In this post I describe a certain categorification of complexity classes with maps between them (rather than mere inclusions). The maps should efficiently map programs of one complexity class (e.g., PSPACE machines) to programs of another (e.g., IP verifiers) in a way which preserves the language computed.

Clearly such a map implies a class inclusion, but when can this be reversed? When can we get an efficient, uniform inclusion of one class into another from knowledge of the mere existence of an inclusion? If we know \(A \subseteq B\) and \(A\) has complete problems (and an efficient reductions to that problem), then basically knowledge of an inclusion will imply a uniform inclusion, since you can just hard code where the complete problem goes to and then use the reduction.

Can this implication be reversed? Do uniform/efficiently-computable inclusions \(A \to B\) imply the existence of complete (with respect to \(B\)-reductions) problems for \(A\)?