Goodwillie's Calculus of Functors and Higher Topos Theory

Abstract

We develop an approach to Goodwillie's calculus of functors using the techniques of higher topos theory. Central to our method is the introduction of the notion of fiberwise orthogonality, a strengthening of ordinary orthogonality which allows us to give a number of useful characterizations of the class of $n$-excisive maps. We use these results to show that the pushout product of a $P_n$-equivalence with a $P_m$-equivalence is a $P_{m+n+1}$-equivalence. Then, building on our previous work, we prove a Blakers-Massey type theorem for the Goodwillie tower. We show how to use the resulting techniques to rederive some foundational theorems in the subject, such as delooping of homogeneous functors.

arXiv comment40 pages, (a slightly modified version of) this paper is accepted for publication by the Journal of Topology

References in corpus (5)

• Calculus III: Taylor Series
• Goodwillie towers and chromatic homotopy: an overview
• Calculus of functors and model categories II
• Calculus of functors and model categories
• A streamlined proof of Goodwillie's n-excisive approximation

Cited by in corpus (10)

• An introduction to univalent foundations for mathematicians
• All $(\infty ,1)$-toposes have strict univalent universes
• A Generalized Blakers-Massey Theorem
• Notes on Clans and Tribes
• A Cubical Language for Bishop Sets
• Left-exact Localizations of $\infty$-Topoi I: Higher Sheaves
• Small object arguments, plus-construction, and left-exact localizations
• Filter Quotients and Non-Presentable $(\infty ,1)$-Toposes
• Tangent $\infty$-categories and Goodwillie calculus
• Left-exact Localizations of $\infty$-Topoi III: The Acyclic Product