I'm a non-mathematician who is interested in differential topology.
If I understand correctly, the existence of exotic $\mathbb{R}^4$ is directly linked to the failure of smooth h-cobordism theorem between 4-manifolds. Exotic $\mathbb{R}^4$s also exist in PL since PL is isomorphic to DIFF in dimension 4, but in TOP they are homeomorphic to the standard $\mathbb{R}^4$.
My question is: could there be a category of manifolds, other than TOP, that can rule out exotic $\mathbb{R}^4$?
For example, we might define an "X-morphism" less strict than diffeomorphism (but stricter than homeomorphism) so that all the exotic $\mathbb{R}^4$s are "X-morphic" to the standard $\mathbb{R}^4$. Or we might have a category of "nice" manifolds, excluding manifolds with fractal structures. Would any of these be a real (and useful) thing in mathematics?