Stable bundles over rig categories

摘要

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 K(R.) Z x |BGL(R)|~+, where GL(R) is the monoidal category of weakly invertible matrices over R. The title refers to the sharper result that BGL(R) is equivalent to BGL(HR). If π_oR is a ring, this is almost formal, and our approach is to replace R by a ring completed version, R, provided by Baas, Dundas, Richter, and Rognes [J. reine angew. Math., to appear] with HR. HR. and π_0R the ring completion of π_0R. The remaining step is then to show that 'stable R-bundles' and 'stable R-bundles' are the same, which is done by a hands-on contraction of a custom-built model for the difference between BGL(R) and BGL(R).