Categorical Structures on Bundle Gerbes and Higher Geometric Prequantisation

Understand

We present a construction of a 2-Hilbert space of sections of a bundle gerbe, a suitable candidate for a prequantum 2-Hilbert space in higher geometric quantisation.

• We introduce a direct sum on the morphism categories in the 2-category of bundle gerbes and show that these categories are cartesian monoidal and abelian.
• Endomorphisms of the trivial bundle gerbe, or higher functions, carry the structure of a rig-category, which acts on generic morphism categories of bundle gerbes.
• We continue by presenting a categorification of the hermitean metric on a hermitean line bundle.