Profunctorial algebras

TL;DR

This paper generalizes Barr's 1970 work using bicategories, extending Set-monads to relations and characterizing ultracategories as algebras of an ultracompletion pseudomonad.

Contribution

It introduces a bicategorical framework for extending pseudomonads to profunctors and characterizes ultracategories as ultraconvergence spaces.

Findings

Every Set-monad induces a pseudomonad on categories with skew extensions to profunctors.

Ultracompletion pseudomonad's pseudoalgebras are ultracategories.

Normalized lax algebras of the profunctorial extension are ultraconvergence spaces.

Abstract

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.