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Join this seminar talk where Giuseppe Rosolini from Università degli Studi di Genova explores the concept of "Ultracompletions" as part of the Orange County Inland Empire (OCIE) Seminar series in History and Philosophy of Mathematics at Chapman University. Discover the evolution of ultracategory notions, from Michael Makkai's original introduction in 1990 for characterizing categories of models of pretoposes, to Jacob Lurie's recent refinements. Learn about the intuitions behind both Makkai's and Lurie's approaches, with examples and applications provided. The presentation introduces an algebraic notion of structured category that encompasses both types of ultracategories, technically described as the "ultracompletion" 2-functor on the 2-category of small categories, extended to a pseudomonad. Understand how this concept relates to existing notions, incorporating recent results by Ali Hamad. This joint work with Richard Garner offers valuable insights into category theory and its applications to mathematical logic.
