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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.)