We give an axiomatic treatment of fixed-point operators in categories. A notion of iteration operator is defined, embodying the equational properties of iteration theories. We prove a general completeness theorem for iteration operators, relying on a new, purely syntactic characterisation of the free iteration theory. We then show how iteration operators arise in axiomatic domain theory. One result derives them from the existence of sufficiently many bifree algebras (exploiting the universal property Freyd introduced in his notion of algebraic compactness) . Another result shows that, in the presence of a parameterized natural numbers object and an equational lifting monad, any uniform fixed-point operator is necessarily an iteration operator. 1. Introduction Fixed points play a central role in domain theory. Traditionally, one works with a category such as Cppo, the category of !-continuous functions between !-complete pointed partial orders. This possesses a least-fixed-point oper...

Toposes and quasi-toposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding "good " quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the "best " regular category (called its regular completion) that embeds it. The second assigns to

We present a simple and workable axiomatization of domain theory within intuitionistic set theory, in which predomains are (special) sets, and domains are algebras for a simple equational theory. We use the axioms to construct a relationally parametric set-theoretic model for a compact but powerful polymorphic programming language, given by a novel extension of intuitionistic linear type theory based on strictness. By applying the model, we establish the fundamental operational properties of the language. 1.

We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ω-continuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, can be used as the basis for a theory of computability, and provides a model of parametric polymorphism.

. We place simple axioms on an elementary topos which suffice for it to provide a denotational model of call-by-value PCF with sum and product types. The model is synthetic in the sense that types are interpreted by their set-theoretic counterparts within the topos. The main result characterises when the model is computationally adequate with respect to the operational semantics of the programming language. We prove that computational adequacy holds if and only if the topos is 1-consistent (i.e. its internal logic validates only true \Sigma 0 1 -sentences). 1 Introduction One axiomatic approach to domain theory is based on axiomatizing properties of the category of predomains (in which objects need not have a "least" element). Typically, such a category is assumed to be bicartesian closed (although it is not really necessary to require all exponentials) with natural numbers object, allowing the denotations of simple datatypes to be determined by universal properties. It is well known...

This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domain-theoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic Zermelo-Fraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambda-calculus with callby-value operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal ” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external ” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1-consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domain-theoretic models.