Morphisms of 2-dimensional structures with applications

This thesis consists of three chapters providing solutions to three problems. All of them involve morphisms of bicategories: lax functors, enriched categories, and categories enriched on two sides. The first problem was to obtain explicit constructions for various 2-categories which represent 2-categorical concepts involving monads and comonads. We considered lax functors (these are the morphisms of bicategories in the sense of Bénabou) between 2-categories C and D and define strictification tensor product for them. Let Lax (C,D) denote the 2-category of lax functors, lax natural transformations and modifications, and [C,D]lnt its full sub-2-category of (strict) 2-functors. Since monads can be seen as lax functors from 1 (the terminal category), the bicategory of monads in D, denoted Mnd(D), is isomorphic to Lax(1,D). A concise way of defining distributive laws is as monads in Mnd(D). The second problem involves enriching in a monoidal category similar to the one used by Lawvere to obtain (generalized) metric spaces. He expressed Cauchy completeness in purely categorical terms which led to the possibility of applying it to an arbitrary base; for example, an ordinary category is Cauchy complete when all its idempotents split. What we do is to obtain spaces of relativistic events as enriched categories and show that they are always Cauchy complete in the categorical sense. We then see this as a more general phenomenon by providing conditions on the base monoidal category which ensure Cauchy completeness of those enriched categories having all idempotents splitting in the underlying category. The splitting condition was not seen in the case of our partially ordered base since the only idempotents are identities. Finally, in order to analyse Cauchy modules for categories enriched in graded and differential graded Abelian groups (GAb and DGAb), we consider two-sided enriched categories between bicategories, forming a tricategory Caten. The construction of DGAb from Ab, which exists in the literature, can be factored via GAb, and we prove that it is an instance of semidirect product of Hopf bimonoids, applicable to an arbitrary base symmetric monoidal category. To extend this approach to the bicategories of modules, we considered a generalization from Hopf bimonoids in a symmetric monoidal category to Hopf comonads in Caten. The crucial property of such comonads is that the forgetful functor creates left Kan extensions,which generalizes creation of duals and cohoms in the monoidal category case, and adjoints in the bicategory case.