Stable bundles over rig categories

TL;DR

This paper proves that virtual 2-vector bundles are classified by the algebraic K-theory of topological K-theory, establishing a geometric cohomology theory with elliptic cohomology complexity.

Contribution

It demonstrates the classification of virtual 2-vector bundles via algebraic K-theory of ring spectra associated to rig categories, extending formal results to well-behaved bimonoidal categories.

Findings

K(HR) is equivalent to Z × |BGL(R)|^+ for well-behaved rig categories R

The approach involves replacing R with a ring completion when π_0R is a ring

Provides a geometric cohomology theory of telescopic complexity similar to elliptic cohomology.

Abstract

The point of this paper is to prove the conjecture that virtual 2-vector bundles are classified by K(ku), the algebraic K-theory of topological K-theory. Hence, by the work of Ausoni and the fourth author, virtual 2-vector bundles give us a geometric cohomology theory of the same telescopic complexity as elliptic cohomology. The main technical step is showing that for well-behaved small rig categories R (also known as bimonoidal categories) the algebraic K-theory space, K(HR), of the ring spectrum HR associated to R is equivalent to Z \times |BGL(R)|^+, where GL(R) is the monoidal category of weakly invertible matrices over R. If \pi_0R is a ring this is almost formal, and our approach is to replace R by a ring completed version provided by [BDRR1] whose \pi_0 is the ring completion of \pi_0R.