Categories and types for axiomatic domain theory

Domain Theory provides a denotational semantics for programming languages and calculi containing fixed point combinators and other so-called paradoxical combinators. This dissertation presents results in the category theory and type theory of Axiomatic Domain Theory. Prompted by the adjunctions of Domain Theory, we extend Benton’s linear/nonlinear dualsequent calculus to include recursive linear types and define a class of models by adding Freyd’s notion of algebraic compactness to the monoidal adjunctions that model Benton’s calculus. We observe that algebraic compactness is better behaved in the context of categories with structural actions than in the usual context of enriched categories. We establish a theory of structural algebraic compactness that allows us to describe our models without reference to enrichment. We develop a 2-categorical perspective on structural actions, including a presentation of monoidal categories that leads directly to Kelly’s reduced coherence conditions. We observe that Benton’s adjoint type constructors can be treated individually, semantically as well as syntactically, using free representations of distributors. We type various of fixed point combinators using recursive types and function types, which we consider the core types of such calculi, together with the adjoint types. We use the idioms of these typings, which include oblique function spaces, to give a translation of the core of Levy’s Call-By-Push-Value. The translation induces call-by-value and call-by-name translations of the core of Plotkin’s Fixed Point Calculus. Following Freyd, we construct a canonical fixed point operation from the algebras provided by the algebraic compactness of our models. Our analysis of Freyd’s construction exposes a remarkable property of morphisms from coalgebras to algebras: morphisms from Gp to s correspond one-for-one to morphisms from p to Hs, where p is a coalgebra for HG and s an algebra for GH. We give an application of this property to the transposition of recursive coalgebras in Taylor’s categorical theory of recursion where G is not left adjoint to H. We develop a theory of parametric transformations corresponding to the uniformity property characterizing canonical fixed points and use this to derive abstract conditions on categories of domains which ensure that the interpretation of fixed point combinators coincides with the canonical fixed point operation. Submitted for the degree of Doctor of Philosophy University of London 2003