An Application of Open Maps to Categorical Logic

This paper is a sequel to [12]. We are here concerned with properties of theories in full first-order intuitionistic logic; the latter correspond under the identification of theories with categories provided by categorical logic (cf. [8] or [ 1 l]), to Heyting pretoposes, i.e. pretoposes with universal quantification of subobjects along morphisms. Using the lattice-theoretic machinery developed in [ 121, we construct a contravariant functor Q, : Pt OP-‘Top from the category of pretoposes to the category of Grothendieck toposes, which sends a morphism of Heyting pretoposes to an open geometric morphism. This functor allows us to deduce from the fact that open geometric surjections are stable under pullback, that conservative morphisms in the category of Heyting pretoposes are stable under pushout. From this it follows easily that every pushout square in that category has the ‘interpolation property’. From the point of view of theories, we thus obtain an essentially very simple, constructive proof of a general form of Craig’s Interpolation Theorem. At the end of the paper we make some remarks about the analogues of these properties for the coherent fragment of intuitionistic logic (i.e. for pretoposes). There are two important ingredients in the construction of the functor @ : Ptop -+Top. The first is the use of ‘indexed lattice theory’ as a bridge between propositional and predicate logic: by indexed lattice theory we mean the pre-order part of indexed category theory (cf. [2]). Specifically, we make use of particular kinds of hyperdoctrines which, following Joyal [5], we call polyadk distributive lattices and polyadic Heyting algebras. The second ingredient is the use of the constructive theor:r of locales in toposes other thanthe topos of sets (cf. [7] in particular): in Section 2 we construct locales and (open) continuous maps in various toposes of presheaves. (Indeed, all the arguments given in this paper can be carried out over an arbitrary base topos with natural number object; in particular we never need to resort to the Completeness Theorem or its equivalents.)