Domain theory for denotational semantics is over thirty years old. There are many variations on the idea and many interesting constructs that have been proposed by many people for realizing a wide variety of types as domains. Generally, the effort has been to create categories of domains that are cartesian closed (that is, have products and function spaces interpreting typed calculus) and permi...

ComK may be defined as the (cartesian closed) category of comonoids in chuK , or equivalently as dictionaries D for which any crossword over D has its main diagonal in D. Com2 resembles Top, ordinary topological spaces. Common to both are the Alexandroff posets and the Scott DCPOs, while the topological space R and the dual DCPO {−∞ < . . . < −2 < −1 < 0} jointly witness the incomparability of ...

Various situations in computer science call for categories that support both cartesian closed and monoidal closed structure. Such situations include (i) models of local state, where the monoidal product describes disjointness of memory, and (ii) treatment of fresh names, as required in models of the π-calculus. I propose a technique to embed the two closed structures into one single structure. ...

When Gödel developed his functional interpretation, also known as the Dialectica interpretation, his aim was to prove (relative) consistency of first order arithmetic by reducing it to a quantifier-free theory with finite types. Like other functional interpretations (e.g. Kleene’s realizability interpretation and Kreisel’s modified realizability) Gödel’s Dialectica interpretation gives rise to ...

Jacobs, B., Comprehension categories and the semantics of type dependency, Theoretical Computer Science 107 (1993) 169-207. A comprehension category is defined as a functor 8: E-+B’ satisfying (a) cod 0 9 is a fibration, and (b) fis Cartesian in E implies that qfis a pullback in B. This notion captures many structures which are used to describe type dependency (like display-map categories (Tayl...

We classify all sub-cartesian closed categories of the category of separable Scott domains. The classification employs a notion of coherence degree determined by the possible inconsistency patterns of sets of finite elements of a domain. Using the classification, we determine all sub-cartesian closed categories of the category of separable Scott domains that contain a universal object. The sepa...

Simpson, A.K., A characterisation of the least-fixed-point operator by dinaturality, Theoretical Computer Science 118 (1993) 301-314. The paper addresses the question of when the least-fixed-point operator, in a Cartesian-closed category of domains, is characterised as the unique dinatural transformation from the exponentiation bifunctor to the identity functor. We give a sufficient condition o...

Exercises 1.2 (a) Composition in RC is well-defined by representatives. (b) A morphism of RC is invertible ⇐⇒ it is a class of strong equivalences ⇐⇒ its counit is an isomorphism. (c) Objects of RC are isomorphic iff they are equivalent categories. (d) RC/T ' Copt(T ) (?). Rigid comparisons embody an important idea from domain theory: approximation. This must have the property that if X ′ appro...