Goodwillie Calculi

TL;DR

This paper explores the relationship between n-excision and n-additivity in Goodwillie calculus, providing new insights into approximations of functors and their properties.

Contribution

It introduces a comparison between n-excision and n-additivity, establishing conditions for their equivalence and constructing related fibrations.

Findings

Established a fibration sequence involving n-additive approximations

Compared new constructions with Goodwillie's original framework

Provided conditions for when different approximations coincide

Abstract

The Goodwillie tower is based on the idea of approximating a functor F by a series of functors satisfying the strong property of "n-excision". In this dissertation, we study a weaker property of "n-additivity" and compare the two. Theorem 9.1, one of the main results in this dissertation, establishes that if $F$ is reasonably good, there is a fibration sequence with the fiber being the realization of a simplicial space built from a cotriple made of iterated cross effects and base space the "discrete" degree $n$ additive approximation to $F$. We also relate the construction given to Goodwillie's construction, and give conditions under which they coincide.