Multifunctorial K-theory is an equivalence of homotopy theories

We show that each of the three K-theory multifunctors from small permutative categories to $$\mathcal {G}_*$$ -categories, -simplicial sets, and connective spectra, is an equivalence homotopy theories. For these multifunctors, we describe explicit inverse functor. As a separate application our general results about pointed diagram categories, observe right-induced theory Bohmann–Osorno {E}_*$$ -categories equivalent simplicial categories.