Sums and covers in categories

A cover of an object X can be seen from two points of view. In the one point of view, it is a suitable sink with codomain X. On the other hand, we can see a cover of X as a collection of parts which add up to make X. These two perspectives come together in the structure of a coproduct. This notion of a cover generalises to a sum structure in the sense of Zurab Janelidze. We study covers and sums at this level, abstracting a number of results and constructions known for coproducts. We examine the dual notion of a product structure and show that these coincide with independence structures of Alex Simpson. We show that morphisms from sum structures to product structures exhibit matrix representations. We study extensivity for sum structures, and focus on extensivity relative to cover-reflecting morphisms. We introduce and explore a new class of morphisms, called monilmorphisms, and use them to construct examples of sum structures exhibiting precisely the said form relative extensivity. We conclude by departing from covers determined by sums, and study Grothendieck topologies as special kinds of functors called forms.

Grantor dc:publisher

Stellenbosch : Stellenbosch University

Year dc:date.issued

2026

Author dc:creator

• Ferguson, Roy Angus

Advisors dc:contributor.advisor

• Janelidze, Zurab
• Hoefnagel, Michael

Language dc:language.iso

en

Repository record dc:identifier.uri

https://scholar.sun.ac.za/handle/10019.1/135984

OAI identifier oai:identifier

oai:scholar.sun.ac.za:10019.1/135984