A diagrammatic approach to constructing fibrant double categories

In [Shu08], Shulman describes a way to construct a fibrant double category from a monoidal bifibration. In this thesis, we take an algebraic approach using indexed categories and string diagrams to better understand this construction and the role that the Beck-Chevalley transformation has within it. We give an explicit calculation of the niche-filling morphism arising from cartesian and opcartesian lifting properties, and we use this to give a more intuitive string-diagrammatic proof of a result on conditions equivalent to the Beck-Chevalley conditions. We give a detailed examination of the construction and make explicit calculations—in string diagrammatic language—of the unit loose 1-cell and the loose composition (left) unitor of the constructed fibrant double category. Motivating examples are given throughout, including proofs that the forgetful functor that maps a G-module V to the group G that acts on it is a weakly Beck-Chevalley and internally closed monoidal bifibration.