Cross effects and calculus in an unbased setting

TL;DR

This paper extends the calculus of functors to a new setting involving subcategories defined by factoring a fixed morphism, introducing cross effects and comparing different tower constructions.

Contribution

It generalizes cotriple Taylor towers and cross effects to functors from subcategories of simplicial model categories, connecting to Goodwillie's calculus.

Findings

The towers agree when D is a spectrum category and F commutes with realizations.

Explicit adjoint pairs of functors for cross effects are identified.

The generalized towers agree at the initial object for non-realization-commuting functors.

Abstract

We study functors F from C_f to D where C and D are simplicial model categories and C_f is the full subcategory of C consisting of objects that factor a fixed morphism f from A to B. We define the analogs of Eilenberg and Mac Lane's cross effects functors in this context, and identify explicit adjoint pairs of functors whose associated cotriples are the diagonals of the cross effects. With this, we generalize the cotriple Taylor tower construction of [10] from the setting of functors from pointed categories to abelian categories to that of functors from C_f to D to produce a tower of functors whose n-th term is a degree n functor. We compare this tower to Goodwillie's tower of n-excisive approximations to F found in [8]. When D is a good category of spectra, and F is a functor that commutes with realizations, the towers agree. More generally, for functors that do not commute with realizations, we show that the terms of the towers agree when evaluated at the initial object of C_f.