Abstract. We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, "functors between two homotopy theories form a homotopy theory", or more precisely that the category of such models has a well-behaved internal hom-object.
We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum L K(2) S 0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E hF 2 where F is a finite subgroup of the Morava stabilizer group and E 2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n = 2 at p = 3 represents the edge of our current knowledge: n = 1 is classical and at n = 2, the prime 3 is the largest prime where the Morava stabilizer group has a p-torsion subgroup, so that the homotopy theory is not entirely algebraic.The problem of understanding the homotopy groups of spheres has been central to algebraic topology ever since the field emerged as a distinct area of mathematics. A period of calculation beginning with Serre's computation of the cohomology of Eilenberg-MacLane spaces and the advent of the Adams spectral sequence culminated, in the late 1970s, with the work of Miller, Ravenel, and Wilson on periodic phenomena in the homotopy groups of spheres and Ravenel's nilpotence conjectures. The solutions to most of these conjectures by Devinatz, Hopkins, and Smith in the middle 1980s established the primacy of the "chromatic" point of view and there followed a period in which the community absorbed these results and extended the qualitative picture of stable homotopy theory. Computations passed from center stage, to some extent, although there has been steady work in the wings
We propose a notion of weak .nCk; n/-category, which we call .nCk; n/-‚-spaces. The .nCk; n/-‚-spaces are precisely the fibrant objects of a certain model category structure on the category of presheaves of simplicial sets on Joyal's category ‚ n . This notion is a generalization of that of complete Segal spaces (which are precisely the .1; 1/-‚-spaces). Our main result is that the above model category is cartesian.18D05; 55U40
We develop a generalization of the theory of Thom spectra using the language of ∞-categories. This treatment exposes the conceptual underpinnings of the Thom spectrum functor: we use a new model of parameterized spectra, and our definition is motivated by the geometric definition of Thom spectra of May-Sigurdsson. For an A∞-ring spectrum R, we associate a Thom spectrum to a map of ∞-categories from the ∞-groupoid of a space X to the ∞-category of free rank one R-modules, which we show is a model for BGL1R; we show that BGL1R classifies homotopy sheaves of rank one R-modules, which we call R-line bundles. We use our R-module Thom spectrum to define the twisted R-homology and cohomology of R-line bundles over a space classified by a map X → BGL1R, and we recover the generalized theory of orientations in this context. In order to compare this approach to the classical theory, we characterize the Thom spectrum functor axiomatically, from the perspective of Morita theory.
Abstract. We describe and compute the homotopy of spectra of topological modular forms of level 3. We give some computations related to the "building complex" associated to level 3 structures at the prime 2. Finally, we note the existence of a number of connective models of the spectrum TMF (Γ0(3)).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.