Ultracompletions

Abstract

Michael Makkai introduced ultracategories for the characterisation theorem of categories of models of pretoposes, an extension to (intuitionistic) first order theories of Stone duality for Boolean algebras. Recently, Jacob Lurie refined that notion.
By means of a pseudomonad, we introduce a notion of structured category which subsumes the two kinds of ultracategories, and compare it with them, providing examples and applications. We show how it can be related to the two existing notions, using also results recently obtained by Ali Hamad.
This is joint work with Richard Garner.