In [12] it is shown that the probabilistic powerdomain of a continuous domain is again continuous. The category of continuous domains, however, is not cartesian closed, and one has to look at subcategories such as RB, the retracts of bifinite domains. [8] offers a proof that the probabilistic powerdomain construction can be restricted to RB. In this paper, we give a counterexample to Graham’s p...

We introduce the notion of quasi-zero-dimensionality as a substitute for the notion of zero-dimensionality, motivated by the fact that the latter behaves badly in the realm of qcb-spaces. We prove that the category QZ of quasi-zero-dimensional qcb0-spaces is cartesian closed. Prominent examples of spaces in QZ are the spaces in the sequential hierarchy of the Kleene-Kreisel continuous functiona...

Quasicontinuity is a generalisation of Scott’s notion of continuous domain, introduced in the early 80s by Gierz, Lawson and Stralka. In this paper we ask which cartesian closed full subcategories exist in qCONT, the category of all quasicontinuous domains and Scottcontinuous functions. The surprising, and perhaps disappointing, answer turns out to be that all such subcategories consist entirel...

Frölicher and Nijenhuis recognized well in the middle of the previous century that the Lie bracket and its Jacobi identity could and should exist beyond Lie algebras. Nevertheless, the conceptual status of their discovery has been obscured by the genuinely algebraic techniques they exploited. The principal objective in this paper is to show that the double dualization functor in a Cartesian clo...

Bistructures are a generalisation of event structures to represent spaces of functions at higher types; the partial order of causal dependency is replaced by two orders, one associated with input and the other output in the behaviour of functions. Bistructures form a categorical model of Girard’s classical linear logic in which the involution of linear logic is modelled, roughly speaking, by a ...

Let CD be the category of all continuous domains and all mappings which preserve directed sups and the way-below relation. That is, a mapping f : P → Q is a morphism of CD if and only if f(sup D) = sup f(D) for any directed set D ⊂ P and f(x) œ f(y) if x œ y for any x, y ∈ P . We shall prove that the category CD is cartesian closed. 1991 Mathematics Subject Classification 06B35.

Using standard analysis only, we present an extension •R of the real field containing nilpotent infinitesimals. On the one hand we want to present a very simple setting to formalize infinitesimal methods in Differential Geometry, Analysis and Physics. On the other hand we want to show that these infinitesimals are also useful in infinite dimensional Differential Geometry, e.g. to study spaces o...

Nominal techniques are based on the idea of sets with a finitelysupported atoms-permutation action. In this paper we consider the idea of sets with a finitely-supported atoms-renaming action (renamings can identify atoms; permutations cannot). We show that these exhibit many of the useful qualities found in traditional nominal techniques; an elementary sets-based presentation, inductive datatyp...

In analogy with the relation between closure operators in presheaf toposes and Grothendieck topologies, we identify the structure in a category with finite limits that corresponds to universal closure operators in its regular and exact completions. The study of separated objects in exact completions will then allow us to give conceptual proofs of local cartesian closure of different categories ...