In this paper I compare two well studied approaches to topological semantics— the domain-theoretic approach, exemplified by the category of countably based equilogical spaces, Equ, and Type Two Effectivity, exemplified by the category of Baire space representations, Rep(B). These two categories are both locally cartesian closed extensions of countably based T0-spaces. A natural question to ask ...

It is well known that to be able to represent continuous functions between domain representable spaces it is critical that the domain representations of the spaces we consider are dense. In this article we show how to develop a representation theory over a category of domains with morphisms partial continuous functions. The raison d’être for introducing partial continuous functions is that by p...

We introduce the new category VRel(H) consisting of H-fuzzy relation spaces and H-fuzzy mappings between them satisfying a certain condition, where the concept of H-fuzzy mapping is the modification of one of fuzzy mapping introduced by Demirci[6]. And we investigate VRel(H) in the sense of a topological universe and show that VRel(H) is Cartesian closed over Set. Moreover, we construct the cat...

We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed categories.

We conservatively extend classical elementary differential calculus to the Cartesian closed category of convergence spaces. By specializing results about the convergence space representation of directed graphs, we use Cayley graphs to obtain a differential calculus on groups, from which we then extract a Boolean differential calculus, in which both linearity and the product rule, also called th...

We recently introduced an extensional model of the pure λcalculus living in a cartesian closed category of sets and relations. In this paper, we provide sufficient conditions for categorical models living in arbitrary cpo-enriched cartesian closed categories to have H∗, the maximal consistent sensible λ-theory, as their equational theory. Finally, we prove that our relational model fulfils thes...

Along the lines of classical categorical type theory for total functions, we establish correspondence results between certain classes of partial equational theories on the one hand and suitable classes of categories having certain finite limits on the other hand. E.g., we show that finitary partial theories with existentially conditioned equations are essentially the same as cartesian categorie...

In [8] and [9] Moisil has introduced the resemblance relations. Following [9] we associate to every resemblance relation an extensive operator which commutes with arbitrary unions of sets. We are leading to consider spaces endowed with such closure operators; we shall call these spaces total tech spaces (TC-spaces). TC-spaces are in one-to-one, onto correspondence with reflexive relations. TC-s...

We define a faithful functor from a cartesian closed category of linearly topologized vector spaces over a field and generalized polynomial functions to the category of “extensional” presheaves over the Lawvere theory of polynomial functions, and show that, under some conditions on the field, this functor is full and preserves the cartesian closed structure.

We present a cartesian closed category of dI-domains with coherence and strongly stable functions which provides a new model of PCF, where terms are interpreted by functions and where, at rst order, all functions are sequential. We show how this model can be reened in such a way that the theory it induces on the terms of PCF be strictly ner than the theory induced by the Scott model of continuo...