Building on Hrushovski's work in [5], we study definable groupoids in stable theories and their relationship with 3-uniqueness and finite internal covers. We introduce the notion of retractability of a definable groupoid (which is slightly stronger than Hrushovski's notion of eliminability), give some criteria for when groupoids are retractable, and show how retractability relates to both 3-uniqueness and the splitness of finite internal covers. One application we give is a new direct method of constructing non-eliminable groupoids from witnesses to the failure of 3-uniqueness. Another application is a proof that any finite internal cover of a stable theory with a centerless liaison groupoid is almost split.

We present definitions of homology groups H n , n ≥ 0, associated to a family of "amalgamation functors". We show that if the generalized amalgamation properties hold, then the homology groups are trivial. We compute the group H 2 for strong types in stable theories and show that any profinite abelian group can occur as the group H 2 in the model-theoretic context.

Boney and Grossberg [BG] proved that every nice AEC has an independence relation. We prove that this relation is unique: In any given AEC, there can exist at most one independence relation that satisfies existence, extension, uniqueness and local character. While doing this, we study more generally properties of independence relations for AECs and also prove a canonicity result for Shelah's good frames. The usual tools of first-order logic (like the finite equivalence relation theorem or the type amalgamation theorem in simple theories) are not available in this context. In addition to the loss of the compactness theorem, we have the added difficulty of not being able to assume that types are sets of formulas. We work axiomatically and develop new tools to understand this general framework.Comment: 33 page

ABSTRACT. Theorem. For each 2 ≤ k < ω there is an Lω 1 ,ω -sentence φ k such that:(1) φ k is categorical in µ if µ ≤ ℵ k−2 ; (2) φ k is not ℵ k−2 -Galois stable; (3) φ k is not categorical in any µ with µ > ℵ k−2 ; (4) φ k has the disjoint amalgamation property;indeed, syntactic first-order types determine Galois types over models of cardinality at mostWe adapt an example of [9]. The amalgamation, tameness, stability results, and the contrast between syntactic and Galois types are new; the categoricity results refine the earlier work of Hart and Shelah and answer a question posed by Shelah in [17].Considerable work (e.g. [14,15,16,7,8,6,18,12,11]) has explored the extension of Morley's categoricity theorem to infinitary contexts. While the analysis in [14,15] applies only to L ω 1 ,ω , it can be generalized and in some ways strengthened in the context of abstract elementary classes.Various locality properties of syntactic types do not generalize in general to Galois types (defined as orbits under an automorphism group) in an AEC [5]; much of the difficulty of the work stems from this difference. One such locality properties is called tameness. Roughly speaking, K is (µ, κ)-tame if distinct Galois types over models of size κ have distinct restrictions to some submodel of size µ. For classes with arbitrarily large models, that satisfy amalgamation and tameness, strong categoricity transfer theorems have been proved [7,8,6,13,4,10]. In particular these results yield categoricity in every uncountable power for a tame AEC in a countable language (with arbitrarily large models satisfying amalgamation and the joint embedding property) that is categorical in any single cardinal above ℵ 2 ([6]) or even above ℵ 1 ([13]).In contrast, Shelah's original work [14,15] showed (under weak GCH) that categoricity up to ℵ ω of a sentence in L ω 1 ,ω implies categoricity in all uncountable cardinalities. Hart and Shelah [9] showed the necessity of the assumption by constructing sentences φ k which were categorical up to some ℵ n but not eventually

We present definitions of homology groups Hn (p), n ≥ 0, associated to a complete type p. We show that if the generalized amalgamation properties hold, then the homology groups are trivial. We compute the group H2(p) for strong types in stable theories and show that any profinite abelian group can occur as the group H2 (p).