Inverse K-Theory is a Pseudo Symmetric Cat -Multifunctor

This chapter proves the main theorem on inverse K-theory. It constructs the pseudo symmetric Cat-multifunctorial inverse K-theory P and a further factorization of P that involves a non-symmetric Cat-multiequivalence. Then it completely unravels the pseudo symmetric Cat-multifunctorial structure of inverse K-theory. It discusses related work in the literature about the multifunctoriality of inverse K-theory. To illustrate the utility of the main theorem, the rest of this chapter discusses the fact that inverse K-theory preserves pseudo symmetric E-infinity-algebras. As preparation it discusses the categorical Barratt-Eccles operad, its coherence, and E-infinity-algebras in Γ-categories. Then it defines pseudo symmetric E-infinity-algebras as pseudo symmetric Cat-multifunctors from the Barratt-Eccles operad. Inverse K-theory sends pseudo symmetric E-infinity-algebras in Γ-categories to those in PermCat^sg. This chapter ends with a detailed unraveling of these structures in Γ-categories and PermCat^sg. This concludes Part III.