nLab
categoricity

In model theory, given a cardinal $\kappa$, a theory is $\kappa$-categorical (or categorical in cardinality $\kappa$), if it has precisely one isomorphism class of models of cardinality $\kappa$. The Morley categoricity theorem says that a first-order theory $T$ with countably many symbols is $\kappa$-categorical for one uncountable cardinal $\kappa$ iff $T$ is categorical in any uncountable cardinality.

• John Baldwin, Categoricity, Amer. Math. Soc. 2011, pdf
• John T. Baldwin, What is a complete theory, talk, pdf
• wikipedia: Morley categoricity theorem

Study of categoricity lead historically to the development of the stability theory in model theory, see also geometric stability theory.