## Abstract

We prove a multiplicative version of the equivariant Barratt-Priddy-Quillen theorem, starting from the additive version proven in [13]. 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 [12].

Original language | English |
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Article number | 108865 |

Journal | Advances in Mathematics |

Volume | 414 |

DOIs | |

State | Published - Feb 1 2023 |

### Bibliographical note

Funding Information:B.J. Guillou was partially supported by Simons Collaboration Grant No. 282316 and NSF grants DMS-1710379 and DMS-2003204 . M. Merling was partially supported by NSF grant DMS- 1709461 / 1850644 , a Simons AMS Travel grant, and NSF CAREER grant DMS-1943925 . A.M. Osorno was partially supported by the Simons Collaboration Grant No. 359449 , the Woodrow Wilson Career Enhancement Fellowship , and NSF grant DMS-1709302 .

Publisher Copyright:

© 2023 Elsevier Inc.

## Keywords

- K-theory
- Multicategories
- Multifunctors
- Multiplicative equivariant infinite loop spaces
- Operads

## ASJC Scopus subject areas

- Mathematics (all)