Atomic Toposes and Countable Categoricity

Abstract

We give a model-theoretic characterization of the class of geometric theories classified by an atomic topos having enough points; in particular, we show that every complete geometric theory classified by an atomic topos is countably categorical. Various results on atomic toposes are also established, and some applications are discussed.

Formal Model Theory & Higher Topology

“…In this section we define categories of saturated objects and study their connection with atomic topoi and categoricity. The connection between atomic topoi and categoricity was pointed out in [Car12]. This section corresponds to a kind of syntax-free counterpart of [Car12].…”

“…The connection between atomic topoi and categoricity was pointed out in [Car12]. This section corresponds to a kind of syntax-free counterpart of [Car12]. In the definition of category of saturated objects we axiomatize the relevant properties of the inclusion ι : Set κ → Set and we prove the following two theorems.…”

Topological Galois theory

“…every geometric sequent which is provable in T is also provable in T'); in order words, they do not have any proper quotients over their signature. Conversely, by the results in [12] and [16] (cf. also [5]), any maximal theory which is finitary or written over a countable signature and has a model in Set is a Galois-type theory.…”

“…We showed in [12] that if T is a complete atomic theory then T is countably categorical (i.e., any two countable models of it are isomorphic), and its unique (up to isomorphism) countable model M, if it exists, satisfies the hypotheses of the theorem. We thus obtain, as a corollary of the theorem, that the classifying topos of T can be represented as the topos Cont(Aut(M)) of continuous Aut(M)-sets, where Aut(M) is the group of automorphisms of M endowed with the topology of pointwise convergence.…”

Universal models and definability

“…, Set] is given by: We recall from [6] that a geometric theory T over a signature is said to have enough models if for every geometric sequent σ over , M σ for all the T-models M in Set implies that σ is provable in T using geometric logic.…”