this paper, we give an abstract 2 categorical characterization of W-types. We calculate these W-types explicitly in some categories of presheaves and sheaves on a site, and in the gluing category or Freyd cover. (We also have an explicit description in the case of Hyland's realizability topos, which will be presented in [17].) These explicit calculations can be formalized in a weak predicative metatheory, and lead to the result that if E is any suitably filtered pretopos with dependent products and W-types, then so is the category of internal sheaves on a site in E (Remark 5.9). Our paper is organized as follows. In Section 2 we review some standard definitions concerning pretoposes and dependent products. In Section 3 we present the categorical definition of the W-construction, and in Section 4 we prove some of its basic functoriality properties; e.g., that it turns coequalizers into equalizers. In Section 5, a construction is presented which to each map between (pre)sheaves of sets associates a sheaf of wellfounded trees, and it is proved that this is in fact the W-type in the category (pre)sheaves of sets (Theorem 5.6). In Section 6, we discuss the W-construction for the Freyd cover. Finally, in Section 7 it is shown how these categorical constructions are not only analogous to but explicitly related to Martin-Lof type theory. 2 Pretoposes and dependent products

The first paper in this thesis presents a machine checked formalisation, in Martin-Löf's type theory, of pointfree topology with applications to domain theory. In the other papers pointfree topology is used in an approach to constructive analysis. The continuum is defined as a formal space from a base of rational intervals. Then the closed rational interval [a, b] is defined as a formal space, in terms of the continuum, and the Heine-Borel covering theorem is proved constructively. The basic definitions for a pointfree approach to functional analysis are given in such a way that the linear functionals from a seminormed linear space to the reals are points of a particular formal space, and in this setting the Alaoglu and the Hahn-Banach theorems are proved in an entirely constructive way. The proofs have been carried out in intensional Martin-Löf type theory with one universe and finitary inductive definitions, and the proofs have also been mechanically checked in an implementation of that system. ...

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

Abstract. Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments.?1. Introduction. In 1963, Paul Cohen introduced the method of forcing to prove the independence of both the axiom of choice and the continuum hypothesis from Zermelo-Fraenkel set theory. It was not long before Saul Kripke noted a connection between forcing and his semantics for modal and

September, 2005 This paper studies categorical models of predicative formal systems, like Aczel’s CZF or Martin-Löf Type Theory. Following the suggestion of Moerdijk and Palmgren that the collection of such categorical models should have the closure properties of toposes, I study the stability under sheaves of five possible axiomatisations. The conclusion will be that all the notions of a “predicative topos ” that I consider, are stable under presheaves, while most are stable under sheaves. This opens up the possibility of using sheaf models for studying predicative theories. 1

. The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces fit together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possible to present the reducibility method using more semantic means, and that a deeper understanding would then be gained. This paper gives mathematical substance to this feeling, by presenting a generalization of the reducibility method based on a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). A key technical ingredient is the introduction a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene's recursive realizability and a variant of Kreisel's modified realizability both fit into this...

Abstract. We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.

Abstract. We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.

The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces t together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possible to present the reducibility method using more semantic means, and that a deeper understanding would then be gained. This paper gives mathematical substance to this feeling, by presenting a generalization of the reducibility method based on a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). A key technical ingredient is the introduction a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene's recursive realizability and a variant of Kreisel's modi ed realizability both t into this framework. We are then able to prove a meta-theorem which shows that if a property of realizers satis es some simple conditions, then it holds for the semantic interpretations of all terms. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed-terms, in particular, strong normalization and con uence. This approach clari es the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. The above approach is applied to the simply-typed-calculus (with types!,,+,and?), and to the second-order (polymorphic)-calculus (with types! and 82), for which it yields a new theorem.