Under one view, there is a mismatch in the comparison between toposes and Abelian categories. Consider enriched category theory over Set and Ab[elian groups]. Over Set: enriched category A = small category, A-action = functor from A to Set (covariant or contra- for right or left action), cat of A-actions = Set^A or Set^A^op, wlog a presheaf topos, "quotient" (by Grothendieck topology) = general Grothendieck topos. Over Ab: enriched category A = ringoid ("ring with several objects"), A-action = right or left module over A, cat of A-actions = Mod-A or A-Mod, "quotient" (by Gabriel topology, a.k.a. hereditary torsion theory) = Grothendieck category, i.e. cocomplete Abelian category in which direct limits are exact and there is a generator. By the Lubkin-Heron-Freyd-Mitchell theorems, Abelian categories embed fully faithfully in Grothendieck categories but are more general. Assuming this parallel Grothendieck toposes || Grothendieck categories is a good one, is there a natural parallel of Abelian categories on the Set-enriched side? Steve Vickers.