05w5078 Workshop in Homotopical Localization and the Calculus of Functors

s of Talks M. Ching Operads and calculus of functors I’ll talk about some aspects of the relationship between the calculus of homotopy functors and the theory of operads. In particular, I’ll describe the operad structure on the derivatives of the identity functor and try to explain how the derivatives of other functors might fit into this framework. C. Casacuberta Continuity of homotopy idempotent functors A functor L in a simplicial model category is called simplicial or continuous if it defines a map from map(X,Y ) → map(LX,LY ) for all X , Y , which is natural and preserves composition and identity. As shown by Farjoun and Hirschhorn, f -localizations can be constructed as continuous functors. Thus, a necessary condition for a homotopy idempotent functor to be equivalent to some f -localization is that it be equivalent to a continuous functor. In joint work with different coauthors, we discuss continuity of homotopy functors in several model categories, with emphasis on simplicial sets, spectra, and groupoids. In the latter, remarkably, continuity is automatic. 4 SCOPE OF WORKSHOP 7 W.G. Dwyer Localization and Calculus I and II A general discussion of the idea of localization in homotopy theory. Followed in part II by specialization to the localization of diagram categories, and further specialization to the case of a particular diagram category associated to the Goodwillie tower. E. Farjoun Open problems and some recent progress in localization and cellularization theory The talk will revisit some of the progress made recently in understanding localization and co-localization functors. We shall list some interesting problems and describe related partial progress. The talks will concentrate mostly on general properties of localization with respect to a map in both algebraic homological algebra and topological categories. T. Goodwillie Introduction to the Calculus of Homotopy Functors, I,II, and III Overview of basic definitions and results (excisive and n-excisive approximations of functors, classification of homogeneous functors, chain rule); key examples; matrix notation. Followed in part II by: more about homogeneous functors, with an emphasis on results which require no information about connectivity. A geometric view of the functor/function analogy. In this view, Top is a variety and functors Top → Spectra are global functions. I will say which categories are the tangent spaces of Top. I will discuss tangent vector fields and more generally tensor fields, in both a coordinate-free way and a coordinate-dependent way. I will show that there are two tangent connections, both of which are flat, and that their difference is the tensor field known as smash product of spectra. I will say something about higher-order jets and about differential operators. I cannot make much sense of differential forms (except 0-forms and 1-forms), but I may talk about them anyway. Applications are work in progress, but I will make sure to at least say something trivial about some nontrivial examples, and maybe something nontrivial about some trivial examples. M. Hovey E(n)∗ − E(n)-comodules I will recap my results with Neil Strickland about the structure of the category of E(n)∗E(n)-comodules (e.g. the Landweber filtration theorem works there as well). I will describe why we need to know more about comodules (derived functors of product in the category of comodules form the E2-term of a spectral sequence converging to the E(n)-homology of a product of spectra; this is relevant for the chromatic splitting conjecture). Then I will describe some new results I have about the honest injective E(n)∗E(n)-comodules. There are only n+ 1 isomorphism classes of indecomposable injectives, and most interestingly, the endomorphism ring of the k-th one is (E(k))(E(k)), where E(k) is the completion of E(k) at Ik. So in the category of E(n)∗E(n)-comodules, you are seeing all the E(k) operations for 0 ≤ k ≤ n, and therefore seeing all the different stabilizer groups Sk for 0 ≤ k ≤ n. This is a good thing, since the relation between the different stabilizer groups is basically what the chromatic splitting conjecture is about. N. Kuhn Periodic homology of infinite loop spaces If E∗ is a homology theory, one can ask to what extent the E∗-homology of an infinite loop space is determined by the E∗-homology of the associated spectrum. Using a combination of the Hopkins-Smith Periodicy Theorem, as packaged in the telescopic functors of Bousfield and me, and Goodwillie calculus, I can give a quite definitive answer to this question when the homology theory is Morava K-theory. There are calculations still to be done that may inform on the Telescope conjecture. A. Mauer-Oats An operad from the derivatives of a monad McClure and Smith have a simple idea that explains how to produce an operad from a functor operad by evaluating on the unit of the smash product. The cross effects of a (reasonably good) monad F are a functoroperad of spaces. We explain the proper way to prolong a multivariate functor to spectra, and use this to produce an operad of symmetric spectra. If a certain problem of cofibrancy can be overcome, the spectra in the operad will be the derivative spectra of F . B. Munson The layers of the embedding tower I will discuss the layers of the embedding tower and their relationship to the obstructions to finding embeddings. 4 SCOPE OF WORKSHOP 8 D. Sinha A pairing between graphs and trees We give an elementary pairing between graphs and trees, which facilitates the study of the Lie operad and free Lie algebras. It arises in topology through both homology of configuration spaces and (conjecturally) in studying Hopf invariants and Whitehead products. We sketch its possible application in using the embedding calculus to define knot invariants, and hope that it might be of interest in the homotopy calculus as well. D. Stanley Complete invariants of t-structures Let R be a Noetherian ring. We give a classification of Bousfield classes on the bounded derived category of R. This also gives complete invariants of t-structures on the same category. We also show that the t-structures on the unbounded derived category of Z-modules do not form a set. I. Volic Embedding calculus and formality of the little cubes operad I will first give a brief introduction to embedding calculus and say how a certain Taylor tower can be assigned to an isotopy functor. Then I will describe joint work with Greg Arone and Pascal Lambrechts in which the central observation is that the stages of the Taylor tower in the case of Emb(M,V ), the space of embeddings of a manifold in a vector space (up to immersions), have the structure of maps of certain modules over the little cubes operad. Using Kontsevich’s formality of this operad, one then concludes that the cohomology spectral sequence for Emb(M,V ) arising from the Taylor tower collapses rationally. In the special case of spaces of knots, this was conjectured by Vassiliev. Additionally, using the interplay between embedding and orthogonal calculus, one also deduces that the rational cohomology of Emb(M,V ) only depends on the rational homotopy type of M when 2dim(M) + 1 Bauer, Kristine (University of Calgary) Casacuberta, Carles (University of Barcelona) Chebolu, Sunil (University of Washington)* Ching, Michael (Massachusetts Institute of Technology)* Chorny, Boris (University of Western Ontario)** Dover, Lynn (University of Alberta)* Dror-Farjoun, Emmanuel (Hebrew University of Jerusalem) Dwyer, William (Notre Dame University) Goodwillie, Tom (Brown University) Gutierrez, Javier (University of Barcelona) Hovey, Mark (Wesleyan University) Krause, Eva (University of Alberta) Kudryavtseva, Elena (University of Calgary/Moscow State University) Kuhn, Nick (University of Virginia) Lambrechts, Pascal (Louvain-la-Neuve) Mauer-Oats, Andrew (Northwestern University)** McCarthy, Randy (University of Illinois at Urbana-Champaign) Munson, Brian (Stanford University)**