Topological Features of Lax Algebras

Abstract

Having as starting point Barr's description of topological spaces as lax algebras for the ultrafilter monad, in this paper we present further topological examples of lax algebras – such as quasi-metric spaces, approach spaces and quasi-uniform spaces – and show that, in a suitable setting, the categories of lax algebras have indeed a topological nature. Furthermore, we generalize to this setting known properties of special categories of lax algebras and, extending the construction of Manes, we describe the Cech–Stone compactification of lax algebras.