Categories of group objects

1 Jul 2002

      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