We give an account of lax orthogonal factorisation systems on order-enriched categories. Among them, we define and characterise the KZ-reflective ones, in a way that mirrors the characterisation of reflective orthogonal factorisation systems. We use simple monads to construct lax orthogonal factorisation systems, such as one on the category of T_0 topological spaces closely related to continuous lattices.
Keywords: lax orthogonal factorization system, lax idempotent monad, order-enriched category, weak factorization system, reflective factorization system, continuous lattice
2020 MSC: Primary 18D05, 18A32. Secondary 55U35
Theory and Applications of Categories, Vol. 35, 2020, No. 36, pp 1379-1423.