  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.
