TY - JOUR

T1 - Multiplicative equivariant K-theory and the Barratt-Priddy-Quillen theorem

AU - Guillou, Bertrand J.

AU - May, J. Peter

AU - Merling, Mona

AU - Osorno, Angélica M.

N1 - Publisher Copyright:
© 2023 Elsevier Inc.

PY - 2023/2/1

Y1 - 2023/2/1

N2 - 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].

AB - 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].

KW - K-theory

KW - Multicategories

KW - Multifunctors

KW - Multiplicative equivariant infinite loop spaces

KW - Operads

UR - http://www.scopus.com/inward/record.url?scp=85148951309&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85148951309&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2023.108865

DO - 10.1016/j.aim.2023.108865

M3 - Article

AN - SCOPUS:85148951309

SN - 0001-8708

VL - 414

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 108865

ER -