Model categories for orthogonal calculus

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 $O\left(n\right)$–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 $O\left(n\right)$–action. This improves upon the classification theorem of Weiss. As an application, we develop a variant of orthogonal calculus by replacing topological spaces with orthogonal spectra.