Multiplicative equivariant $K$-theory and the Barratt-Priddy-Quillen theorem

TL;DR

This paper extends the equivariant Barratt-Priddy-Quillen theorem to a multiplicative setting, introducing new operadic multicategories and an equivariant infinite loop space machine to produce structured ring spectra.

Contribution

It develops a multiplicative equivariant infinite loop space machine and operadic multicategories, enabling a multiplicative version of the equivariant BPQ theorem.

Findings

Proves a multiplicative equivariant BPQ theorem.

Constructs a new operadic multicategory framework.

Establishes a multifunctor from symmetric monoidal G-categories to G-spectra.

Abstract

We prove a multiplicative version of the equivariant Barratt-Priddy-Quillen theorem, starting from the additive version proven in arXiv:1207.3459. The proof uses a multiplicative elaboration of an additive equivariant infinite loop space machine that manufactures orthogonal $G$-spectra from symmetric monoidal $G$-categories. The new machine produces highly structured associative ring and module $G$-spectra from appropriate multiplicative input. It relies on new operadic multicategories that are of considerable independent interest and are defined in a general, not necessarily equivariant or topological, context. Most of our work is focused on constructing and comparing them. We construct a multifunctor from the multicategory of symmetric monoidal $G$-categories to the multicategory of orthogonal $G$-spectra. With this machinery in place, we prove that the equivariant BPQ theorem can be lifted to a multiplicative equivalence. That is the heart of what is needed for the presheaf reconstruction of the category of $G$-spectra in arXiv:1110.3571.