Two-vector Bundles Define a Form of Elliptic Cohomology

We prove that for well-behaved small commutative rig categories (aka. symmetric bimoidal categories) R the algebraic K-theory space of the K-theory spectrum, HR, of R is equivalent to K0(π0R)× |BGL(R)| where GL(R) is the monoidal category of weakly invertible matrices over R. In particular, this proves the conjecture from [BDR] that K(ku) is the K-theory of the 2-category of complex 2-vector spaces. Hence, the work of the fourth author and Christian Ausoni on K(ku) [A, AR] shows that the theory of virtual 2-vector bundles as in [BDR, Theorem 4.10] qualifies as a form of elliptic cohomology theory.