1,720,988 research outputs found
Rigidity of the -local stable homotopy category
Get PDFWe investigate a new case of rigidity in stable homotopy theory which is the rigidity of the K(1)-local stable homotopy category Ho(LK(1)Sp) at p=2. In other words, we show that recovering higher homotopy information by just looking at the triangulated structure of Ho(LK(1)Sp) is possible, which is a property that only a few interesting stable model categories are known to possess
Franke's Realization Functor and Monoidal Products
Get PDFIn 1996, Jens Franke in an unpublished paper states that the homotopy category of E(1)-local spectra is equivalent as a triangulated category to D1(A), the derived category of quasi-periodic cochain complexes of period 1 for primes p ≥ 3. This is Franke's realization functor R: D1(A) → Ho(L1Sp). However, Irakli Patchkoria spotted gaps in the proof of J.Franke that were filled in a series of papers and put in a firm ground that for primes p ≥ 5 Franke's realization functor is a triangulated equivalence. The categories D1(A) and v are in fact tensor-triangulated, that is, both categories posses a monoidal structure that are compatible with the triangulated structure. In this thesis we prove that Franke's realization functor commutes with the monoidal products up to a natural isomorphism, that is, R i is tensor triangulated functor
The Derived Deligne Conjecture
Get PDFWe study the operad of derived A∞-algebras from a new point of view in order to find a derived version of the Deligne conjecture. We start by defining the brace structure on an operad of graded R-modules using operadic suspension, which we describe in depth for the first time as a functor, and use it to define A∞-algebra structures on certain operads, with the endomorphism operad as our main example. This construction provides us with an operadic context from which A∞-algebras arise in a natural way and allows us to generalize the Lie algebra structure on the Hochschild complex of an A∞-algebra. Next, we generalize these con structions to operads of bigraded R-modules, introducing a totalization functor. This allows us to generalize a Lie algebra structure on the to tal complex of a derived A∞-algebra. This construction and the use of some enriched categories allow us to obtain new versions of the Deligne conjecture
Rigidity and exotic models for v1-local G-equivariant stable homotopy theory
Get PDFWe prove that the v1-local G-equivariant stable homotopy category for G a finite group has a unique G-equivariant model at p=2. This means that at the prime 2 the homotopy theory of G-spectra up to fixed point equivalences on K-theory is uniquely determined by its triangulated homotopy category and basic Mackey structure. The result combines the rigidity result for K-local spectra of the second author with the equivariant rigidity result for G-spectra of the first author. Further, when the prime p is at least 5 and does not divide the order of G, we provide an algebraic exotic model as well as a G-equivariant exotic model for the v1-local G-equivariant stable homotopy category, showing that for primes p≥5 equivariant rigidity fails in general
Arbeitsgemeinschaft mit aktuellem Thema: Modern Foundations for Stable Homotopy Theory: Mathematisches Forschungsinstitut Oberwolfach Report No. 46/2005, organised/edited by John Rognes (Oslo) and Stefan Schwede (Bonn)
Get PDFIn recent years, spectral algebra or stable homotopical algebra over structured ring spectra has become an important new direction in stable homotopy theory. This workshop provided an introduction to structured ring spectra and applications of spectral algebra, both within homotopy theory
and in other areas of mathematics
Bousfield localisations along Quillen bifunctors and applications
Get PDFWe describe left and right Bousfield localisations along Quillen adjunctions of two variables. These localised model structures can be used to define Postnikov sections and homological localisations of arbitrary model categories, and to study the homotopy limit model structure on the category of sections of a left Quillen presheaf of localised model structures. We obtain explicit results in this direction in concrete examples of towers and fiber products of model categories. In particular, we prove that the category of simplicial sets is Quillen equivalent to the homotopy limit model structure of its Postnikov tower, and that the category of symmetric spectra is Quillen equivalent to the homotopy fiber product of its Bousfield arithmetic square. For spectral model categories, we show that the homotopy fiber of a stable left Bousfield localisation is a stable right Bousfield localisation
On the algebraic classification of K-local spectra
Get PDFIn 1996, Jens Franke proved the equivalence of certain triangulated categories possessing an Adams spectral sequence. One particular application of this theorem is that the K(p)-local stable homotopy category at an odd prime can be described as the derived category of an abelian category. We explain this proof from a topologist's point of view
Rigidity and Exotic Models for the <i>K</i>-local Stable Homotopy Category
Get PDFWhen two stable model categories C and D are Quillen equivalent, their homotopy categories Ho(C) and Ho(D) are equivalent as triangulated categories. But is the converse also true? For the stable homotopy category Ho(Sp), i.e., the homotopy category of spectra, there is the following result by Stefan Schwede: Rigidity Theorem (Schwede `05) Let C be a stable model category, and f: Ho(Sp) -> Ho(C) an equivalence of triangulated categories. Then the underlying model categories Sp and C are Quillen equivalent, i.e., Ho(Sp) is ''rigid''. Next, we ask the question if this is also true for Bousfield localisations of the stable homotopy category at a generalised cohomology theory. This thesis treats the case of Ho(L_K Sp), i.e. the stable homotopy category localised at 2-local complex K-theory K_(2) and comes to the conclusion that Ho(L_K Sp) is rigid: K_(2)-local Rigidity Theorem (Roitzheim) Let C be a stable model category, and f: Ho(L_K Sp) -> Ho(C) an equivalence of triangulated categories. Then the underlying model categories L_K Sp and C are Quillen equivalent, i.e., Ho(L_1 Sp) is ''rigid''. However, for odd primes p, the K_(p)-local stable homotopy category is not rigid, and we discuss a counterexample given by Jens Franke
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