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 ...

We define a constructive topos to be a locally cartesian closed pretopos. The terminology is supported by the fact that constructive toposes enjoy a relationship with constructive set theory similar to the relationship between elementary toposes and (impredicative) intuitionistic set theory. This paper elaborates upon one aspect of the relationship between constructive toposes and constructive ...

Is there any cartesian-closed category of continuous domains that would be closed under Jones and Plotkin’s probabilistic powerdomain construction? This is a major open problem in the area of denotational semantics of probabilistic higher-order languages. We relax the question, and look for quasi-continuous dcpos instead. We introduce a natural class of such quasi-continuous dcpos, the !QRB-dom...

It was already known that the category of T 0 topological spaces is not itself cartesian closed, but can be embedded into the cartesian closed categories FIL of lter spaces and EQU of equilogical spaces where the latter embeds into the cartesian closed category ASSM of assemblies over algebraic lattices. Here, we rst clarify the notion of lter space|there are at least three versions FIL a FIL b...

One natural way to generalize Domain Theory is to replace partially ordered sets by categories. This kind of generalization has recently found application in the study of concurrency. An outline is given of the elegant mathematical foundations which have been developed. This is specialized to give a construction of cartesian closed categories of domains, which throws light on standard presentat...