Subject: Abelian-topos (AT) categories

From: categories <[email-masked]>
Date: Sun, 2 Nov 1997 14:53:51 -0400 (AST)

Date: Sun, 02 Nov 1997 06:52:02 -0800
From: Vaughan R. Pratt <[email-masked]>

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

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From: categories <[email-masked]>
Date: Mon, 3 Nov 1997 15:44:30 -0400 (AST)

Date: Mon, 3 Nov 1997 10:00:59 -0500 (EST)
From: Peter Freyd <[email-masked]>

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.

---

From: categories <[email-masked]>
Date: Tue, 4 Nov 1997 08:17:20 -0400 (AST)

Date: Mon, 03 Nov 1997 21:57:20 -0800
From: Vaughan R. Pratt <[email-masked]>

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

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From: categories <[email-masked]>
Date: Wed, 5 Nov 1997 17:35:26 -0400 (AST)

Date: Wed, 5 Nov 1997 11:11:52 +0000
From: Steven Vickers <[email-masked]>

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.

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From: categories <[email-masked]>
Date: Thu, 6 Nov 1997 16:36:17 -0400 (AST)

Date: Thu, 6 Nov 1997 10:39:35 GMT
From: Michael Barr <[email-masked]>

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

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From: categories <[email-masked]>
Date: Fri, 7 Nov 1997 14:45:36 -0400 (AST)

Date: Thu, 6 Nov 1997 16:38:20 -0500 (EST)
From: Peter Freyd <[email-masked]>

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.