Equivariant Weiss Calculus

TL;DR

This paper develops an equivariant version of Weiss calculus for finite groups, enabling analysis of functors with group actions through Taylor approximations, derivatives, and spectra, with new notions of restriction and fixed points.

Contribution

It introduces an equivariant Weiss calculus framework for finite groups, incorporating group representations, spectra, and fixed-point notions, extending classical calculus of functors.

Findings

Established equivariant Taylor approximations and derivatives.

Compared fixed-point and restriction functors with their Taylor approximations.

Developed a theory connecting group actions with functor calculus.

Abstract

In this paper, we introduce an equivariant analog of Weiss calculus of functors for all finite group $\mathrm{G}$. In our theory, Taylor approximations and derivatives are index by finite dimensional $\mathrm{G}$-representations, and homogeneous layers are classified by orthogonal $\mathrm{G}$-spectra. Further, our framework permits a notion of restriction as well as a notion of fixed-point at the level of Weiss functors. We establish various results comparing Taylor approximations and derivatives of fixed-point (resp. restrictions) functors to that of the fixed-point (resp. restrictions) of Taylor approximations and derivatives.