structure in model theory
type in model theory
stability in model theory, geometric stability theory
definable set, definable groupoid
In model theory, given a cardinal κ\kappa, a theory is κ\kappa-categorical (or categorical in cardinality κ\kappa), if it has precisely one isomorphism class of models of cardinality κ\kappa.
The Morley categoricity theorem says that a first-order theory TT with countably many symbols is κ\kappa-categorical for one uncountable cardinal κ\kappa iff TT is categorical in any uncountable cardinality.
Study of categoricity lead historically to the development of the stability theory in model theory, see also geometric stability theory.
John Baldwin, Categoricity, Amer. Math. Soc. 2011, pdf
John T. Baldwin, What is a complete theory, talk, pdf
wikipedia: Morley categoricity theorem