|dc.description.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 start
by briefly recalling the construction of the 2-category of bundle gerbes, with minor alterations
that allow us to endow morphisms with additive structures. The morphisms in the
resulting 2-categories are investigated in detail. We introduce a direct sum on morphism
categories 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, a categorised ring, and we show that generic morphism categories
of bundle gerbes form module categories over this rig-category.
We continue by presenting a categorification of the hermitean bundle 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 the acting
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
the bundle gerbe under consideration. We show that the resulting categories of sections
carry a rig-module structure over the category of nite-dimensional Hilbert spaces with its
canonical direct sum and tensor product. 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, with its additive and module structures, 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 the 2-category of 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.||en_US