Interpolation categories for homology theories

For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by proving the existence of truncated versions of resolution or $E_2$-model structures. Examples of functors fitting in our framework are given by every generalized homology theory represented by a ring spectrum satisfying the Adams-Atiyah condition. The constructions are closely related to the modified Adams spectral sequence and give a very conceptual approach to the associated moduli problem and obstruction theory. As application we establish an isomorphism between certain E(n)-local Picard groups and some Ext-groups.Comment: 40 pages, corrected version of second part of the replaced version, first part will appear sepparately as "Truncated resolution model structures", to appear in JPA

Calculus of functors and model categories II

This is a continuation, completion, and generalization of our previous joint work with B. Chorny. We supply model structures and Quillen equivalences underlying Goodwillie's constructions on the homotopy level for functors between simplicial model categories satisfying mild hypotheses.Comment: 47 pages. Version 3 contains a new introduction and further small changes, based on suggestions of the referee. To appear in "Algebraic and Geometric Topology

Goodwillie's Calculus of Functors and Higher Topos Theory

We develop an approach to Goodwillie's calculus of functors using the techniques of higher topos theory. Central to our method is the introduction of the notion of fiberwise orthogonality, a strengthening of ordinary orthogonality which allows us to give a number of useful characterizations of the class of $n$-excisive maps. We use these results to show that the pushout product of a $P_n$-equivalence with a $P_m$-equivalence is a $P_{m+n+1}$-equivalence. Then, building on our previous work, we prove a Blakers-Massey type theorem for the Goodwillie tower. We show how to use the resulting techniques to rederive some foundational theorems in the subject, such as delooping of homogeneous functors.Comment: 40 pages, (a slightly modified version of) this paper is accepted for publication by the Journal of Topolog