Constructing Monoidal Double Categories of Bimodules

In this thesis we construct a monoidal fibred double category (also known as a proarrow equipment or framed bicategory) of cocommutative Hopf algebras, algebra morphisms, and bimodules in an arbitrary closed symmetric monoidal category V with sufficient coequalisers. That is, we construct a double category of cocommutative Hopf algebras and bimodules where the data of the strict 1-cells can be recovered from the loose 1-cells. We provide some background on enriched categories, bicategories, and double categories. We will also provide background about Hopf algebras and modules taken in an enriched category V and prove several results about Hopf algebras and modules using intuitive string diagrammatic proofs in the general setting of an enriched category. We introduce the idea of a monoidal bifibration using the example of a functor from a category of representations to the category of groups and then show how this can be used to construct a monoidal fibred double category. We generalise this construction to modules and cocommutative Hopf algebras in a closed symmetric monoidal category V and explicitly construct a monoidal fibred double category of Hopf algebras and modules. We show this is equivalent to a double category of Hopf algebras and bimodules to explicitly describe the data of the constructed double category.