Suppose $\mathcal L \subseteq \mathcal L’$ are first-order languages, $\kappa$ is a cardinal, and $T’$ is a theory in $\mathcal L’$ that is $\kappa$-categorical. Let $T = T’ \restriction \mathcal L$. Is $T$ $\kappa$-categorical?


If $|\mathcal L’| = \kappa = \aleph\_0$, then I can show the answer is yes using the Ryll-Nardzewski Theorem, which says that a countable theory is $\aleph\_0$-categorical iff for each $n$, the number of $n$-types is finite.



Is categoricity retained when reducing the language?
The answer is no, one can lose categoricity in a reduct of a theory. Consider the following example.


Consider the theory $T$ describing a bijection between two disjoint infinite
predicates $f:A\to B$. So a model consists of two disjoint parts, the
$A$-part and the $B$-part, and a bijection $f$ between them. The
language is $\{f,A,B\}$.


This theory is categorical in every cardinality. But if we restrict
the theory to its consequences in the language with the two predicates $\{A,B\}$
and without the bijection, then it is just the theory of two disjoint infinite predicates, which is no longer categorical in
uncountable powers, since one predicate could have a different
cardinality than the other.


