Abstract We develop the theory of Kim-independence in the context of NSOP $_{1}$ theories satisfying the existence axiom. We show that, in such theories, Kim-independence is transitive and that -Morley sequences witness Kim-dividing. As applications, we show that, under the assumption of existence, in a low NSOP $_{1}$ theory, Shelah strong types and Lascar strong types coincide and, additionally, we introduce a notion of rank for NSOP $_{1}$ theories.

Abstract In this paper we study the relativized Lascar Galois group of a strong type. The group is a quasi-compact connected topological group, and if in addition the underlying theory T is G-compact, then the group is compact. We apply compact group theory to obtain model theoretic results in this note. For example, we use the divisibility of the Lascar group of a strong type to show that, in a simple theory, such types have a certain model theoretic property that we call divisible amalgamation. The main result of this paper is that if c is a finite tuple algebraic over a tuple a, the Lascar group of $\operatorname {stp}(ac)$ is abelian, and the underlying theory is G-compact, then the Lascar groups of $\operatorname {stp}(ac)$ and of $\operatorname {stp}(a)$ are isomorphic. To show this, we prove a purely compact group-theoretic result that any compact connected abelian group is isomorphic to its quotient by every finite subgroup. Several (counter)examples arising in connection with the theoretical development of this note are presented as well. For example, we show that, in the main result above, neither the assumption that the Lascar group of $\operatorname {stp}(ac)$ is abelian, nor the assumption of c being finite can be removed.

Abstract We study the notion of weak canonical bases in an NSOP $_{1}$ theory T with existence. Given $p(x)=\operatorname {tp}(c/B)$ where $B=\operatorname {acl}(B)$ in ${\mathcal M}^{\operatorname {eq}}\models T^{\operatorname {eq}}$ , the weak canonical base of p is the smallest algebraically closed subset of B over which p does not Kim-fork. With this aim we firstly show that the transitive closure $\approx $ of collinearity of an indiscernible sequence is type-definable. Secondly, we prove that given a total $\mathop {\smile \hskip -0.9em ^| \ }^K$ -Morley sequence I in p, the weak canonical base of $\operatorname {tp}(I/B)$ is $\operatorname {acl}(e)$ , if the hyperimaginary $I/\approx $ is eliminable to e, a sequence of imaginaries. We also supply a couple of criteria for when the weak canonical base of p exists. In particular the weak canonical base of p is (if exists) the intersection of the weak canonical bases of all total $\mathop {\smile \hskip -0.9em ^| \ }^K$ -Morley sequences in p over B. However, while we investigate some examples, we point out that given two weak canonical bases of total $\mathop {\smile \hskip -0.9em ^| \ }^K$ -Morley sequences in p need not be interalgebraic, contrary to the case of simple theories. Lastly we suggest an independence relation relying on weak canonical bases, when T has those. The relation, satisfying transitivity and base monotonicity, might be useful in further studies on NSOP $_1$ theories .

Abstract In this article, we prove that if a countable non- ${\aleph _0}$ -categorical NSOP1 theory with nonforking existence has finitely many countable models, then there is a finite tuple whose own preweight is ω. This result is an extension of a theorem of the author on any supersimple theory.

Abstract We classify, in a nontrivial amenable collection of functors, all 2-chains up to the relation of having the same 1-shell boundary. In particular, we prove that in a rosy theory, every 1-shell of a Lascar strong type is the boundary of some 2-chain, hence making the 1st homology group trivial. We also show that, unlike in simple theories, in rosy theories there is no upper bound on the minimal lengths of 2-chains whose boundary is a 1-shell.

Abstract 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).

Abstract In this paper, we continue the construction done in [3], so that under model-4-CA or 4-CA, given a bounded quadrangleCinduced from a group configuration, we build a canonical hyperdefinable homogeneous space equivalent toC. WhenCis principal, we can choose the homogeneous space principal as well.

Abstract An amalgamation basepin a simple theory isstably definableif its canonical base is interde-finable with the set of canonical parameters for the ϕ-definitions ofpas ϕ ranges through all stable formulae. A necessary condition for stably definability is given and used to produce an example of a supersimple theory with stable forking having types that are not stably definable. This answers negatively a question posed in [8]. A criterion for and example of a stably definable amalgamation base whose restriction to the canonical base is not axiomatised by stable formulae are also given. The examples involve generic relations over non CM-trivial stable theories.

Abstract Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimpleT. canonical base of an amalgamation class is the union of names ofψ-definitions of ,ψranging over stationaryL-formulas in . Also, we prove that the same is true with stable formulas for an 1-based theory having elimination of hyperimaginaries. For such a theory, the stable forking property holds, too.

In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical supersimple theories. Motivated by methods there, we prove the existence of canonical bases (in a suitable sense) for types in any simple theory. This is done in Section 2. In general these canonical bases will (as far as we know) exist only as “hyperimaginaries”, namely objects of the forma/Ewhereais a possibly infinite tuple andEa type-definable equivalence relation. (In the supersimple, ω-categorical case, these reduce to ordinary imaginaries.) So in Section 1 we develop the general theory of hyperimaginaries and show how first order model theory (including the theory of forking) generalises to hyperimaginaries. We go on, in Section 3 to show the existence and ubiquity of regular types in supersimple theories, ω-categorical simple structures and modularity is discussed in Section 4. It is also shown here how the general machinery of simplicity simplifies some of the general theory of smoothly approximable (or Lie-coordinatizable) structures from [2]. Throughout this paper we will work in a large, saturated modelMof a complete theoryT. All types, sets and sequences will have size smaller than the size ofM. We will assume that the reader is familiar with the basics of forking in simple theories as laid out in [4] and [6]. For basic stability-theoretic results concerning regular types, orthogonality etc., see [1] or [9].

Abstract LetTbe a countable, small simple theory. In this paper, we prove that for suchT, the notion of Lascar strong type coincides with the notion of strong type, over an arbitrary set.