[mlochbaum]

The structure is pretty similar. If you could encode array structures into categories (part of my problem is I don't know how to do this effectively: are the structures objects, or categories?), then my take is that a structural function that extracts some part of an array should give a functor G. Then a right adjoint ⌾G puts the omitted structure back. But I don't think this can work, because the definition of an adjoint says morphisms in the target of G exactly correspond to morphisms in its source, when ⌾G is used to find the codomain. This doesn't match up: if G extracts structure, then it should collapse some functions together, when they only differ in what they do to the parts of the array that get left out.

Most of my problem seems to be that the definition of Under really depends on applying functions to particular arguments. So it's hard to see how to set up categories to capture the necessary properties. But I don't know if it's actually hard to do or just confusing.