A size structure is a class of discrete morphisms, called discrete-small or just small, satisfying certain closure axioms that directly generalize the axioms for classes of small maps in algebraic set theory.
It improves on the notion of classifying discrete opfibration by giving a notion of smallness for maps that are not necessarily opfibrations. In particular, we can say that an object is locally small if is small, though it is definitely not an opfibration. We may ask that the small maps in a size structure which are opfibrations have a classifier, but it is not clear whether knowing the small opfibrations suffices to determine all the small maps. It is possible, however, that by using the 2-internal logic, from a classifying discrete opfibration satisfying suitable axioms we could canonically define a general notion of “small” and construct a size structure.
(to be written…)