Groupoids, Imaginaries and Internal Covers

The questions this manuscript addresses arose in the course of an investigation of the imaginary sorts in ultraproducts of p-adic fields. These were shown to be understandable given the imaginary sorts of certain finite-dimensional vector spaces over the residue field. The residue field is pseudo-finite, and the imaginary elements there were previously studied, and shown in fact to be eliminable over an appropriate base. It remains therefore to describe the imaginaries of finite-dimensional vector spaces over a field F , given those of F . I expected this step to be rather easy; but it turned out to become easy only after a number of issues, of interest in themselves, are made clear. Let T be a first-order theory. A correspondence is established between internal covers of models of T and definable groupoids within T . Internal covers were recognized as central in the study of totally categorical structures, but nevertheless remained mysterious; it was not clear how to describe the possible T ′ from the point of view of T . We give an account of this here, in terms of groupoids in place of equivalence relations. This description permits the view of the cover as a generalized imaginary sort. This seems to be a useful language even for finite covers, though there the situation is rather well-understood, cf. [8]. We concentrate on finite generalized imaginaries, and describe a connection between elimination of imaginaries and higher amalgamation principles within the algebraic closure of an independent n-tuple. The familiar imaginaries of T eq correspond to 3-amalgamation, as was understood for some time for stable and simple theories, and finite generalized imaginaries correspond to 4-amalgamation. This brings out ideas present in some form in [6], [3], [7], [8]. In particular, 4-amalgamation always holds for stable theory T , if “algebraic closure” is taken to include generalized imaginaries. We also relate uniqueness of namalgamation to existence of n+1-amalgamation; using “all” finite imaginaries (not necessarily arising from groupoids) we show that n-amalgamation exists and is unique for all n. Adding an automorphism to the language to obtain a Robinson theory T σ has the effect of shifting the amalgamation dimension by one; n-amalgamation in the expanded language corresponds to n+1-amalgamation for T . Thus ordinary imaginaries of T σ can be understood, given generalized imaginaries of T . We thus find a strong relation between four things: covers, failure of uniqueness for 3amalgamation, imaginaries of T , and definable groupoids. A clear continuation to n = 4 would be interesting. Returning to the original motivation, this is used to describe the imaginaries for systems of finite-dimensional vector spaces over fields, and especially over pseudo-finite fields (Theorem 4.12).