Cosimplicial models for the limit of the Goodwillie tower

TL;DR

This paper introduces cosimplicial models for the intermediate stages of Goodwillie's Taylor tower, establishing an equivalence that simplifies the computation of homotopy inverse limits of functors.

Contribution

It presents a new cosimplicial model for the intermediate constructions in Goodwillie's calculus, linking them to cosimplicial resolutions and simplifying inverse limit calculations.

Findings

New cosimplicial model for $ ext{T}_n F$

Equivalence between inverse limits of towers and cosimplicial resolutions

Simplified construction for homotopy inverse limits of Taylor towers

Abstract

We call attention to the intermediate constructions $\T_n F$ in Goodwillie's Calculus of homotopy functors, giving a new model which naturally gives rise to a family of towers filtering the Taylor Tower of a functor. We also establish a surprising equivalence between the homotopy inverse limits of these towers and the homotopy inverse limits of certain cosimplicial resolutions. This equivalence gives a greatly simplified construction for the homotopy inverse limit of the Taylor tower of a functor $F$ under general assumptions.