Domains occur among spaces as strict algebras among lax

Abstract

Whereas Alan Day showed that thecontinuous latticesare the algebras of afilter monadonSet, we employ the theory oflax algebras(as developed by Barr, Pisani, Clementino, Hofmann, Tholen, Seal and others) to broaden this characterisation to a description of the wider class ofcontinuous dcposas algebras of a lax filter monad. Building on an axiomatisation of topological spaces through convergence as lax algebras of a lax extension of the filter monad to a category of relations, we show that those topological spaces whose associated lax algebra is in fact astrictalgebra are what M. Erné called theC-spaces. ThesoberC-spaces are precisely the continuous dcpos under the Scott topology, and we discuss how the possibly little-known C-spaces, which have been studied by B. Banaschewski, J. D. Lawson, R.-E. Hoffmann, M. Erné and G. Wilke, very directly capture an essential topological notion of approximation inherent in the continuous dcpos, and hence provide a natural topological concept of domain.