Hypotheses that imply total categoricity

Question

Some theories are totally categorical.

Are there some hypotheses that do imply the total categoricity of a theory?

Answer 1

In the comments, dav11 suggests $\aleph_0$-categorical + $\aleph_1$ categorical, by Morley's theorem. Using the Baldwin-Lachlan characterization of uncountably categorical theories, totally categorical is also equivalent to $\aleph_0$-categorical + $\omega$-stable + no Vaughtian pairs.

For a sufficient condition that encompasses the case of infinite-dimensional vector spaces over a finite field, you might like strongly minimal + locally finite.

Strongly minimal means that for any model $M$, any definable subset of $M$ is finite or cofinite.

Locally finite means that $\mathrm{acl}(A)$ is finite whenever $A$ is finite.

Why is this sufficient? If $T$ is strongly minimal, then already it is $\aleph_1$-categorical. And in a strongly minimal theory, there is a unique non-algebraic type over any set. So if $\mathrm{acl}(A)$ is finite, then there are only finitely many $1$-types over $A$ (the types of the elements of $\mathrm{acl}(A)$ together with the unique non-algebraic type). So there are finitely many $1$-types over every finite set, which is equivalent to $\aleph_0$-categoricity.

Answer 2

A countable theory is uncountably categorical if and only if it is $\omega$-stable and has no Vaughtian pairs, and it is $\aleph_0$-categorical if and only if $S_n(\emptyset)$ is finite for every $n$.

I don't recall any alternative description of having no Vaughtian pairs, but I believe $\omega$-stable $\omega$-categorical theories can be alternatively described as those for which $S_1(A)$ is finite or countable whenever $A$ is finite or countable (respectively).