It is well known that one can build models of full higher-order dependent type theory (also called the calculus of constructions) using partial equivalence relations (PERs) and assemblies over a partial combinatory algebra (PCA). But the idea of categories of PERs and ERs (total equivalence relations) can be applied to other structures as well. In particular, we can easily dene the category of ERs and equivalencepreserving continuous mappings over the standard category Top 0 of topological T 0 -spaces; we call these spaces (a topological space together with an ER) equilogical spaces and the resulting category Equ. We show that this category|in contradistinction to Top 0 |is a cartesian closed category. The direct proof outlined here uses the equivalence of the category Equ to the category PEqu of PERs over algebraic lattices (a full subcategory of Top 0 that is well known to be cartesian closed from domain theory). In another paper with Carboni and Rosolini (cited herein) a more abstract categorical generalization shows why many such categories are cartesian closed. The category Equ obviously contains Top 0 as a full subcategory, and it naturally contains many other well known subcategories. In particular, we show why, as a consequence of work of Ershov, Berger, and others, the Kleene-Kreisel hierarchy of countable functionals of nite types can be naturally constructed in Equ from the natural numbers object N by repeated use in Equ of exponentiation and binary products. We also develop for Equ notions of modest sets (a category equivalent to Equ) and assemblies to explain why a model of dependent type theory is obtained. We make some comparisons of this model to other, known models. 1

) Steven Awodey 1 Lars Birkedal 2y Dana S. Scott 2z 1 Department of Philosophy, Carnegie Mellon University 2 School of Computer Science, Carnegie Mellon University April 15, 1999 Abstract This work is a step toward developing a logic for types and computation that includes both the usual spaces of mathematics and constructions and spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes, which we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we study first axiomatically, and then by deriving a modal calculus as its internal logic. The resulting framework is intended as a setting for the logical and categorical study of relative computability. 1 Introduction We report here on the current status of research on the Logic of Types and Computation at Carnegie Mellon University [SAB + ]. The general goal of this research program is to develop a logical fra...

. We show that dependent sums and dependent products of continuous parametrizations on domains with dense, codense, and natural totalities agree with dependent sums and dependent products in equilogical spaces, and thus also in the realizability topos RT(P!). Keywords: continuous functionals, dependent type theory, domain theory, equilogical spaces. 1 Introduction Recently there has been a lot of interest in understanding notions of totality for domains [3, 23, 4, 18, 21]. There are several reasons for this. Totality is the semantic analogue of termination, and one is naturally interested in understanding not only termination properties of programs but also how notions of program equivalence depend on assumptions regarding termination [21]. Another reason for studying totality on domains is to obtain generalizations of the nite-type hierarchy of total continuous functionals by Kleene and Kreisel [11], see [8] and [19] for good accounts of this subject. Ershov [7] showed how the Klee...

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

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this paper) and it was described by means of tripos theory right from the beginnings of that theory, see, e.g., [17, Section 1.5, item (ii)]. Recently there has been a renewed interest in Relative Realizability, both in Thomas Streicher's "Topos for Computable Analysis" [18] and in [2, 1, 4]. The idea is, that instead of doing realizability with one partial combinatory algebra A one uses an inclusion of partial combinatory algebras A ] ` A (such that there are combinators k; s 2 A ] which also serve as combinators for A), the principal point being that "(A ] -) computable" functions may also act on data (in A) that need not be computable

This is the third in a series of papers on algebraic set theory, the aim of which is to develop a categorical semantics for constructive set theories, including predicative ones, based on the notion of a “predicative category with small maps”. 1 In the first paper in this series [8] we discussed how these predicative categories

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We compare realizability models over partial combinatory algebras by embedding them into sheaf toposes. We then use the machinery of Grothendieck toposes and geometric morphisms to study the relationship between realizability models over di#erent partial combinatory algebras. This research is part of the Logic of Types and Computation project at Carnegie Mellon University under the direction of Dana Scott. 1

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