Filtered model of a fibration and rational obstruction theory

For a Serre fibration a filtered model is defined and obstructions to the homotopy equivalence of two fibrations (maps), to the section problem and to the extension problem in rational homotopy theory are given.     

A model for the homotopy theory of homotopy theory

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.

     

The homotopy theory of simplicial props

The category of (colored) props is an enhancement of the category of colored operads, and thus of the category of small categories. In this paper, the second in a series on ‘higher props,’ we show that the category of all small colored simplicial props admits a cofibrantly generated model category structure. With this model structure, the forgetful functor from props to operads is a right Quillen functor.     

The homotopy theory of fusion systems

We define and characterize a class of -complete spaces which have many of the same properties as the -completions of classifying spaces of finite groups. For example, each such has a Sylow subgroup , maps for a -group are described via homomorphisms , and is isomorphic to a certain ring of ``stable elements' in . These spaces arise as the ``classifying spaces' of certain algebraic objects which we call ``-local finite groups'. Such an object consists of a system of fusion data in , as formalized by L. Puig, extended by some extra information carried in a category which allows rigidification of the fusion data.

     

The milnor-moore theorem in tame homotopy theory

LetX be a 1-connected space with Moore loop space ΩX. By a well-known theorem of J. W. Milnor and J. C. Moore [7] the Hurewicz homomorphism induces an isomorphism of Hopf algebrasU(π_*(ΩX) ⊗Q)→H _*(ΩX;Q). HereU(−) denotes the universal enveloping algebra and the Lie bracket on π_*(ΩX) ⊗Q is given by the Samelson product. Assume now thatX is the geometric realization of anr-reduced simplicial set,r≥3. LetL _X be a differential graded free Lie algebra over ℤ describing the tame homotopy type ofX according to the theory of [4]. Then the main result of the present paper is the construction of a sequence of morphisms of differential graded algebras betwenU(L _X ) and the algebraC U _*(ΩX)z of normalized cubical chains on ΩX such that the induced morphisms on homology with coefficientsR _k are isomorphismsH _r-1+l (U(L _x );R _k ) ≅H _r-1+l C U _*(ΩX);R _k ) forl≤k; hereR ₀⊆R ₁⊆… is a tame ring system, i. e.R _k )⊑Q and each primep with 2p−3≤k is invertible inR _k . However, it is no longer true that the Pontrjagin algebraH _≤r−1+k (ΩX; R _k ) of ΩX in degrees ≤r−1+k is determined by π_*(ΩX) or by a cofibrant (-fibrant) modelM of π_*(ΩX) as will be shown by an example. But there is a filtration onH _≤r−1+k (ΩX; R _k ) such that the associated graded algebra is isomorphic toH _≤r−1+k (U(M); R _k ).This will be proved by using a filtered Lie algebra model ofX constructed from a bigraded model of π_*(ΩX). Supported by a CNRS grant and PROCOPE Supported by PROCOPE     

Three models for the homotopy theory of homotopy theories

Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the “homotopy theory” of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a “homotopy theory of homotopy theories”. In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to the homotopy theory arising from the model category structure on simplicial categories. Thus, any of these three categories with the respective weak equivalences could be considered a model for the homotopy theory of homotopy theories. One of them in particular, Rezk’s complete Segal space model category structure on the category of simplicial spaces, is much more convenient from the perspective of making calculations and therefore obtaining information about a given homotopy theory.     

Simplicial chains over a field andp-local homotopy theory

The author was partially supported by the National Science Foundation and the Alfred P. Sloan Foundation     

The Serre spectral sequence of a multiplicative fibration

In a fibration we show that finiteness conditions on force the homology Serre spectral sequence with -coefficients to collapse at some finite term. This in particular implies that as graded vector spaces, is ``almost' isomorphic to . One consequence is the conclusion that is elliptic if and only if and are.

     

The homotopy theory of bialgebras over pairs of operads

We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in two steps. In the first step, we equip coalgebras over an operad with a cofibrantly generated model category structure. In the second step we use the adjunction between bialgebras and coalgebras via the free algebra functor. This result allows us to do classical homotopical algebra in various categories such as associative bialgebras, Lie bialgebras or Poisson bialgebras in chain complexes.     

The stack of formal groups in stable homotopy theory

We construct the algebraic stack of formal groups and use it to provide a new perspective onto a recent result of M. Hovey and N. Strickland on comodule categories for Landweber exact algebras. This leads to a geometric understanding of their results as well as to a generalisation.     

Rational -equivariant homotopy theory

We give an algebraicization of rational -equivariant homotopy theory. There is an algebraic category of `` -systems' which is equivalent to the homotopy category of rational -simply connected -spaces. There is also a theory of ``minimal models' for -systems, analogous to Sullivan's minimal algebras. Each -space has an associated minimal -system which encodes all of its rational homotopy information, including its rational equivariant cohomology and Postnikov decomposition.

     

Endofiniteness in stable homotopy theory

We study endofinite objects in a compactly generated triangulated category in terms of ideals in the category of compact objects. Our results apply in particular to the stable homotopy category. This leads, for example, to a new interpretation of stable splittings for classifying spaces of finite groups.