This paper is concerned with models of SDT encompassing traditional categories of domains used in denotational semantics [7,18], showing that the synthetic approach generalises the standard theory of domains and suggests new problems to it. Consider a (locally small) category of domains D with a (small) dense generator G equipped with a Grothendieck topology. Assume further that every cover in G is effective epimorphic in D. Then, by Yoneda, D embeds fully and faithfully in the topos of sheaves on G for the canonical topology, which thus provides a set-theoretic universe for our original category of domains. In this paper we explore such a situation for two traditional categories of domains and, in particular, show that the Grothendieck toposes so arising yield models of SDT. In a subsequent paper we will investigate intrinsic characterizations, within our models, of these categories of domains. First, we present a model of SDT embedding the category !-Cpo of posets with least upper bounds of countable chains (hence called !-complete) and

Solid theoretical foundations of object oriented databases (OODBs) are still missing. The work reported in this paper contains results on a formally founded object oriented datamodel (OODM) and is intended to contribute to the development of a uniform mathematical theory of OODBs. A clear distinction between objects and values turns out to be essential in the OODM. Types and classes are used to structure values and objects repectively. This can be founded on top of any underlying type system. We outline different approaches to type systems and their semantics and claim that OODB theory on top of arbitrary type systems leads to type theory with topos-theoretically defined semantics. On this basis the known solutions to the problems of unique object identification and genericity can be generalized. It turns out that extents of classes must be completely representable by values. Such classes are called value-representable. As a consequence object identifiers degenerate to a pure...

Conceptual modelling requires a solid mathematical theory of concepts concerning the collection of concepts used in a specific, but broad enough field. The field considered in this paper is database modelling. Here object orientation in the widest sense has been identified as a unifying conceptual umbrella that encompasses all relevant datamodels. The theory of object oriented databases has brought to light the fundamental distinction between the concepts of objects and values and correspondingly types and classes. This can be founded on top of any underlying type system. Thus, expressiveness of a datamodel basically depends on the type concept, from which the other concepts can be derived. In order to achieve a uniform mathematical theory we outline different type systems and their semantics and claim that OODB theory on top of arbitrary type systems leads to type theory with topos-theoretically defined semantics. On this basis the known solutions to the problems of unique ...

Introduction In [2] Martin Hyland has proposed the notion of "S-replete object" relative to a given object S as the appropriate generalisation of predomain for the purposes of Synthetic Domain Theory (SDT). The aim of SDT is to provide a constructive logical framework for reasoning about domains and thus about meanings of functional programs where all functions between predomains are continuous and all endofunctions on domains have least fixpoints. Furthermore, SDT allows one to consider predomains as a full reflective subcategory of the ambient category of constructive sets. For this reason the ambient category is assumed to be a model of some sufficiently strong impredicative constructive type theory which will be used as the internal language for the ambient category of constructive sets in order to replace complicated external arguments by simpler proofs in the

We give various internal descriptions of the category !-Cpo of !-complete posets and !-continuous functions in the model H of Synthetic Domain Theory introduced in [8]. It follows that the !-cpos lie between the two extreme synthetic notions of domain given by repleteness and well-completeness. Introduction Synthetic Domain Theory aims at giving a few simple axioms to be added to an intuitionistic set theory in order to obtain domain-like sets. The idea at the core of this study was proposed by Dana Scott in the late 70's: domains should be certain "sets" in a mathematical universe where domain theory would be available. In particular, domains would come with intrinsic notions of approximation and passage to the limit with respect to which all functions will be continuous. Various suggestions for the notion of domain (typically within a set-theoretic universe given by an elementary topos with natural numbers object [17]) appeared in the literature, e.g. in [11, 26, 10, 23, 20, 16]. A...

The object of study of the present paper may be considered as a model, in an elementary topos with a natural numbers object, of a non-classical variation of the Peano arithmetic. The new feature consists in admitting, in addition to the constant (zero) s0 2 N and the unary operation (the successor map) s1 : N ! N, arbitrary operations su : N u ! N of arities u `between 0 and 1'. That is, u is allowed to range over subsets of a singleton set.

We provide an internal characterization of the category!-Cpo of!-complete posets and!-continuous functions within the model H of SDT recently introduced by the authors. It follows that!-cpos lie between the two extreme synthetic notions of domain given by repleteness and well-completeness.