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 ${\mathcal{G}}_{\ast }$-categories, ${\mathcal{G}}_{\ast }$-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 ${\mathcal{E}}_{\ast }$-categories is equivalent to the homotopy theory of pointed simplicial categories.

20 pages. Final version. To appear in Journal of Homotopy and Related Structures