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
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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 -