Orientations on 2-vector bundles and determinant gerbes (Q2857593)

In [\textit{C. Ausoni} et al., Doc. Math., J. DMV 13, 795--801 (2008; Zbl 1203.19002)], a half magnetic monopole was discovered. This describes an obstruction to the existence of a continuous map \(K(ku) \to B(ku^* )\) 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 \(K(Z,3) \to K(ku)\); and any sensible definition of determinant like should produce such a retract. In the present paper, this obstruction is precisely described using monoidal categories. By a result of \textit{N. A. Baas} et al. [J. Topol. 4, No. 3, 623--640 (2011; Zbl 1227.19002)], \(K(ku) \) classifies 2-vector bundles. The author thus defines the notion of oriented 2-vector bundles, which removes the obstruction by the magnetic monopole. He uses 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, he calls this a determinant gerbe map.