A similar problem arises with the "comprehension scheme" when we try to set up an adjunction between modules (actions, presheaves) and "fibrations" . This does not seem routine when not only the fibers but also the base and the hypothetical total are to be, for example , Ab-enriched. Is there a coherent approach to this ? Note that for example the "trivial extension" idea of adjointing infinitesimals serves well as a way of making modules into rings (additive categories) but does not appear as any kind of cousin to the notion of fibered category.
A situation which I mentioned earlier in an entirely different context might be relevant here. It is described most conceptually in ``Differential homological algebra'' by J. C. Moore, in the Nice 1970 congress proceedings. There, a fibration under a monoid G and over a comonoid B is described as an object E with an action of G and a coaction of B which are compatible; there is a further condition which expresses principality of the fibration. The condition actually forces E to be (isomorphic to) G (x) B, so one can fix all objects and let only morphisms vary. In the cartesian situation every B has a unique comonoid structure and a coaction of B on E amounts to a map E -> B so that one recovers the ``standard'' principal fibration concept, at least when G is a group. An intermediate example is given by taking homology of the cartesian example; there, the monoid side is in addition a Hopf object, i.~e. has itself a compatible comonoid structure. However the general theory works without this extra structure. A ``comprehension scheme''-like feature is the fact that (under certain conditions) for every monoid G there is a universal E and B (the bar construction) and for every B there is a universal E and G (the cobar construction). In the abelian situation, for a dg algebra G and a dg coalgebra B, there is a one-to-one correspondence between the E as above and the so called twisting cochains G -> B. All of the above seems to correspond to groupoids which are a product of a one object groupoid and a discrete groupoid. There is a notion of Hopf algebroid which is more general. Does anybody know about bar-cobar-like constructions in this case?