Theories of presheaf type
Journal of Symbolic Logic (JSL)
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 , pp. 381-386, and the textbook of Adámek–Rosický  for additional information: Proposition 0.1. For a
... position 0.1. For a category , the following properties are equivalent: (i) is a finitely accessible category in the sense of Makkai–Paré , i.e., it has filtered colimits and a small dense subcategory of finitely presentable objects ii) 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 .