Multifunctorial $K$-Theory is an Equivalence of Homotopy Theories

TL;DR

This paper proves that three different $K$-theory multifunctors from small permutative categories to various homotopical categories are all equivalences of homotopy theories, providing explicit inverse functors and relating different homotopy frameworks.

Contribution

It establishes the equivalence of multiple $K$-theory constructions as homotopy theories and describes explicit inverse functors, advancing understanding of their interrelations.

Findings

Each $K$-theory multifunctor is an equivalence of homotopy theories.

Explicit homotopy inverse functors are constructed for each multifunctor.

The homotopy theory of Bohmann-Osorno $ ext{E}_*$-categories is equivalent to pointed simplicial categories.

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.