Unitary Functor Calculus with Reality

Abstract

We construct a calculus of functors in the spirit of orthogonal calculus, which is designed to study "functors with reality" such as the Real classifying space functor, $B{U}_{\mathbb{R}}(-)$. The calculus produces a Taylor tower, the $n$-th layer of which is classified by a spectrum with an action of $C_2⋉U(n)$. We further give model categorical considerations, producing a zig-zag of Quillen equivalences between spectra with an action of $C_2⋉U(n)$ and a model structure on the category of input functors which captures the homotopy theory of the $n$-th layer of the Taylor tower.

27 pages, v.2 accepted version