Model Categories for Orthogonal Calculus

TL;DR

This paper reformulates orthogonal calculus using model categories, clarifying the role of O(n)-equivariance, and classifies n-homogeneous functors via spectra with O(n)-actions, extending Weiss's work.

Contribution

It introduces a model category framework for orthogonal calculus, providing clearer results and a new classification of n-homogeneous functors through spectra with O(n)-actions.

Findings

Developed model structures for polynomial and homogeneous functors

Classified n-homogeneous functors via spectra with O(n)-action

Extended orthogonal calculus to orthogonal spectra

Abstract

We restate the notion of orthogonal calculus in terms of model categories. This provides a cleaner set of results and makes the role of O(n)-equivariance clearer. Thus we develop model structures for the category of n-polynomial and n-homogeneous functors, along with Quillen pairs relating them. We then classify n-homogeneous functors, via a zig-zag of Quillen equivalences, in terms of spectra with an O(n)-action. This improves upon the classification of Weiss. As an application, we develop a variant of orthogonal calculus by replacing topological spaces with orthogonal spectra.