Functor Calculus for under Categories and the De Rham Complex

Goodwillie’s calculus of homotopy functors associates a tower of polynomial approximations, the Taylor tower, to a functor of topological spaces over a fixed space. We define a new tower, the varying center tower, for functors of categories with a fixed initial object, such as algebras under a fixed ring spectrum. We construct this new tower using elements of the Taylor tower constructions of Bauer, Johnson, and McCarthy for functors of simplicial model categories, and show how the varying center tower differs from Taylor towers in terms of the properties of its individual terms and convergence behavior. We prove that there is a combinatorial model for the varying center tower given as a proequivalence between the varying center tower and towers of cosimplicial objects; this generalizes Eldred’s cosimplicial models for finite stages of Taylor towers. As an application, we present models for the de Rham complex of rational commutative ring spectra due to Rezk on the one hand, and Goodwillie and Waldhausen on the other, and use our result to conclude that these two models will be equivalent when extended to E∞-ring spectra.