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...

In categorical semantics, there have traditionally been two approaches to modelling environments, one by use of finite products in cartesian closed categories, the other by use of the base categories of indexed categories with structure. Each requires modifications in order to account for environments in call-by-value programming languages. There have been two more general definitions along bot...

However, the replete construction is of a very general nature and most of its properties are independent of the axioms for Σ and of E being a topos. We consider the replete construction in the more general setting of a cartesian closed B-fibration p: C → B, where B is a category with finite products. This encompasses the case of a cartesian closed category (i.e. p: C → 1) and that of a locally ...

There are two main approaches to obtaining “topological” cartesian-closed categories. Under one approach, one restricts to a full subcategory of topological spaces that happens to be cartesian closed — for example, the category of sequential spaces. Under the other, one generalises the notion of space — for example, to Scott’s notion of equilogical space. In this paper we show that the two appr...

In the context of semi-abelian categories, several additional conditions have been considered in order to obtain a closer group-like behavior. Among them are locally algebraic cartesian closedness and algebraic coherence. The recent notion of S-protomodular categories, whose main examples are the category of monoids and, more generally, categories of monoids with operations and Jónsson-Tarski v...

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 may be also useful in infinite dimensional Differential Geometry, e.g. to study space...

Seely’s paper Locally cartesian closed categories and type theory contains a well-known result in categorical type theory: that the category of locally cartesian closed categories is equivalent to the category of Martin-Löf type theories with Π,Σ, and extensional identity types. However, Seely’s proof relies on the problematic assumption that substitution in types can be interpreted by pullback...

In [12] it is shown that the probabilistic powerdomain of a continuous domain is again continuous. The category of continuous domains, however, is not cartesian closed, and one has to look at subcategories such as RB, the retracts of bifinite domains. [8] offers a proof that the probabilistic powerdomain construction can be restricted to RB. In this paper, we give a counterexample to Graham’s p...

We introduce the notion of quasi-zero-dimensionality as a substitute for the notion of zero-dimensionality, motivated by the fact that the latter behaves badly in the realm of qcb-spaces. We prove that the category QZ of quasi-zero-dimensional qcb0-spaces is cartesian closed. Prominent examples of spaces in QZ are the spaces in the sequential hierarchy of the Kleene-Kreisel continuous functiona...

Quasicontinuity is a generalisation of Scott’s notion of continuous domain, introduced in the early 80s by Gierz, Lawson and Stralka. In this paper we ask which cartesian closed full subcategories exist in qCONT, the category of all quasicontinuous domains and Scottcontinuous functions. The surprising, and perhaps disappointing, answer turns out to be that all such subcategories consist entirel...