Lax naturality through enrichment

Abstract

We develop the relationship between algebraic structure and monads enriched over the monoidal bicloscd category LocOrd of small locally ordered categories, with closed structure given by Lax( A, B). We state the theorem, give a series of examples, and incorporate an account of sketches and contravariance into the theory. This was motivated by C.A.R. Hoare's use of category theoretic structures to model data refinement.