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Register a new userMultiplicative equivariant $K$-theory and the Barratt-Priddy-Quillen theorem
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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.
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We show that Mandells inverse $K$-theory functor is a categorically-enriched multifunctor. In particular, it preserves algebraic structures parametrized by operads. As applications, we describe how ring categories, bipermutative categories, braided ring categories, and $E_n$-monoidal categories arise as the images of inverse $K$-theory.
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