Ultracategories as colax algebras for a pseudo-monad on CAT

Abstract

We show a result inspired by a conjecture by Shulman claiming that ultracategories as defined by Lurie are normal colax algebras for a certain pseudo-monad on the category of categories CAT. Such definition allows us to regard left and right ultrafunctors as defined by Lurie as instances of lax/colax algebras morphisms.

Version 4. Fixed typos and inconsistencies, clarified proofs and diagrams