Ultracategories

The notion of ultracategory was introduced by Michael Makkai in a paper in APAL in 1990 for the characterisation of categories of models of pretoposes, an ample extension to (intuitionistic) first order theories of Stone duality for Boolean algebras, providing a kind of Stone duality for first order theories -- aka conceptual completeness. Recently, Jacob Lurie refined that notion in unpublished notes producing another approach to the duality for pretoposes -- the two notions of ultracategory appear to be different, though no separating example has been produced yet. In the talk, we shall give intuitions about Makkai's and Lurie's notions, providing examples and applications. Then we shall introduce an algebraic notion of structured category which subsumes the two kinds of ultracategories mentioned above -- technically, the "ultracompletion" 2-functor on the 2-category of small categories, and extend it to a pseudomonad. Next we show how it relates to the two existing notions. This is joint work with Richard Garner.