Equivariant stable categories for incomplete systems of transfers

TL;DR

This paper develops incomplete equivariant stable categories using $N_ abla$ operads, extending the structure of equivariant homotopy theory to include incomplete transfer systems, with applications to units and Picard spaces.

Contribution

It introduces a new operadic framework for incomplete equivariant stable categories, generalizing existing stable homotopy theory to incomplete transfer systems.

Findings

Constructed incomplete equivariant stable categories from $N_ abla$ operads.

Established structural properties like the tom Dieck splitting in this setting.

Applied the framework to equivariant units and Picard space examples.

Abstract

In this paper, we construct incomplete versions of the equivariant stable category; i.e., equivariant stabilization of the category of $G$-spaces with respect to incomplete systems of transfers encoded by an $N_\infty$ operad $\mathcal{O}$. These categories are built from the categories of $\mathcal{O}$-algebras in $G$-spaces. Using this operadic formulation, we establish incomplete versions of the usual structural properties of the equivariant stable category, notably the tom Dieck splitting. Our work is motivated in part by the examples arising from the equivariant units and Picard space functors.