Additivity and Fiber Sequences for Combinatorial K-Theory
Maru Sarazola, Brandon T. Shapiro
TL;DR
This work addresses limitations in the ACGW framework that impede obtaining a spectrum-valued $K$-theory and an iterated $S_ullet$-construction by proposing ECGW categories with a restricted notion of squares. It introduces relative ECGW categories and proves a Waldhausen-style $S_ullet$-construction that yields a $K$-theory spectrum satisfying an analogue of Additivity, along with Fibration and Localization analogues. The framework is demonstrated on examples including exact categories, extensive categories, algebraic varieties, and polytopes up to scissors congruence, highlighting its broad applicability to combinatorial and geometric settings. Overall, the results extend combinatorial K-theory beyond classical exact categories, enabling robust motivic-type measurements and connections to non-additive contexts while preserving a spectrum-valued invariant through the ECGW construction.
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 Dévissage 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.