Kan injectivity in order-enriched categories

Abstract

Continuous lattices were characterised by Martín Escardó as precisely those objects that are Kan-injective with respect to a certain class of morphisms. In this paper we study Kan-injectivity in general categories enriched in posets. As an example, ω-CPO's are precisely the posets that are Kan-injective with respect to the embeddings ω ↪ ω + 1 and 0 ↪ 1. For every class H of morphisms, we study the subcategory of all objects that are Kan-injective with respect to H and all morphisms preserving Kan extensions. For categories such as Top_0 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, and we present a class of continuous mappings in Top_0 for which Kan-injectivity does not yield a monadic category.