If a Quillen model category can be specified using a certain logical syntax (intuitively, ``is algebraic/combinatorial enough''), so that it can be defined in any category of sheaves, then the satisfaction of Quillen's axioms over any site is a purely formal consequence of their being satisfied over the category of sets. Such data give rise to a functor from the category of topoi and geometric morphisms to Quillen model categories and Quillen adjunctions.Comment: 30 page

We investigate properties of accessible categories with directed colimits and their relationship with categories arising from Shelah's Abstract Elementary Classes. We also investigate ranks of objects in accessible categories, and the effect of accessible functors on ranks.cardinals λ such that L is λ-categorical and, at the same time, a proper class of cardinals λ such that L is not λ-categorical.If this K has directed colimits then the categoricity conjecture fails for the class (4). By 4.12, however, this simple trick does not help for categories in class (3) of our hierarchy.Remark 6.4. Shelah's conjecture seems to be unknown even for finitely accessible categories, lying at level (1) of the hierarchy. It would be interesting to understand whether the exquisite Galois-theoretic machinery of [3] can be brought to bear implications in this setting.

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].

We introduce a notion of 'cover of level n' for a topological space, or more generally any Grothendieck site, with the key property that simplicial homotopy classes computed along the filtered diagram of n-covers biject with global homotopy classes when the target is an n-type. When the target is an Eilenberg-MacLane sheaf, this specializes to computing derived functor cohomology, up to degree n, via simplicial homotopy classes taken along n-covers. Our approach is purely simplicial and combinatorial.

We introduce the notion of λ-equivalence and λ-embeddings of objects in suitable categories. This notion specializes to L ∞λ -equivalence and L ∞λ -elementary embedding for categories of structures in a language of arity less than λ, and interacts well with functors and λ-directed colimits. We recover and extend results of Feferman and Eklof on "local functors" without fixing a language in advance. This is convenient for formalizing Lefschetz's principle in algebraic geometry, which was one of the main applications of the work of Eklof.