Group completion in the K-theory and Grothendieck-Witt theory of proto-exact categories

TL;DR

This paper explores the algebraic K-theory and Grothendieck-Witt theory of proto-exact categories, especially those related to -linear structures, establishing analogues of classical theorems connecting these theories via group completion.

Contribution

It introduces new analogues of Quillen and Schlichting theorems for proto-exact categories, linking their K-theory and Grothendieck-Witt theories through the hermitian Q-construction and group completion.

Findings

Established analogues of Quillen and Schlichting theorems.

Connected K-theory and Grothendieck-Witt theory via group completion.

Focused on -linear proto-exact categories.

Abstract

We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories, with a particular focus on classes of examples of $\mathbb{F}_1$-linear nature. Our main results are analogues of theorems of Quillen and Schlichting, relating the $K$-theory or Grothendieck-Witt theories of proto-exact categories defined using the (hermitian) $Q$-construction and group completion.