Carsten Butz

Topological representation of sheaf cohomology of sites

For a site & (with enough points), we construct a topological space X(&) and a full embedding ϕ* of the category of sheaves on & into those on X(&) (i.e., a morphism of toposes ϕ:Sh (X(&)) →Sh(&)). The embedding will be shown to induce a full embedding of derived categories, hence isomorphisms H*(&,A) = H*(X(&), ϕ*A) for any Abelian sheaf A on &. As a particular case, this will give for any scheme Y a topological space X(Y) and a functorial isomorphism between the étale cohomology H*(Yét,A) and the ordinary sheaf cohomology H*(X((Y),),ϕ*A), for any sheaf A for the étale topology on Y.

An elementary definability theorem for first order logic

In this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language , then, clearly, any definable subset S ⊂ M (i.e., a subset S = { a ∣ M ⊨ φ( a )} defined by some formula φ) is invariant under all automorphisms of M . The same is of course true for subsets of M n defined by formulas with n free variables. Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M , in which precisely the T -provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula . Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning Boolean valued models.

A topological completeness theorem

Abstract. We prove a topological completeness theorem for infinitary geometric theories with respect to sheaf models. The theorem extends a classical result of Makkai and Reyes, stating that any topos with enough points has an open spatial cover. We show that one can achieve in addition that the cover is connected and locally connected.

Représentation de topos par des espaces topologiques

Resume Soit τ un topos avec assez de points. Nous demontrons l’existence d’un espace topologique X τ ayant les proprietes suivantes : premierement, τ est equivalent a la categorie Fais G ( X τ ) des faisceaux equivariants pour un groupoide topologique G ⇉ X τ (Theoreme A); de plus, le morphisme de topos associe Fais( X τ ) → τ induit des isomorphismes en cohomologie (ou un plongement pleinement fidele des categories derivees) (Theoreme B).

Syntax and Semantics of the logic L_omega omega^lambda

In this paper we study the logic L_omega omega^lambda , 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_omega omega^lambda with respect to models in Grothendieck toposes, which can be sharpened to completeness with respect to Heyting valued models. The logic L_omega omega^lambda is the strongest for which Heyting valued completeness is known. Finally, we relate the logic to locally connected geometric morphisms between toposes.