Comparing the orthogonal and homotopy functor calculi

TL;DR

This paper compares Goodwillie's homotopy functor calculus and Weiss's orthogonal calculus, showing their towers agree for analytic functors and establishing a relationship between their model categories.

Contribution

It provides a comparison of the two calculi, demonstrating their equivalence for analytic functors and connecting their model category frameworks.

Findings

The towers agree up to weak equivalence for analytic functors.

The Weiss Taylor tower of BO converges.

A commutative diagram of Quillen functors relates the two calculi.

Abstract

Goodwillie's homotopy functor calculus constructs a Taylor tower of approximations to F, often a functor from spaces to spaces. Weiss's orthogonal calculus provides a Taylor tower for functors from vector spaces to spaces. In particular, there is a Weiss tower associated to the functor which sends a vector space V to F evaluated at the one-point compactification of V. In this paper, we give a comparison of these two towers and show that when F is analytic the towers agree up to weak equivalence. We include two main applications, one of which gives as a corollary the convergence of the Weiss Taylor tower of BO. We also lift the homotopy level tower comparison to a commutative diagram of Quillen functors, relating model categories for Goodwillie calculus and model categories for the orthogonal calculus.