Title:

Topological C*categories

Tensor C*categories are the result of work to recast the fundamental theory of operator algebras in the setting of category theory, in order to facilitate the study of higherdimensional algebras that are expected to play an important role in a unified model of physics. Indeed, the application of category theory to mathematical physics is itself a highly active field of research. C*categories are the analogue of C*algebras in this context. They are defined as normclosed selfadjoint subcategories of the category of Hilbert spaces and bounded linear operators between them. Much of the theory of C*algebras and their invariants generalises to C*categories. Often, when a C*algebra is associated to a particular structure it is not completely natural because certain choices are involved in its definition. Using C*categories instead can avoid such choices since the construction of the relevant C*category amounts to choosing all suitable C*algebras at once. In this thesis we introduce and study C*categories for which the set of objects carries topological data, extending the present body of work, which exclusively considers C*categories with discrete object sets. We provide a construction of Ktheory for topological C*categories, which will have applications in widening the scope of the BaumConnes conjecture, in index theory, and in geometric quantisation. As examples of such applications, we construct the C*categories of topological groupoids, extending the familiar constructions of Renault.
