Day [75] showed that the category of continuous lattices and maps which preserve directed joins and arbitrary meets is the category of algebras for a monad over Set, Sp or Pos, the free functor being the set of filters of open sets. Separately, Berry [78] constructed a cartesian closed category whose morphisms preserve directed joins and connected meets, whilst Diers [79] considered similar fun...

Infinite contexts and their corresponding lattices are of theoretical and practical interest since they may offer connections with and insights from other mathematical structures which are normally not restricted to the finite cases. In this paper we establish a systematic connection between formal concept analysis and domain theory as a categorical equivalence, enriching the link between the t...

The aim of this paper is to establish some Cartesian closed categories which are between the two Cartesian closed categories: SLP (the category of L-domains and stable functions) and DI (the full subcategory of SLP whose objects are all dI-domains). First we show that the exponentials of every full subcategory of SLP are exactly the spaces of stable functions. Then we prove that the full subcat...

From the analogue of Böhm’s Theorem proved for the typed lambda calculus, without product types and with them, it is inferred that every cartesian closed category that satisfies an equality between arrows not satisfied in free cartesian closed categories must be a preorder. A new proof is given here of these results, which were obtained previously by Richard Statman and Alex K. Simpson. Mathema...

The inspiration for this paper is a result proved by Michael Smyth which states that Gordon Plotkin's category SFP is the largest cartesian closed category of domains. Although this category is easily enough motivated from concepts in domain theory and category theory, it is clearly harder to describe and less \elementary" than the most popular categories of domains for denotational semantics. ...

Berry's category of dI-domains with stable functions is a relatively intricate , yet elegant framework for semantics of programming languages. Despite over fteen years of work in the area, the exact reasons for dis-tributivity (Axiom d) and nitariness (Axiom I) have not been fully ex-plicated. This paper shows that Axiom d and Axiom I are important when one works within the realm of Scott-domai...

We show that a version of Martin-Löf type theory with extensional identity, a unit type N1,Σ,Π, and a base type is a free category with families (supporting these type formers) both in a 1and a 2-categorical sense. It follows that the underlying category of contexts is a free locally cartesian closed category in a 2-categorical sense because of a previously proved biequivalence. We then show th...

It is well known that one can build models of full higher-order dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relations) can be applied to other structures as well. In particular, we can easily define the category o...