Theories of presheaf type

Abstract

Let us say that a geometric theory T is of presheaf type if its classifying topos 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–Moerdijk [13], pp. 381-386, and the textbook of Adámek–Rosický [1] for additional information:Proposition 0.1. For a category , the following properties are equivalent:(i) is a finitely accessible category in the sense of Makkai–Paré [14], i.e., it has filtered colimits and a small dense subcategory of finitely presentable objectsii) is equivalent to Pts, the category of points of some presheaf topos(iii) is equivalent to the free filtered cocompletion (also known as Ind-) of a small category .(iv) is equivalent to Mod(T) for some geometric theory of presheaf type.Moreover, if these are satisfied for a given , then the —in any of (i), (ii) and (iii)—can be taken to be the full subcategory of consisting of finitely presentable objects. (There may be inequivalent choices of , as it is in general only determined up to idempotent completion; this will not concern us.)This seems to completely solve the problem of identifying when T is of presheaf type: check whether Mod(T) is finitely accessible and if so, recover the presheaf topos as Set-functors on the full subcategory of finitely presentable models. There is a subtlety here, however, as pointed out (probably for the first time) by Johnstone [10].

General facts on the Scott Adjunction

“…It was shown in [Bek04][Cor 2.2] that F cannot be of presheaf type, and thus ǫ F cannot be an equivalence of categories.…”

Cellular objects and Shelah's singular compactness theorem

“…Proof. We will define a smooth chain A : (1). Given that the diagrams F A(β) and B(β), β < α, are isomorphic: for successor α, since F -structures extend along morphisms, one can select…”

“…This theory can be axiomatized by card(fp B) many sentences of the logic L card(fp B) + , ω 0 , cf. [11] 4.4.3 and 3.2.3, or [1] (2.1) through (2.5) for an explicit set of formulas. Condition (6) now follows as in the proof of the downwards Löwenheim-Skolem theorem; see e.g.…”

“…We build heavily on the results of [12]. By [12] 4.14 and 4.15 (1), an object of B is Xcellular if and only if it belongs to the image of F . Let D 1 : A 1 → B and D 2 : A 2 → B be objects of A and m a morphism of A determined by a downward-closed embedding m : A 1 → A 2 .…”

Completeness of cocompletions

“…In fact, the attempt to generalize this latter condition to a fully categorical context lead the first-named author to consider one of the notions proposed here and to relate it to the question when the completion of a category under finite colimits has finite limits [12]. More recently such conditions were related to the question when the classifying topos of a geometric theory is a presheaf topos [3] and when the theory of flat functors on a category admits a coherent axiomatization [4].…”