Algebraic $K$-theory for squares categories

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.

Final version. Contains several fixes and expository changes from the first version. We are thankful to an anonymous referee whose suggestions greatly improved the paper

References in corpus (3)

• A universal characterization of higher algebraic K-theory
• A descent principle for compactly supported extensions of functors
• A K-theory spectrum for cobordism cut and paste groups

Cited by in corpus (2)

• A genuine $G$-spectrum for the cut-and-paste $K$-theory of $G$-manifolds
• Segal K-theory factors through Waldhausen categories