halperin and Thomas. With only three hours to devote to such a broad and rich subject, it was difficult to choose among the numerous possible topics to present. Based on the subjects covered in the first week of this summer school, I decided that the goal of this course should be to establish carefully the founda- tions of rational homotopy theory, then to treat more superficially one of its most important tools, the Sullivan model. Finally, I provided a brief summary of the ex- tremely fruitful interactions between rational homotopy theory and local algebra, in the spirit of the summer school theme “Interactions between Homotopy Theory and Algebra.” I hoped to motivate the students to delve more deeply into the subject themselves, while providing them with a solid enough background to do so with relative ease. As these lecture notes do not constitute a history of rational homotopy theory, I have chosen to refer the reader to [18], instead of to the original papers, for the proofs of almost all of the results cited, at least in Sections 1 and 2. The reader interested in proper attributions will find them in [18] or [24]. The author would like to thank Luchezar Avramov and Srikanth Iyengar, as well as the anonymous referee, for their helpful comments on an earlier version of this article. -
Stable Model Categories Are Categories of Modules 1
STABLE MODEL CATEGORIES ARE CATEGORIES OF MODULES STEFAN SCHWEDE AND BROOKE SHIPLEY Abstract: A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard’s work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R. 1. Introduction The recent discovery of highly structured categories of spectra has opened the way for a new wholesale use of algebra in stable homotopy theory. In this paper we use this new algebra of spectra to characterize stable model categories, the settings for doing stable homotopy theory, as categories of highly structured modules. This characterization also leads to a Morita theory for equivalences between categories of highly structured modules. -
On Relations Between Adams Spectral Sequences, with an Application to the Stable Homotopy of a Moore Space
Journal of Pure and Applied Algebra 20 (1981) 287-312 0 North-Holland Publishing Company ON RELATIONS BETWEEN ADAMS SPECTRAL SEQUENCES, WITH AN APPLICATION TO THE STABLE HOMOTOPY OF A MOORE SPACE Haynes R. MILLER* Harvard University, Cambridge, MA 02130, UsA Communicated by J.F. Adams Received 24 May 1978 0. Introduction A ring-spectrum B determines an Adams spectral sequence Ez(X; B) = n,(X) abutting to the stable homotopy of X. It has long been recognized that a map A +B of ring-spectra gives rise to information about the differentials in this spectral sequence. The main purpose of this paper is to prove a systematic theorem in this direction, and give some applications. To fix ideas, let p be a prime number, and take B to be the modp Eilenberg- MacLane spectrum H and A to be the Brown-Peterson spectrum BP at p. For p odd, and X torsion-free (or for example X a Moore-space V= So Up e’), the classical Adams E2-term E2(X;H) may be trigraded; and as such it is E2 of a spectral sequence (which we call the May spectral sequence) converging to the Adams- Novikov Ez-term E2(X; BP). One may say that the classical Adams spectral sequence has been broken in half, with all the “BP-primary” differentials evaluated first. There is in fact a precise relationship between the May spectral sequence and the H-Adams spectral sequence. In a certain sense, the May differentials are the Adams differentials modulo higher BP-filtration. One may say the same for p=2, but in a more attenuated sense.