First, we briefly recall the main definitions of the theory of Information Bases and Translations. These mathematical structures are the basis to construct the cartesian closed category InfBas, which is equivalent to the category ScDom of Scott Domains. Then, we will show that all the definitions and the proof of all the properties that one needs in order to show that InfBas is indeed a cartesi...

Exercises 1.2 (a) Composition in RC is well-defined by representatives. (b) A morphism of RC is invertible ⇐⇒ it is a class of strong equivalences ⇐⇒ its counit is an isomorphism. (c) Objects of RC are isomorphic iff they are equivalent categories. (d) RC/T ' Copt(T ) (?). Rigid comparisons embody an important idea from domain theory: approximation. This must have the property that if X ′ appro...

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 give two classes of sound and complete models for the computational-calculus, or c-calculus. For the rst, we generalise the notion of cartesian closed category to that of closed Freyd-category. For the second, we generalise simple indexed categories. The former gives a direct semantics for the computational-calculus. The latter corresponds to an idealisation of stack-based intermediate langu...

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...