We give a systematic category theoretic axiomatics for modelling data refinement in call by value programming languages. Our leading examples of call by value languages are extensions of the computational -calculus, such as FPC and languages for modelling nondeterminism, and extensions of the first order fragment of the computational -calculus, such as a CPS language. We give a category theoretic account of the basic setting, then show how to model contexts, then arbitrary type and term constructors, then signatures, and finally data refinement. This extends and clarifies Kinoshita and Power's work on lax logical relations for call by value languages.

We introduce -algebras and -algebras as semantic domains for data refinement of imperative programming languages. The functorial semantics of -calculus is given by using the adjunction between the category of -algebras and the category of small locally ordered categories. We define the notion of upward and downward simulation between the interpretations of atomic commands, and refinements between the interpretations of commands, by using lax or oplax transformations. We show that the adjunction extends to an enriched adjunction in the sense of lax, and which provide the fundamental machinery in the proof of soundness and completeness of the downward simulation with respect to the refinement.