Umberto Tarantino: 2-dimensional topologies via profunctorial algebras

Abstract: In his 1970 paper, Barr described how to extend a monad on Set to a skew monad acting on relations, and characterized in terms of weak pullbacks when the extension is itself a monad. As a key application, he then showed how topological spaces can be recovered as lax algebras for the relational extension of the ultrafilter monad β, thus lifting Manes’ characterization of compact Hausdorff spaces as β-algebras to the relational setting. In this talk, I will describe a bicategorical version of both results. First, identified the role of relations in an arbitrary bicategory with two-sided codiscrete cofibrations, I will present an extension result for pseudomonads on a bicategory to skew monads on its bicategory of two-sided codiscrete cofibrations, characterizing in terms of exact squares when the extension is itself a pseudomonad. Then, focusing on the case of CAT, I will show how ultraconvergence spaces — a 2-dimensional analogue of topological spaces — can be recovered as suitable lax algebras for the skew extension of the ultracompletion pseudomonad, whose (ordinary) algebras are ultracategories. This talk is based on joint work with Quentin Aristote.