This article 1 has two parts: In the rst part, we present some general results about xpoint objects. The minimal categorical structure required to model soundly the equational type theory which combines higher order recursion and computation types (introduced by 4]) is shown to be precisely a let-category possessing a xpoint object. Functional completeness for such categories is developed. We a...

We present a model construction for the Calculus of Constructions (CC) where all dependencies are carried out in a set-theoretical setting. The Soundness Theorem is proved and as a consequence of it Strong Normalization for CC is obtained. Some other applications of our model constructions are: showing that CC 4Classical logic is consistent (by constructing a model for it} and showing that the ...

We give a new characterisation of morphisms that are definable by the interpretation of the simply typed lambda calculus with sums in any bi-Cartesian closed category. The ⊤⊤-closure operator will be used to construct the category in which the collection of definable morphisms at sum types can be characterised as the coproducts of such collections at lower types.

We build a cartesian closed category, called Cho, based on event structures. It allows an interpretation of higher-order stateful concurrent programs that is refined and precise: on the one hand it is conservative with respect to standard Hyland-Ong games when interpreting purely functional programs as innocent strategies, while on the other hand it is much more expressive. The interpretation o...

We present a variant of the calculus of deductive systems developed in [5, 6], and give a generalization of the Curry-Howard-Lambek theorem giving an equivalence between the category of typed lambda-calculi and the category of cartesian closed categories and exponentialpreserving morphisms that leverages the theory of generalized categories [13]. We discuss potential applications and extensions.

It is shown how many techniques of categorical domain theory can be expressed in the general context of topical categories (where \topical" means internal in the category Top of Grothendieck toposes with geometric morphisms). The underlying topos machinery is hidden by using a geometric form of constructive mathematics, which enables toposes as \generalized topological spaces" to be treated in ...

We study exponentiability of homomorphisms in varieties of universal algebras close to classical ones. After describing an “almost folklore” general result, we present a purely algebraic proof of “étale implies exponentiable”, alternative to the topologically motivated proof given in one of our previous papers, in a different context. We prove that only isomorphisms are exponentiable homomorphi...

We consider a symmetric monoidal closed category V = (V ,⊗, I, [−,−]) together with a regular injective object Q such that the functor [−, Q] : V → V op is comonadic and prove that in such a category, as in the monoidal category of abelian groups, a morphism of commutative monoids is an effective descent morphism for modules if and only if it is a pure monomorphism. Examples of this kind of mon...

This paper presents an attempt to characterize synchronous stream functions within the framework of co-iteration and to use this characterization in building a compiler for (higher order and recursive) synchronous dataow programs. First length-preserving functions are considered and we show that streams equipped with such functions form a Cartesian-closed category. Then this point of view is ex...

This is the second in a series of papers detailing the author’s investigations into the intensional type theory of Martin-Löf, as described in Nordström et al. (1990). The first of these papers, Garner (2009), investigated syntactic issues relating to its dependent product types. The present paper is a contribution to its categorical semantics. Seely (1984) proposed that the correct categorical...