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.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2019

Author dc:creator

• Ghiorzi, Enrico

Advisor dc:contributor.advisor

• Hyland, J. Martin E.

• category theory
• internal categories
• enriched categories
• effective topos
• modest sets

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/bb44c613-d300-4529-9421-d5df6fc2f50d/download
• https://creativecommons.org/licenses/by/4.0/

Language dc:language

en

Author Identifier

0000-0001-5983-6230

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/298234