A conjecture of Smyth is discussed which says that if D and [D → D] are effectively algebraic directed-complete partial orders with least element (cpo’s), then D is an effectively strongly algebraic cpo, where it was not made precise what is meant by an effectively algebraic and an effectively strongly algebraic cpo. Notions of an effectively strongly algebraic cpo and an effective SFP domain a...

Inspired by a construction of Escardó, Lawson, and Simpson, we give a general construction of C-generated objects in a topological construct. When C consists of exponentiable objects, the resulting category is Cartesian-closed. This generalizes the familiar construction of compactly-generated spaces, and we apply this to Krishnan’s categories of streams and prestreams, as well as to Haucourt st...

We prove that every finitary polynomial endofunctor of a category C has a final coalgebra if C is locally Cartesian closed, has finite disjoint coproducts and a natural number object. More generally, we prove that the category of coalgebras for such an endofunctor has all finite limits.

Suppose in an arithmetic unverse we have two predicates φ and ψ for natural numbers, satisfying a base case φ(0) → ψ(0) and an induction step that, for generic n, the hypothesis φ(n) → ψ(n) allows one to deduce φ(n+ 1) → ψ(n+ 1). Then it is already true in that arithmetic universe that (∀n)(φ(n) → ψ(n)). This is substantially harder than in a topos, where cartesian closedness allows one to form...

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. ...