The problems faced by the foundationalist programmes of the last century included trying to decide:
- Which is the one true logic on which all mathematics should be based? See also, Brouwer-Hilbert controversy.
- Which are the true principles of the universe of sets? See also, Axiom of Determinacy, Generalized Continuum Hypothesis.
As I see it, categorial logic provides a solution. Rather than trying to answer these questions, we can study functors (etc.) that map problems around from one "version" of mathematics into another. The question then becomes not, "Which is the one true foundation?" but rather, "How can I map this problem elsewhere so that it becomes easier to solve?"
At least, that's the impression I'm under. Does the above paragraph accurately portray what category theorists with an interest in the foundations are actually trying to achieve? And if so, to what extent has it actually been achieved?
Foundations deals with finding a good theory (in some type of language) which is good for express all the mathematical objects and have enough axioms to prove the existence of all well known construction such as the one of natural numbers, spaces, and so on.
For long time was thought that set theory was the only possible foundation for mathematics, and so that all the mathematics can be written in terms of sets.
Category theory and in particular categorical logic have shown that's not the case. Indeed category theorists have shown that using the language of topos theory is possible to codify almost (if not all) the mathematics in every topos, also those different from sets.
So categorical logic didn't change the problems (or question in foundation), rather it answered those question, just in a rather different way respect to the classical one, which is based on set theory.
p.s. Just as a notice from category theory originated also homotopy type theory which can be considered another big step in foundation that aims to provide that all the mathematics can be written not just in the language of a topos but in the language of a higher topos. But I'm not too expert to be able to say more for now, maybe someone else could add something about it :)