Kan injectivity in order-enriched categories

Continuous lattices were characterised by Mart́ın Escardó as precisely the objects that are Kan-injective w.r.t. a certain class of morphisms. We study Kan-injectivity in general categories enriched in posets. An example: ω-CPO’s are precisely the posets that are Kan-injective w.r.t. the embeddings ω →֒ ω + 1 and 0 →֒ 1. For every class H of morphisms we study the subcategory of all objects Kan-injective w.r.t. H and all morphisms preserving Kan-extensions. For categories such as Top0 and Pos we prove that whenever H is a set of morphisms, the above subcategory is monadic, and the monad it creates is a Kock-Zöberlein monad. However, this does not generalise to proper classes: we present a class of continuous mappings in Top0 for which Kan-injectivity does not yield a monadic category. Dedicated to the memory of Daniel M. Kan (1927–2013)