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 form a/E where a is a possibly infinite tuple and E a 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 model M of a complete theory T. All types, sets and sequences will have size smaller than the size of M. 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].
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.
We give definitions that distinguish between two notions of indiscernibility
for a set $\{a_\eta \mid \eta \in \W\}$ that saw original use in \cite{sh90},
which we name \textit{$\s$-} and \textit{$\n$-indiscernibility}. Using these
definitions and detailed proofs, we prove $\s$- and $\n$-modeling theorems and
give applications of these theorems. In particular, we verify a step in the
argument that TP is equivalent to TP$_1$ or TP$_2$ that has not seen
explication in the literature. In the Appendix, we exposit the proofs of
\citep[{App. 2.6, 2.7}]{sh90}, expanding on the details.Comment: submitte
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.