Two-Vector Bundles Define a Form of Elliptic Cohomology (original) (raw)

We prove that for well-behaved small rig categories R (also known as bimonoidal categories) the algebraic K-theory space, K(HR), of the K-theory ring spectrum of R is equivalent to K(R) ≃ Z × |BGL(R)|+, where GL(R) is the monoidal category of weakly invertible matrices over R. To achieve this, we solve the long-standing problem of group completing within the context of rig categories. More precisely, we construct an additive group completion ¯ R of R that retains the multiplicative structure, i.e., that remains a rig category. 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 Christian Ausoni and the fourth author on K(ku) (AR, A) shows that the theory of virtual 2-vector bundles as in (BDR, Theorem 4.10) qualifies as a form of elliptic cohomology. In telescopic complexity 0, 1 and ∞ there are cohomology theories that possess a geometric definition: de Rham cohomology of manifolds is given...