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arXiv · 0909.1742

Stable bundles over rig categories

Nils A. Baas·Bjorn Ian Dundas·Birgit Richter·John Rognes
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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.

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Nils A. Baas, Bjorn Ian Dundas, Birgit Richter, John Rognes. 2009-09-09. Stable bundles over rig categories. https://doi.org/10.1112/jtopol%2Fjtr016

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