The over-topos at a model

Abstract

With a model of a geometric theory in an arbitrary topos, we associate a site obtained by endowing a category of generalized elements of the model with a Grothendieck topology, which we call the antecedent topology. Then we show that the associated sheaf topos, which we call the over-topos at the given model, admits a canonical totally connected morphism to the given base topos and satisfies a universal property generalizing that of the colocalization of a topos at a point. We first treat the case of the base topos of sets, where global elements are sufficient to describe our site of definition; in this context, we also introduce a geometric theory classified by the over-topos, whose models can be identified with the model homomorphisms towards the (internalizations of the) model. Then we formulate and prove the general statement over an arbitrary topos, which involves the stack of generalized elements of the model. Lastly, we investigate the geometric and 2-categorical aspects of the over-topos construction, exhibiting it as a bilimit in the bicategory of Grothendieck toposes. NotationThe notation employed in the paper will be standard; in particular, − We shall denote by Set the category of sets (within a fixed model of set theory).− Given a geometric theory T, we shall denote by (C T , J T ) its geometric syntactic site and by Set[T] its classifying topos, which, as is well-known, can be represented as Sh(C T , J T ) (for background on classifying toposes, the reader may refer to [2]).− For any geometric theory T, we denote by T[E] the category of T-models in a geometric category E and model homomorphisms between them. For any E, we have an equivalence T[E] ≃ Cart J T (C T , E) between T[E] and the category of cartesian (that is, finite-limitpreserving) functors C T → E which are J T -continuous (that is, which send J T -covering families to covering families in E). The functor C T → E corresponding to a T-model M in E will be denoted by F M ; it sends any geometric formula { x.φ} over the signature of T to its interpretation [[ x.φ]] M in M , and acts accordingly on arrows (for more details, see, for instance, Chapter 1 of [2]).