Sober topological spaces valued in a quantale

The notion of sobriety is extended to the realm of topological spaces valued in a commutative and unital quantale, via an adjunction between a category of quantale modules and the category of quantale-valued topological spaces. Relations between such sober spaces and quantitative domains based on flat ideals are investigated.Comment: 22 pages. arXiv admin note: text overlap with arXiv:2102.0293