Segal models for equivariant incomplete infinite loop spaces

TL;DR

This paper develops models for equivariant infinite loop spaces using equivariant $ ext{Gamma}$-spaces, linking universe choices to transfer systems and Segal conditions, and applies these to construct Segal $K$-theory for normed permutative categories.

Contribution

It introduces equivariant analogs of $ ext{Gamma}$-spaces for modeling equivariant infinite loop spaces and explores their various categorical forms.

Findings

Established a correspondence between universe choices and transfer systems.

Defined equivariant $ ext{Gamma}$-spaces interpolating between known categories.

Constructed Segal $K$-theory for normed permutative categories.

Abstract

We model equivariant infinite loop spaces indexed on incomplete universes via suitable equivariant analogs of $\Gamma$-spaces. The choice of universe dictates a transfer system which in turn dictates the Segal condition on equivariant $\Gamma$-spaces. Equivariant $\Gamma$-spaces themselves come in different but equivalent guises interpolating between categories $\Gamma$ as defined by Segal and $\Gamma_G$ as defined by Shimakawa. The main application is the construction of Segal $K$-theory of normed permutative categories.