Surely a natural name for these "abelian-topos" categories would be abelian-topos categories, AT categories for short. (I barely grasp them, I only found out what abelian categories were a week ago, when someone on sci.math asked about them and I looked them up and answered him because I wanted to know too. It struck me as interesting that they had so much in common with toposes, whence my question about intersecting their respective theories. I made language a parameter because it seemed intuitively obvious that a strong enough language would make the models of that intersection the union of the respective classes. But as Peter points out this is already a triviality for any theory closed under classical disjunction, which I still can't believe I didn't know.) On the question of the right language for defining AT, I fully agree with Peter that universal Horn sentences are the appropriate logical strength of language. (I take back what I said before about "universal" not making a difference. Peter showed that for the Horn theory, i.e. allowing existential quantification, the models are representable as products AxT of an abelian category A with a pretopos T. Presumably the weaker universal Horn theory admits in addition the appropriate subcategories thereof.) So would it be correct to say that this makes AT a quasivariety in CAT with functors preserving the signature Peter is using? What I'm less clear about than the logical strength is the choice of signature. In particular what about the closed structure? Without the closed structure we have Peter's AxT representation theorem, with the theory finitely axiomatized to boot. Presumably this remains unchanged by the introduction of closed structure: we retain only those categories AxT such that A and T independently admit closed structure, and can finitely axiomatize the separate closed structure of each in terms of the associated retractions A and T (ugh, overloading), thereby axiomatizing the joint closed structure. For example the tensor unit I of AxT will be (Z,1) (Z the tensor unit of A), with AI = (Z,0) != I (except for pure abelian categories) and TI = (0,1) = 1 != I (except for pure pretoposes). But for objects a,b of A and t,u of T, the tensor product (a,t)@(b,u) will be (a@b,txu), with A(a@b,txu) = (a@b,0) = (a,0)@(b,0) = A(a,t)@A(b,u) and T(a@b,txu) = (0,txu) = (0,t)@(0,u) = T(a,t)@T(b,u). (So the retractions preserve tensor product but not tensor unit---TI just gives the terminator, but AI furnishes a new constant.) So should there perhaps be two classes, the AT categories and the closed AT categories? The latter would add I and @ to the signature (and presumably \aleph\lambda\rho, bless them). For the closed AT categories, TX can be neatly defined as 1@X, with the T-type objects identified as those having only one map to I (Mike Barr pointed out to me that strictness of 0, maps to 0 only from initial objects, would do this job), and with a topos defined as a closed AT category for which I is terminal (or 0 is strict). But although people seem comfortable working with toposes as opposed to pretoposes, what about abelian closed categories (if that's the right word order)? Despite Ab itself being a closed category, I don't see much discussion of closed structure for abelian categories. Why is this? Am I just reading the wrong stuff, or is the closed structure of abelian categories boring, or what? Or is FinAb (abelian but not abelian closed---no suitable tensor unit) too desirable to discard in this way? (What comparably interesting pretoposes are so lost? Not the topos FinSet.) Does the requirement of being closed kill off too many desirable abelian categories? I would have thought lack of closed structure would greatly impair the utility of a category, however beautiful its objects might be. I'm interested in these classes, especially those whose categories admit closed structure, because toposes sit at or near the left (geometric or discrete) end of what I've been calling the Stone gamut, while abelian categories sit near the middle. AT categories offer an entirely different approach to the Chu construction for mixing categories from strategic positions on the Stone gamut. The Chu construction Chu(V,k) mixes two closed categories, V and V\op, symmetrically positioned about the center of the Stone gamut, to yield all categories "in between" V and V\op (and "all"--certainly all small--categories period when V and V\op are at the outermost points, viz. Set and Set\op). In contrast AT categories mix categories from the far left (represented by Set) and the center (represented by Ab) to get a qualitatively different effect that I'm not sure how to relate to the Chu construction but which seems in some vague sense dual to it. One such sense is as follows. AT is the quasivariety generated just by Set and Ab alone. So in this sense at least it is the smallest quasivariety spanning the left half of the Stone gamut, assuming that the quasivariety generated by either Set or Ab alone does not span the gamut but crowds around their two respective positions on the gamut, left and middle. Include the duals of AT categories (not necessarily expanded to a quasivariety, see below) and now you cover the whole gamut in this minimal sense. On the other hand the various comprehensiveness properties I've been pointing out for the concrete subcategories of Chu(Set,K) for large enough K (including all small categories, even when concreteness is a requirement unlike the comprehensiveness results of the 1960's) make "sub-Chu" maximal over the Stone gamut. Vague question. The minimality of AT and the maximality of Chu is a very weak sort of duality, analogous to the minimal structure of sets vs. the maximal structure of Boolean algebras. Is there a more formal duality here, analogous to the duality of Set and CABA? That the retracts AX and TX seem to be dual notions, being respectively coreflective and reflective, gives them some of the flavor of Chu. But is AT itself dual to Chu in any categorical sense? More precise question. Is the quasivariety generated by all three of Set, Ab, and Set\op (aka CABA) finitely axiomatizable? And if not, does using Bool instead of CABA help or hinder? Vaughan