Multifunctorial K-theory is an equivalence of homotopy theories

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.