A streamlined proof of Goodwillie's n-excisive approximation

TL;DR

This paper provides a concise proof of a crucial lemma in Goodwillie's calculus, simplifying the demonstration that the P_nF construction yields an n-excisive functor.

Contribution

It offers a shorter, more efficient proof of a key lemma in Goodwillie's calculus, enhancing understanding of n-excisive functors.

Findings

Simplified proof of Lemma 1.9 from Goodwillie's Calculus III

Clarifies the construction of n-excisive functors

Strengthens the theoretical foundation of Goodwillie's calculus

Abstract

We give a shorter proof of Lemma 1.9 from Goodwillie, "Calculus III", which is the key step in proving that the construction P_nF gives an n-excisive functor.