Internal enriched categories
This dissertation introduces and develops the theory of internal enriched categories, arising from the internalization of the theory of enriched categories. Given an internal monoidal category V in an ambient category E, we define the notions of V-enriched category, functor and natural transformation. We then develop such theory, which presents many of the good properties of standard enriched category theory. Notably, under suitable conditions, the category of internal V-enriched categories and their functors is monoidal closed. Internal enriched categories admit a notion of internal weighted limit, analo- gously to how internal categories admit internal limits. Such theory of limits constitutes a major focus point in the dissertation and yields fundamental results such as the adjoint functor theorem. It is observed that internal categories are intrinsically small and some of them are non-trivial examples of small complete categories, whereas the only standard small complete categories are complete lattices. As a consequence, the internal theory is better behaved than that of standard categories, particularly in relation with size issues, while still featuring interesting examples. Moreover, to frame it into a wider context, the notion of internal enriched category is compared with related notions from the literature, such as those of indexed enriched category and enriched generalized multicategory. It turns out that internal enriched categories are indeed strongly connected with such other notions, thus providing a novel approach to–and, possibly, insight into–other topics in category theory.