The category of locally compact locales over any elementary topos is characterised by means of the axioms of abstract Stone duality (monadicity of the topology, considered as a selfadjoint exponential Σ(−), and Scott continuity, Fφ = ∃`. F (λn. n ∈ `)∧∀n ∈ `. φn), together with an “underlying set” functor that is right adjoint to the inclusion of the full subcategory of overt discrete objects (...

We obtain semantic characterizations, holding for any Grothendieck site (C, J), for the models of a theory classified by a topos of the form Sh(C, J) in terms of the models of a theory classified by a topos [C,Set]. These characterizations arise from an appropriate representation of flat functors into Grothendieck toposes based on an application of the Yoneda Lemma in conjunction with ideas fro...

The class of functors known as discrete Conduché fibrations forms a common generalization of discrete fibrations and discrete opfibrations, and shares many of the formal properties of these two classes. F. Lamarche [7] conjectured that, for any small category B, the category DCF/B of discrete Conduché fibrations over B should be a topos. In this note we show that, although for suitable categori...

We define and study a Weil-étale topos for any regular, proper scheme X over Spec(Z) which has some of the properties suggested by Lichtenbaum for such a topos. In particular, the cohomology with R̃-coefficients has the expected relation to ζ(X , s) at s = 0 if the Hasse-Weil L-functions L(h(XQ), s) have the expected meromorphic continuation and functional equation. If X has characteristic p the...

A generating family in a category C is a collection of objects {Ai|i ∈ I} such that if for any subobject Y // m //X, every Ai f //X factors through m, then m is an isomorphism – i.e. the functors C(Ai, ) are collectively conservative. In this paper, we examine some circumstances under which subobjects of 1 form a generating family. Objects for which subobjects of 1 do form a generating family a...

Let us say that a geometric theory T is of presheaf type if its classifying topos B[T ] is (equivalent to) a presheaf topos. (We adhere to the convention that geometric logic allows arbitrary disjunctions, while coherent logic means geometric and finitary.) Write Mod(T ) for the category of Set-models and homomorphisms of T . The next proposition is well known; see, for example, MacLane–Moerdij...

Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell’s “many-topoi” view and modal-structuralism. Surprisingly, a com...