The $K$-theory of assemblers

Abstract

In this paper we introduce the notion of an assembler, which formally encodes "cutting and pasting" data. An assembler has an associated $K$-theory spectrum, in which $\pi_0$ is the free abelian group of objects of the assembler modulo the cutting and pasting relations, and in which the higher homotopy groups encode further geometric invariants. The goal of this paper is to prove structural theorems about this $K$-theory spectrum, including analogs of Quillen's localization and d\'evissage theorems. We demonstrate the uses of these theorems by analysing the assembler associated to the Grothendieck ring of varieties and the assembler associated to scissors congruence groups of polytopes.