Lawvere's notion of completeness for quantale-enriched categories has been extended to the theory of lax algebras under the name of L-completeness. In this paper we introduce the corresponding morphism concept and examine its properties. We explore some important relativized topological concepts like separatedness, denseness, compactness and compactification with respect to L-complete morphisms. Moreover, we show that separated L-complete morphisms belong to a factorization system.
Keywords: Completeness, compactness, lax algebra, module, proper map, injectivity, fibrewise sober, factorization system
2010 MSC: 18A05, 18A20, 18A32, 18B30, 18C15, 18D20, 54A20, 54B30, 54C10
Theory and Applications of Categories, Vol. 27, 2013, No. 12, pp 242-262.