It is shown that every topos with enough points is equivalent to the classifying topos of a topological groupoid. 1 De nitions and statement of the result We recall some standard de nitions 1, 5, 9. A topos is a category E which i s equivalent to the category of sheaves of sets on a small site. Equivalently, E is a topos i it satis es the Giraud axioms 1 , p. 303. The category of sets S is a topos, and plays a role analogous to that of the one point space in topology. In

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Abstract. We show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi). As our main result, we obtain a complete axiomatization of the intuitionistic set theory validated by all such interpretations. Since every elementary topos is equivalent to one carrying a dssi, we thus obtain a first-order set theory whose associated categories of sets are exactly the elementary toposes. In addition, we show that the full axiom of Separation is validated whenever the dssi is superdirected. This gives a uniform explanation for the known facts that cocomplete and realizability toposes provide models for Intuitionistic Zermelo-Fraenkel set theory (IZF). §1. Introduction. The notion of elementary topos abstracts from the structure of the category of sets. The abstraction is sufficiently general that elementary toposes encompass a rich collection of other very different categories, including categories that have arisen in fields as diverse as algebraic geometry, algebraic topology, mathematical logic, and combinatorics. Nonetheless, elementary toposes retain many of the essential features of the category of sets. In particular, elementary toposes possess an internal logic, which is a form of higher-order type theory, see e.g., [11,13,9]. This logic allows one to reason with objects of the topos as if they were abstract sets in the sense of [12]; that is, as if they were unstructured collections of elements. Although the reasoning supported in this way is both powerful and natural, it differs in several respects from the set-theoretic reasoning available in the familiar first-order set theories, such as Zermelo-Fraenkel set theory (ZF).A first main difference between the internal logic and ZF is: 1. Except in the special case of boolean toposes, the underlying internal logic of a topos is intuitionistic rather than classical. Many toposes of mathematical interest are not boolean. Thus the use of intuitionistic logic is unavoidable. Moreover, as fields such as synthetic differential geometry and synthetic domain theory demonstrate, the non-validity of classical logic has mathematical applications. In these areas, intuitionistic logic offers the opportunity of working consistently with convenient but

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Using recent results in topos theory, two systems of higher-order logic are shown to be complete with respect to sheaf models over topological spaces-so-called "topological semantics". The first is classical higherorder logic, with relational quantification of finitely high type; the second system is a predicative fragment thereof with quantification over functions between types, but not over arbitrary relations. The second theorem applies to intuitionistic as well as classical logic.