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
I've had trouble with starting a reply to Vaughan's most recent posting, the one headed, "Abelian-topos (AT) categories". It makes the wide gap that's come into being all too painful. Vaughan wrote, 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? And that was after he had written, I only found out what abelian categories were a week ago. For a long time Category Theory existed (say, in the Mathematical Reviews) as a subset of Homological Algebra -- which is a way of saying that category theory was abelian category theory. (I can't remember a new result in the theory of abelian categories in the last quarter-century. I do remember, alas, a bunch of new announcements of such results.) I have trouble with Vaughan's phrase "the closed structure" on an abelian category. There can be many. The category of finite- dimensional complex representations of a compact group has a distinguished symmetric monoidal closed structure. If they are viewed just as categories then for any two compact groups with the same number of conjugacy classes the categories are isomorphic. If viewed just as monoidal closed categories then the necessary and sufficient condition that they be isomorphic is that the groups have isomorphic character tables. On the other hand, if viewed as a _symmetric_ monoidal closed categories, one can recover the group from the category. If you want a specific example consider the two non-abelian groups of order eight, the dihedral and the quaternian. In each case the plain category of representations is the 5-fold cartesian power of the category of finite-dimensional complex vector spaces. As monoidal closed categories they are isomorphic (but not isomorphic with the 5-fold cartesian power of the closed monoidal category of finite- dimensional complex vector spaces). As symmetric monoidal closed categories they are different. Anyway, there's a whole body of material. A lot of it is now viewed as standard in a number of (non-categorical) subjects and as for all successfull branches of category theory the theory of abelian categories is no longer considered to be a branch of category theory. Back to pratt cats: if one wants to axiomatize those categories that are products of abelian cats and topoi and not worry about the intervening families the axioms can be made quite simple. After saying that it's a regular category with a coterminator contained in its terminator, I'd start with the P-E-l-r-/\ structure as in my last post, define TX as the image of rX and prove it to be the correflection of X into the full subcategory of type-T objects. 0xX -> X is easily seen to be the correflection of X into the full subcategory of type-A objects. Then the axiom that these two correflections yield a coproduct decomposition for each object allows one to prove quickly that the category is the cartesian product of the two correfletive subcategories. The type-T objects clearly form a topos. All that's needed now is a couple of axioms to make the type-A objects abelian.
I appreciate that there are people on the list with more years of experience with abelian categories than I have days. AC's don't seem to have penetrated much into computer science, and I have no idea whether they need to. But the finite axiomatizability of the quasivariety generated by Set and Ab definitely has my attention. And the fact that toposes and abelian categories, so far apart intuitively (sets vs. abelian groups?), are brought to within so short an axiom of each other by the definition of AT cats, has the potential to make abelian categories much more relevant to fans of toposes. Peter (and privately Fred Linton and Mike Barr) have answered my question about what I was naively calling "abelian closed". "Abelian" and "cartesian" are not interchangeable adjectives inasmuch as the latter describes the tensor product in the context of "cartesian closed" while the former names a quasivariety. While I was aware of the distinction, I was hoping that abelian categories as the models of the universal Horn theory of Ab, combined with Ab having closed structure, would somehow make the juxtaposition "abelian closed" meaningful, but the examples show this to be wishful thinking. And Peter's define TX as the image of rX removes any motivation to define TX as 1@X. (Meanwhile I've reconciled myself to TX as the pushout of the projections of 0xX.)
After saying that it's a regular category with a coterminator contained ^^^^^^^ [Is "effective" not needed? -v] in its terminator, I'd start with the P-E-l-r-/\ structure as in my last post, and prove it to be the correflection of X into the full subcategory of type-T objects. 0xX -> X is easily seen to be the correflection of X into the full subcategory of type-A objects. Then the axiom that these two correflections yield a coproduct decomposition for each object allows one to prove quickly that the category is the cartesian product of the two correfletive subcategories. The type-T objects clearly form a topos. All that's needed now is a couple of axioms to make the type-A objects abelian.
Not just finitely axiomatizable but beautifully so. Vaughan
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.
In reply to Steve Vickers' post, I have a few comments. First off, it was not Lubkin-Heron-Freyd-Mitchell who proved the full embedding theorem. The first three proved only a faithful functor into set (and subject to some smallness condition), while Mitchell showed the full embedding theorem. And not merely into a Grothendieck abelian category, but one with a small projective generator. The analogue would be an embedding into a set-valued functor category. Unfortunately, a beautiful observation of Makkai's shows that that is impossible. Makkai pointed out that under a full embedding that preserves finite limits, finite sums and epis the boolean algebra of complemented subobjects, which is classified by maps into 1 + 1, would have to be preserved. But in a functor category that lattice is complete and atomic (that is, completely distributive), so that fact, which is not true for toposes in general, becomes a necessary condition for the existence of an embedding. (Is it sufficient?) There is, of course, a full embedding theorem for small exact categories, but that is a lot less than a topos. But is true that additive + exact = abelian, so maybe that is also a good analogy. Michael
The result that Steve Vickers cited -- that every small abelian cateogory can be fully embedded into a Grothendieck category -- actually must have come before anything proved by Lubkin, Heron, Freyd or Mitchell. I'm sure that Grothendieck knew about the canonical representation of a small abelian category into its category of abelian pre-canonical sheaves.
participants (5)
-
Michael Barr -
Peter Freyd -
Steven Vickers -
Vaughan R. Pratt -
Vaughan R. Pratt