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Multifunctorial $K$-Theory is an Equivalence of Homotopy Theories (2205.08401v3)
Published 17 May 2022 in math.AT, math.CT, and math.KT
Abstract: We show that each of the three $K$-theory multifunctors from small permutative categories to $\mathcal{G}*$-categories, $\mathcal{G}$-simplicial sets, and connective spectra, is an equivalence of homotopy theories. For each of these $K$-theory multifunctors, we describe an explicit homotopy inverse functor. As a separate application of our general results about pointed diagram categories, we observe that the right-induced homotopy theory of Bohmann-Osorno $\mathcal{E}_$-categories is equivalent to the homotopy theory of pointed simplicial categories.