Additivity and Fiber Sequences for Combinatorial K-Theory

TL;DR

This paper develops a new categorical framework called ECGW categories that generalizes existing structures for algebraic K-theory, enabling new constructions and theorems such as the Additivity, Fibration, and Localization theorems.

Contribution

It introduces ECGW categories, modifies axioms from Campbell and Zakharevich, and establishes a K-theory spectrum with key properties and broad applicability.

Findings

ECGW categories enable an S_bullet-construction similar to Waldhausen's.

The framework satisfies an analogue of the Additivity Theorem.

Relative ECGW categories satisfy Fibration and Localization theorems.

Abstract

The (A)CGW categories of Campbell and Zakharevich show how finite sets and varieties behave like the objects of an exact category for the purpose of algebraic $K$-theory. These structures admit a well-behaved Q-construction akin to Quillen's, and satisfy analogues of the D\'evissage and Localization theorems. In this work, we modify Campbell and Zakharevich's axioms to obtain a framework called ECGW categories that allows for an $S_\bullet$-construction akin to Waldhausen's, and show how it produces a K-theory spectrum which satisfies an analogue of the Additivity Theorem. We also define a notion of ``relative ECGW categories'' which have weak equivalences determined by a subcategory of acyclic objects satisfying minimal conditions; these satisfy analogues of the Fibration and Localization Theorems that generalize previous versions in the literature. We illustrate these results with examples including exact categories, extensive categories, algebraic varieties, and polytopes up to scissors congruence.