Isn't there supposed to be a nice three-way equivalence between certain systems of logic, category theory, and lambda calculus? Presumably something that lies beyond the scope of CT also lies beyond the scope of those other systems.
I can see ways to represent lambda calculus constructs using CT and viceversa. I guess that 'systems of logic' includes lambda calculus, or systems with equivalent semantics; so yeah, I can believe there's a connection at some level.
The (one?) problem with CT is that it, by definition, puts too much importance on the relation between things via composition, which is a powerful paradigm in itself, but also quite restrictive. 'Structure' is not usually encoded like that in a natural way.
CT is just part of a larger framework that is currently being unveiled (!!!); but, to be quite frank, people who focus mostly on CT seem to be missing the forest for the trees.
