Given a category C, one may form another category, call it Group(C), with objects the group objects in C and morphisms homomorphisms of groups objects in C. (Of course the objects of Group(C) need not constitute "groups" in the ordinary sense - for example, take C as the homotopy category.) Can one give necessary and sufficient conditions on a(n abstract) category D for isomorphism to the Group(C) of some C? for equivalence to the Group(C) of some C? for isomorphism/equivalence to a subcategory of the Group(C) of some C? to a full subcategory? Suitable hypotheses should make the passage from C to Group(C) functorial. In what contexts does such a functor have adjoints? Thanks for any help! David Feldman