Weight (stability theory)

Work in a stable theory (or more generally a simple theory) $$T$$. The preweight of a complete type $$\operatorname{tp}(a/C)$$ is defined to be the supremum of the cardinals $$\kappa$$ such that there is some $$C$$-independent set $$\{b_\lambda : \lambda < \kappa\}$$ such that $$a$$ forks with $$b_\lambda$$ for every $$\lambda$$, i.e., $$a \not \downarrow_C b_\lambda$$ for every $$\lambda$$. This is well-defined, and in fact the preweight of $$\operatorname{tp}(a/C)$$ is bounded above by the $$\kappa$$ appearing in the local character of forking (which is $$\aleph_0$$ for superstable theories).

If $$p$$ is a stationary type, the weight of $$p$$ is defined to be the largest weight of any non-forking extension of $$p$$. Types of Morley rank 1, or more generally, Lascar rank 1 have weight 1. More generally, regular types have weight 1.

Weight is generalized to simple theories in a straightforward way. Weight is generalized to NIP theories by the notion of dp-rank, and is generalized to NTP2 theories by the notion of burden.

Superstable theories have plenty of weight 1 types, in some sense… (Every type is domination equivalent to a product of weight 1 type. Also, every type has finite weight.)