Orientations on 2-vector Bundles and Determinant Gerbes - Kragh, Thomas (MATHEMATICA SCANDINAVICA, 2013)
In a paper from 2009, a half magnetic monopole was discovered by Ausoni, Dundas, and Rognes. This describes an obstruction to the existence of a continuous map K(ku) → B(ku^*) with determinant like properties. This magnetic monopole is in fact an obstruction to the existence of a map from K(ku) to K(Z,3), which is a retract of the natural map K(Z,3) → K(ku); and any sensible definition of determinant like should produce such a retract. In this paper we describe this obstruction precisely using monoidal categories. By a result from 2011 by Baas, Dundas, Richter and Rognes K(ku) classifies 2-vector bundles. We thus define the notion of oriented 2-vector bundles, which removes the obstruction by the magnetic monopole. We use this to define an oriented K-theory of 2-vector bundles with a lift of the natural map from K(Z,3). It is then possible to define a retraction of this map and since K(Z,3) classifies complex gerbes we call this a determinant gerbe map.