Multifunctorial Inverse $K$-Theory

TL;DR

This paper demonstrates that Mandell's inverse $K$-theory functor is a categorically-enriched multifunctor that preserves algebraic structures, providing new insights into the origin of ring categories.

Contribution

It establishes the multifunctorial nature of inverse $K$-theory and its ability to preserve algebraic structures parametrized by non-symmetric operads.

Findings

Inverse $K$-theory is a categorically-enriched non-symmetric multifunctor.

The functor preserves algebraic structures parametrized by non-symmetric operads.

Ring categories can be described as images of inverse $K$-theory.

Abstract

We show that Mandell's inverse $K$-theory functor is a categorically-enriched non-symmetric multifunctor. In particular, it preserves algebraic structures parametrized by non-symmetric operads. As applications, we describe how ring categories arise as the images of inverse $K$-theory.