On n-excisive functors of module categories

We give a new construction for the n-th Taylor polynomial, in the sense of Goodwillie calculus, for homotopy functors from spectra to spectra. We then use this model to classify n-excisive functors of module categories of functors with smash product (FSPs) by bi-modules of explicit FSPs. Introduction: In [Cal3], T. Goodwillie constructs a Taylor tower for functors from spectra to spectra. The layers of this construction are called n-homogeneous functors and he classifies these by use of the n-th cross effects of a functor. In this way, the layers of the tower become reasonably computable by using an explicit algorithm applied to the n-th cross effect of a functor. Our first goal in this paper is to establish a generalization of this process to not only obtain the layers of the Tower, but the tower itself. It should be pointed out, however, that our construction only works for functors of spectra while Goodwillie’s classification of the layers also works in more general settings like homotopy functors of spaces. The idea for this construction arose from a conversation with G. Arone when he explained to me the Taylor tower of ΣΩ (from spectra to spectra). In order to state the first result, we need to make some definitions. Let Mn be the dual of the category of non-empty finite sets and surjective morphisms (one object {1, . . . , n} for each n ∈ N). For a pointed space X we let X be the functor from Mn to pointed spaces defined by U ∈ Mn 7→ X ∧U and f : V → U 7→ X : X −→ X X (x1 ∧ . . . ∧ xu) = (xf(1) ∧ . . . ∧ xf(v)) (the map induced by diagonals). For F a functor from spectra to itself we define ĉrF—a functor from Mn to functors of spectra. For U ∈ Mn, ĉr F is equivalent to Goodwillie’s |U |-cross effect but defined dually using products and homotopy cofibers (see section 1 for details). For F a homotopy functor of spectra which is stably n-excisive we prove: Theorem (4.6): PnF ( ) ≃ hocolimk MapMn((S ), ĉrF (S∧ )). In the second part of the paper we use this result to give a classification of n-excisive functors of module categories. A functor with smash product (FSP) is a model for ring spectra which is very convenient for explicit constructions (for example, they are used to define the Hochschild homology of ring spectra in [Bö]). One can easily work with the category of modules of an FSP and they inherit many of the useful properties that chain † This work was supported by National Science Foundation grant # 1-5-30943