Ring completion of rig categories

TL;DR

This paper constructs a natural additive group completion for rig categories, transforming them into ring categories while preserving multiplicative structures, and applies this to algebraic K-theory conjectures.

Contribution

It provides a method to complete rig categories into ring categories, enabling new applications in algebraic K-theory.

Findings

Constructed a natural additive group completion for rig categories.

Retained multiplicative structure in the completion, forming a ring category.

Applied the construction to prove a conjecture relating algebraic K-theory of ku and V.

Abstract

We offer a solution to the long-standing problem of group completing within the context of rig categories (also known as bimonoidal categories). Given a rig category R we construct a natural additive group completion R' that retains the multiplicative structure, hence has become a ring category. If we start with a commutative rig category R (also known as a symmetric bimonoidal category), the additive group completion R' will be a commutative ring category. In an accompanying paper we show how this can be used to prove the conjecture from [BDR] that the algebraic K-theory of the connective topological K-theory spectrum ku is equivalent to the algebraic K-theory of the rig category V of complex vector spaces.