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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.

The contravariant powerset, and its generalisations ΣX to the lattices of open subsets of a locally compact topological space and of recursively enumerable subsets of numbers, satisfy the Euclidean principle that φ ∧ F (φ) =φ ∧ F (⊤). Conversely, when the adjunction Σ (−) ⊣ Σ (−) is monadic, this equation implies that Σ classifies some class of monos, and the Frobenius law ∃x.(φ(x) ∧ ψ) =(∃x.φ(x)) ∧ ψ) for the existential quantifier. In topology, the lattice duals of these equations also hold, and are related to the Phoa principle in synthetic domain theory. The natural definitions of discrete and Hausdorff spaces correspond to equality and inequality, whilst the quantifiers considered as adjoints characterise open (or, as we call them, overt) and compact spaces. Our treatment of overt discrete spaces and open maps is precisely dual to that of compact Hausdorff spaces and proper maps. The category of overt discrete spaces forms a pretopos and the paper concludes with a converse of Paré’s theorem (that the contravariant powerset functor is monadic) that characterises elementary toposes by means of the monadic and Euclidean properties together with all quantifiers, making no reference to subsets.

In this paper we study the logic L !! , which is first order logic extended by quantification over functions (but not over relations). We give the syntax of the logic, as well as the semantics in Heyting categories with exponentials. Embedding the generic model of a theory into a Grothendieck topos yields completeness of L !! with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting valued models. The logic L !! is the strongest for which Heyting valued completeness is known. Finally, we relate the logic to locally connected geometric morphisms between toposes.

Abstract. Let M = (M, m, u) be a monad and let (MX, m) be the free M-algebra on the object X. Consider an M-algebra (A, a), a retraction r: (MX, m) → (A, a) and a section t: (A, a) → (MX, m) of r. The retract (A, a) is not free in general. We observe that for many monads with a ‘combinatorial flavor ’ such a retract is not only a free algebra (MA0, m), but it is also the case that the object A0 of generators is determined in a canonical way by the section t. We give a precise form of this property, prove a characterization, and discuss examples from combinatorics, universal algebra, convexity and topos theory. 1.

introduction

Abstract. In this article we prove the following: A topos with a point is connected atomic if and only if it is the classifying topos of a localic group, and this group can be taken to be the locale of automorphisms of the point. We explain and give the necessary definitions to understand this statement. The hard direction in this equivalence was first proved in print in [4], Theorem 1, Section 3, Chapter VIII, and it follows from a characterization of atomic topoi in terms of open maps and from a theory of descent for morphisms of topoi and locales. We develop our version and our proof of this theorem, which is completely independent of descent theory and of any other result in [4]. Here the theorem follows as an straightforward consequence of a direct generalization of the fundamental theorem of Galois. In Proposition I of “Memoire sur les conditions de resolubilite des equations par radicaux”, Galois established that any intermediate extension of the splitting field of a polynomial with rational coefficients is the fixed field of its galois group. We first state and prove the (dual) categorical interpretation of of this statement, which is a theorem about atomic sites with a representable point. These developments correspond exactly to Classical Galois Theory. In the general case, the point determines a proobject and it becomes (tautologically) prorepresentable. We state and prove the, mutatus mutatis, prorepresentable version of Galois theorem. In this case the classical group of automorphisms has to be replaced by the localic group of automorphisms. These developments form the content of a theory that we call Localic Galois Theory.

The goals of this paper are (1) to compare the Galois groupoid that appears naturally in the construction of the fundamental groupoid of a topos E bounded over an arbitrary base topos S given by Bunge (1992), with the formal Galois groupoid defined by Janelidze (1990) in a very general setting given by a pair of adjoint functors, and (2) to discuss a good notion of covering morphism of a topos E over S which is general enough to include, in addition to the covering projections determined by the locally constant objects, also the unramified morphisms of topos theory given by those local homeomorphisms which are at the same time complete spreads in the sense of Bunge-Funk (1996, 1998).

INTRODUCTION By an S-valued distribution on a topos E bounded over a base topos S it is meant here a cocontinuous S-indexed functor : E ! S. Since introduced by F. W. Lawvere in 1983, considerable progress has been made in the study of distributions on toposes from a variety of viewpoints [19, 15, 24, 5, 6, 12, 7, 8, 9]. However, much work still remains to be done in this area. The purpose of this paper is to deepen our understanding of topos distributions by exploring a (dual) lattice-theoretic notion of distribution algebra. We characterize the distribution algebras in E relative to S as the S-bicomplete S-atomic Heyting algebras in E . As an illustration, we employ distribution algebras explicitly in order to give an alternative description of the display locale (complete spread) of a distribution [10, 12, 7].

We present some new findings concerning branched covers in topos theory. Our discussion involves a particular subtopos of a given topos that can be described as the smallest subtopos closed under small coproducts in the including topos. Our main result is a description of the covers of this subtopos as a category of fractions of branched covers, in the sense of Fox [10], of the including topos. We also have some new results concerning the general theory of KZ-doctrines, such as the closure under composition of discrete fibrations for a KZ-doctrine, in the sense of Bunge and Funk [6].