We show that Kim-forking satisfies existence in all NSOP1 theories.

We show that Kim-forking satisfies existence in all NSOP_1 theories.

In this paper we study the Lascar group over a hyperimaginary e. We verify that various results about the group over a real set still hold when the set is replaced by e. First of all, there is no written proof in the available literature that the group over e is a topological group. We present an expository style proof of the fact, which even simplifies existing proofs for the real case. We further extend a result that the orbit equivalence relation under a closed subgroup of the Lascar group is type-definable. On the one hand, we correct errors appeared in the book, "Simplicity Theory" [6, 5.1.14-15] and produce a counterexample. On the other, we extend Newelski's Theorem in "The diameter of a Lascar strong type" [12] that `a G-compact theory over a set has a uniform bound for the Lascar distances' to the hyperimaginary context. Lastly, we supply a partial positive answer to a question raised in "The relativized Lascar groups, type-amalgamations, and algebraicity" [4, 2.11], which is even a new result in the real context.

AbstractWe 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.

We study Kim-independence over arbitrary sets. Assuming that forking satisfies existence, we establish Kim's lemma for Kim-dividing over arbitrary sets in an NSOP$_{1}$ theory. We deduce symmetry of Kim-independence and the independence theorem for Lascar strong types.

We apply compact group theory to obtain some model-theoretic results about the relativized Lascar Galois group of a strong type.

AbstractWe 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 .

Tree properties are introduced by Shelah, and it is well-known that a theory has TP (the tree property) if and only if it has TP$_1$ or TP$_2$. In any simple theory (i.e., a theory not having TP), forking supplies a good independence notion as it satisfies symmetry, transitivity, extension, local character, and type-amalgamation. Shelah also introduced SOP$_n$ ($n$-strong order property). Recently it is proved that in any NSOP$_1$ theory (i.e. a theory not having SOP$_1$) holding nonforking existence, Kim-forking also satisfies all the mentioned independence properties except base monotonicity (one direction of transitivity). These results are the sources of motivation for this paper. Mainly, we produce type-counting criteria for SOP$_2$ (which is equivalent to TP$_1$) and SOP$_1$. In addition, we study relationships between TP$_2$ and Kim-forking, and obtain that a theory is supersimple iff there is no countably infinite Kim-forking chain.

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.

AbstractIn 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.

A graph parameter f with a values in a ring R is required to be invariant under graph isomorphisms. f induces an equivalence relation on graphs, where two graphs G 1 , G 2 are f -equivalent if f ( G 1 ) = f ( G 2 ). Two graph parameters f,g have the same distinctive power (are d.p.-equivalent) if they induce the same equivalence relation. Typical

We give an explicit description of the homology group Hn(p) of a strong type p in any stable theory under the assumption that for every non-forking extension q of p the groups Hi(q) are trivial for 2≤i<n. The group Hn(p) turns out to be isomorphic to the automorphism group of a certain part of the algebraic closure of n independent realizations of p; it follows from the authors' earlier work that such a group must be abelian. We call this the “Hurewicz correspondence” by analogy with the Hurewicz Theorem in algebraic topology.

Let $p$ be a strong type of an algebraically closed tuple over $B=\acl^{\eq}(B)$ in any theory $T$. Depending on a ternary relation $\indo^*$ satisfying some basic axioms (there is at least one such, namely the trivial independence in $T$), the first homology group $H^*_1(p)$ can be introduced, similarly to \cite{GKK1}. We show that there is a canonical surjective homomorphism from the Lascar group over $B$ to $H^*_1(p)$. We also notice that the map factors naturally via a surjection from the `relativised' Lascar group of the type (which we define in analogy with the Lascar group of the theory) onto the homology group, and we give an explicit description of its kernel. Due to this characterization, it follows that the first homology group of $p$ is independent from the choice of $\indo^*$, and can be written simply as $H_1(p)$. As consequences, in any $T$, we show that $|H_1(p)|\geq 2^{\aleph_0}$ unless $H_1(p)$ is trivial, and we give a criterion for the equality of stp and Lstp of algebraically closed tuples using the notions of the first homology group and a relativised Lascar group. We also argue how any abelian connected compact group can appear as the first homology group of the type of a model.

We prove that in a rosy theory, the n-uniqueness of a strong type p implies the strong n-uniqueness of p. In addition we study some of the boundary properties of p.

Let T be a stable theory. It was shown in [5] that one can define the notions of homology groups attached to a stationary type of T . It was also shown that if T fails to have an amalgamation property called 3-uniqueness, then for some stationary type p the homology group H2(p) has to be a nontrivial abelian profinite group. The goal of this paper is to show that for any abelian profinite group G there is a stable (in fact, categorical) theory and a stationary type p such that H2(p) ∼= G.

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.

In this article, we show that the first homology group of strong type $H_1(p)$ is well-defined for any strong type $p$ in any theory, and this group is given by the quotient of automorphism group $G$ of $p$ by the normal subgroup of automorphisms fixing each orbit on $p(M)$ under the action of the derived group $G'$. We also suggest a candidate of Lascar group localized at $p$, the quotient of $G$ by the normal subgroup of automrophisms fixing all Lascar equivalence class of arbitrary length of realizations of p, which does not depend on the choice of monster model. Using this Lascar group, we show that there is a canonical epimorphism from this Lascar group to the first homology group for a strong type in an arbitrary theory.

We show that in a stable first-order theory, the failure of higher-dimensional type amalgamation can always be witnessed by algebraic structures which we call n-ary polygroupoids. This generalizes a result of Hrushovski that failures of 4-amalgamation in stable theories are witnessed by definable groupoids (which are 2-ary polygroupoids in our terminology). The n-ary polygroupoids are definable in a mild expansion of the language (adding a unary predicate for an infinite Morley sequence).

We present definitions of homology groups 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 in this context, the class of possible groups H_2 is precisely the profinite abelian groups.