A more sophisticated approach is to enrich the classification by remembering the isomorphisms. More precisely, on any base "B" one can consider the category of families on "B" with only isomorphisms between families taken as morphisms. One then considers the fibred category which assigns to any space "B" the groupoid of families over "B". The use of these "categories fibred in groupoids" to describe a moduli problem goes back to Grothendieck (1960/61). In general, they cannot be represented by schemes or even algebraic spaces, but in many cases, they have a natural structure of an algebraic stack.