Here's the setup: you have a first-order theory T, in a countable language L for simplicity. Let k be a cardinal and suppose T is k-categorical. This means that, for any two models
M,N |= T
of cardinality k, there is an isomorphism f : M --> N.
Supposing all this happens inside of ZFC, let's say I change the underlying model of ZFC, e.g by restricting to the constructible sets, or by forcing new sets in. I would like to understand what happens to the k-categoricity of T.
I'll assume the set theory doesn't change so drastically that we lose L or T. Then, a priori, a bunch of things may happen:
(i) We may lose all isomorphisms between a pair of models M,N of cardinality k; (ii) Some models that used to be of cardinality k may no longer have bijections with k; (iii) k may become a different cardinal, meaning new cardinals may appear below it, or others may disappear by the introduction of new bijections; (iv) some models M, or k itself, may disappear as sets, leading to a new set being seen as "the new k".
Overall, nearly every aspect of the phrase "T is k-categorical" may be affected. How likely is it to still be true? Do some among (i)-(iv) not matter, or is there some cancellation of effects? (Say, maybe all isomorphisms M-->N disappear, but so do all bijections between N and k?)