Generalization of Algebraic Operations via Enrichment

Abstract

In this dissertation we examine enrichment relations between categories of dual structure and we sketch an abstract framework where the theory of fibrations and enriched category theory are appropriately united. We initially work in the context of a monoidal category, where we study an enrichment of the category of monoids in the category of comonoids under certain assumptions. This is induced by the existence of the universal measuring comonoid, a notion originally defined by Sweedler in vector spaces. We then consider the fibred category of modules over arbitrary monoids, and we establish its enrichment in the opfibred category of comodules over arbitrary comonoids. This is now exhibited via the existence of the universal measuring comodule, introduced by Batchelor. We then generalize these results to their `many-object' version. In the setting of the bicategory of V-enriched matrices, we investigate an enrichment of V-categories in V-cocategories as well as of V-modules in V-comodules. This part constitutes the core of this treatment, and the theory of fibrations and adjunctions between them plays a central role in the development. The newly constructed categories are described in detail, and they appropriately fit in a picture of duality, enrichment and fibrations as in the previous case. Finally, we introduce the concept of an enriched fibration, aimed to provide a formal description for the above examples. Related work in this direction, though from a different perspective and with dissimilar outcomes, has been realized by Shulman. We also discuss an abstraction of this picture in the environment of double categories, concerning categories of monoids and modules therein.