View : 581 Download: 0
Non-commutative groupoids obtained from the failure of 3-uniqueness in stable theories
- Title
- Non-commutative groupoids obtained from the failure of 3-uniqueness in stable theories
- Authors
- Kim, Byunghan; Kim, SunYoung; Lee, Junguk
- Ewha Authors
- 김선영
- SCOPUS Author ID
- 김선영
- Issue Date
- 2020
- Journal Title
- FUNDAMENTA MATHEMATICAE
- ISSN
- 0016-2736
1730-6329
- Citation
- FUNDAMENTA MATHEMATICAE vol. 249, no. 1, pp. 47 - 70
- Keywords
- generic groupoid; 3-uniqueness; fundamental group
- Publisher
- POLISH ACAD SCIENCES INST MATHEMATICS-IMPAN
- Indexed
- SCIE; SCOPUS
- Document Type
- Article
- Abstract
- Given an arbitrary connected groupoid G with its vertex group G(a), if G(a) is a central subgroup of a group F, then there is a canonical extension F = G circle times F of G in the sense that Ob(G) = Ob(F), Mor(G) subset of Mor(F), and F is isomorphic to all the vertex groups of F. From the failure of 3-uniqueness of a strong type p over A = acl(eq)(A) in a stable theory T, a canonical finitary connected commutative groupoid G with the binding group G was A-type-definably constructed by John Goodrick and Alexei Kolesnikov (2012). In this paper we take a certain (possibly non-commutative) automorphism group F where G is embedded centrally (so inducing t(a) : G(a) -> Z(F)), and show that the abstract groupoid G circle times F lives A-invariantly in models of T. More precisely, we A-invariantly construct a connected groupoid T, isomorphic to G circle times F as abstract groupoids, satisfying the following: (1) Ob(F) = Ob(G), and Mor(F) and composition maps are A-invariant (i.e., described by infinite disjunctions of conjunctions of formulas over A), so that an A-automor-phism of a model of T induces a groupoid automorphism of F. (2) There is an A-invariant faithful functor I : G -> F which is the identity on the objects, and I(G(a)) = i(a) o t(a), where i(a) is a canonical group isomorphism from F onto a vertex group F-a of F. An automorphism group approximated by the vertex groups of the non-commutative groupoids is suggested as a "fundamental group" of the strong type p.
- DOI
- 10.4064/fm755-7-2019
- Appears in Collections:
- 자연과학대학 > 수학전공 > Journal papers
- Files in This Item:
There are no files associated with this item.
- Export
- RIS (EndNote)
- XLS (Excel)
- XML