  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 behaviour. We
prove that there is a combinatorial model for the varying center tower given as
a pro-equivalence 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 $E_{\infty}$-ring spectra.
