Topological Features of Lax Algebras

Having as starting point Barr’s description of topological spaces as lax algebras for the ultrafilter monad [2], 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 [12], we describe the Čech-Stone compactification of lax algebras. Mathematics Subject Classification: 18C20, 18A40, 18B30, 54B30, 54A20, 54E15.