Amalgamation functors and boundary properties in simple theories

Abstract

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.

Non-commutative groupoids obtained from the failure of 3-uniqueness in stable theories

“…Proof. The proof will be similar to that of Proposition 2.22 in [2]. Note firstly that due to Claim 2.2, Z(G a ) acts on X as an obvious manner.…”

“…More precisely, from a symmetric witness to the failure of 3-uniqueness in a stable theory, we construct a new groupoid F whose "vertex groups" Mor F (a, a) need not be abelian. In fact, we will show that Mor G (a, a) ≤ Z(Mor F (a, a)), where G is the commutative groupoid constructed in [1] and [2]. We may call F a non-commutative groupoid constructed from the symmetric witness.…”

“…Originally, 3-uniqueness is defined functorially in [5], [2], but as we will not use amalgamation notion the following equivalent definition would suffice in this note. Definition 1.1.…”

“…[2] We say the fixed complete type p has 3-uniqueness over A if whenever {a 0 , a 1 , a 2 } is an A-independent set of realizations of p, and for 0 ≤ i < j ≤ 2, σ ij ∈ Aut(a ij /a i a j ), then σ 01 ∪ σ 02 ∪ σ 12 is also an elementary map. [2] Let a, b, c |= p be independent over A. Then p has 3-uniqueness over A iff ab = dcl(ab).…”

Un critère simple

“…Il en déduit que la théorie est simple, une notion introduite par Shelah comme généralisation de la stabilité. L'amalgamation généralisée a été traitée par la suite de façon homologique [16]. À partir d'une configuration de groupe dans une théorie simple, l'amalgamation généralisée permet d'améliorer la construction dans [3] pour obtenir un groupe hyperdéfinissable [22].…”