Profunctorial algebras
This provides a bicategorical framework for extending monads to profunctors, with applications in categorifying topological spaces, but it is incremental as it builds directly on Barr's foundational work.
The paper generalizes Barr's 1970 work by extending pseudomonads in bicategories to skew monads on two-sided discrete fibrations, characterizing when these become pseudomonads via exact squares, and applies this to show that Set-monads induce pseudomonads on categories with profunctor extensions, leading to a characterization of ultraconvergence spaces as normalized lax algebras of the ultracompletion pseudomonad.
We provide a bicategorical generalization of Barr's landmark 1970 paper, in which he describes how to extend Set-monads to relations and uses this to characterize topological spaces as the relational algebras of the ultrafilter monad. With two-sided discrete fibrations playing the role of relations in a bicategory, we first describe how to extend pseudomonads on a bicategory to skew monads on its bicategory of two-sided discrete fibrations, and we characterize in terms of exact squares when these extensions are themselves pseudomonads. As a wide class of examples, we show that every Set-monad induces a pseudomonad on the 2-category of categories admitting a skew extension to profunctors, and in a few relevant cases we introduce suitable quotients also extending to profunctors. Among the latter, we then focus on the ultracompletion pseudomonad, whose pseudoalgebras are ultracategories: we characterize the normalized lax algebras of its profunctorial extension as ultraconvergence spaces, a recently-introduced categorification of topological spaces.