Title: Categorical Structures on Bundle Gerbes and Higher Geometric Prequantisation,
Abstract: 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. This is
achieved by introducing a functorial dual that extends the dual of vector
bundles to morphisms of bundle gerbes, and constructing a two-variable
adjunction for the aforementioned rig-module category structure on morphism
categories. Its right internal hom is the module action, composed by taking the
dual of higher functions, while the left internal hom is interpreted as a
bundle gerbe metric. Sections of bundle gerbes are defined as morphisms from
the trivial bundle gerbe to a given bundle gerbe. The resulting categories of
sections carry a rig-module structure over the category of finite-dimensional
Hilbert spaces. A suitable definition of 2-Hilbert spaces is given, modifying
previous definitions by the use of two-variable adjunctions. We prove that the
category of sections of a bundle gerbe fits into this framework, thus obtaining
a 2-Hilbert space of sections. In particular, this can be constructed for
prequantum bundle gerbes in problems of higher geometric quantisation. We
define a dimensional reduction functor and show that the categorical structures
introduced on bundle gerbes naturally reduce to their counterparts on hermitean
line bundles with connections. In several places in this thesis, we provide
examples, making 2-Hilbert spaces of sections and dimensional reduction very
explicit.