Combinatorial models for Taylor polynomials of functors

TL;DR

This paper introduces a new combinatorial tower, the varying center tower, for functors of categories with fixed initial objects, extending Goodwillie's calculus and providing models for de Rham complexes of rational ring spectra.

Contribution

It defines the varying center tower, constructs a pro-equivalence with cosimplicial towers, and generalizes Eldred's models, enhancing understanding of polynomial approximations in homotopy theory.

Findings

The varying center tower differs from traditional Taylor towers in properties and convergence.

A combinatorial pro-model for the varying center tower is established.

Models for de Rham complexes of rational ring spectra are extended and shown to be equivalent for $E_{inite}$-ring spectra.

Abstract

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.