Algebraic $K$-theory for squares categories

TL;DR

This paper introduces squares $K$-theory, a new formalism that generalizes classical $K$-theory relations using square diagrams, with applications to manifolds, varieties, and motivic measures.

Contribution

It develops a novel squares $K$-theory formalism that extends traditional relations and applies it to geometry and motivic measures.

Findings

Defines $K_0$ for squares categories satisfying a four-term relation.

Provides examples in $K$-theory of manifolds and varieties.

Constructs a derived motivic measure in $K$-theory of homotopy sheaves.

Abstract

In this paper we introduce a new formalism for $K$-theory, called squares $K$-theory. This formalism allows us to simultaneously generalize the usual three-term relation $[B] = [A] + [C]$ for an exact sequence $A \hookrightarrow B \twoheadrightarrow C$ or for a subtractive sequence $A\hookrightarrow B \leftarrow C$, by defining $K_0$ of a squares category to satisfy a four-term relation $[A]+[D]= [C] + [B]$ for a ``good'' square diagram with these corners. Examples that rely on this formalism are $K$-theory of smooth manifolds of a fixed dimension and $K$-theory of (smooth and) complete varieties. Another application we give of this theory is the construction of a derived motivic measure taking value in the $K$-theory of homotopy sheaves.