Multifunctorial K -theory is an equivalence of homotopy theories

We show that each of the three K -theory multifunctors from small permutative categories to 𝒢_* -categories, 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 ℰ_* -categories is equivalent to the homotopy theory of pointed simplicial categories.