In this paper we give an isomorphic description of the category of non-Archimedian approach spaces as a category of lax algebras for the ultrafilter monad and an appropriate quantale. Non-Archimedean approach spaces are characterised as those approach spaces having a tower consisting of topologies. We study topological properties p, for p compactness and Hausdorff separation along with low-separation properties, regularity, normality and extremal disconnectedness and link these properties to the condition that all or some of the level topologies in the tower have p. A compactification technique is developed based on Shanin's method.
Keywords: Lax algebra, quantale, non-Archimedean approach space, quasi-ultrametric space, initially dense object, topological properties in $(\beta, P_\wedge$-Cat, compactification
2010 MSC: 18C15, 18C20, 54A05, 54B30, 54E99
Theory and Applications of Categories, Vol. 32, 2017, No. 41, pp 1454-1484.