151. Multiplicative equivariant $K$-theory and the Barratt-Priddy-Quillen theorem
- Author
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Bertrand J. Guillou, J. Peter May, Mona Merling, and Angélica M. Osorno
- Subjects
Mathematics::K-Theory and Homology ,General Mathematics ,Mathematics::Category Theory ,Mathematics - K-Theory and Homology ,Primary 19D23, 19L47, 55P48, Secondary 18D20, 18D40, 18M65, 55P91, 55U40 ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Mathematics - Category Theory ,Category Theory (math.CT) ,K-Theory and Homology (math.KT) ,Mathematics - Algebraic Topology ,Mathematics::Algebraic Topology - 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.
- Published
- 2021
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