Unitary calculus: model Categories and convergence

TL;DR

This paper develops a unitary calculus framework using model categories, introducing unitary spectra and analyzing convergence through weakly polynomial functors, extending orthogonal calculus to complex geometry.

Contribution

It constructs the unitary analogue of orthogonal calculus with model categories, introduces unitary spectra, and studies convergence via weakly polynomial functors.

Findings

Unitary spectra model the stable homotopy category for unitary calculus.

A zig-zag of Quillen equivalences links unitary spectra with unitary group actions.

Weakly polynomial functors facilitate the analysis of convergence in the Taylor tower.

Abstract

We construct the unitary analogue of orthogonal calculus developed by Weiss, utilising model categories to give a clear description of the intricacies in the equivariance and homotopy theory involved. The subtle differences between real and complex geometry lead to subtle differences between orthogonal and unitary calculus. To address these differences we construct unitary spectra - a variation of orthogonal spectra - as a model for the stable homotopy category. We show through a zig-zag of Quillen equivalences that unitary spectra with an action of the $n$-th unitary group models the homogeneous part of unitary calculus. We address the issue of convergence of the Taylor tower by introducing weakly polynomial functors, which are similar to weakly analytic functors of Goodwillie but more computationally tractable.