Date: Mon, 27 Oct 1997 16:09:44 -0400 (AST)
Subject: Topos-abelian categories that are neither
Date: Mon, 27 Oct 1997 10:19:51 -0800
From: Vaughan Pratt
Toposes and abelian categories are striking for the number of
elementary properties they have in common (monic+epi = iso, mono/epi is
a unique factorization system, etc. etc.) and the paucity of their
common models, namely just the final category.
This observation prompts the following questions.
1. Are there any other pairs of large and/or useful classes of
categories whose respective theories have so much in common yet whose
models have so little in common?
2. What can be said of the class of those categories having all the
elementary properties common to toposes and abelian categories? In
particular does it contain anything other than toposes and abelian
categories? And if so, does this outcome change when the language is
extended to say second order logic?
To the extent that both toposes and abelian categories share much
pleasant structure, the models of the intersection of their theories,
for a suitable choice of language, would seem to be a nice class in its
own right.
Vaughan Pratt
Date: Thu, 30 Oct 1997 22:33:29 -0400 (AST)
Subject: pratt cat
Date: Thu, 30 Oct 1997 16:39:48 -0500 (EST)
From: Peter Freyd
Vaughan wrote:
To the extent that both toposes and abelian categories share much
pleasant structure, the models of the intersection of their
theories, for a suitable choice of language, would seem to be a nice
class in its own right.
There are several variations on the topic. I'll look first at the
theory obtained by starting with
finite completeness,
coterminator,
pushouts for pairs of maps one of which is monic,
pushouts for kernel pairs
then adjoining all universal Horn sentences in these predicates that
hold for both the category of abelian groups and the category of sets
(or, if prefered, hold for all abelian categories and all pre-topoi).
We will see that this theory can be finitely axiomatized and that the
representations of its models into the categories of abelian groups
and sets are collectively faithful. This will be the "first task.
The representation will be done by finding a single faithful
representation into the product of an abelian category and a topos and
using known theorems thereon. By a representation I here mean a
functor that preserves finite limits, coterminators, pushouts of pairs
of maps one of which is monic and pushouts of kernel pairs. (It
follows that a representation preserves epimorphisms, finite
coproducts and images.)
If we expand the theory to include all positive AE sentences it
remains finitely axiomatizable, and the only models are those
categories that are a product of an abelian category and a pre-topos.
This will be the "second task".
After that, I'll add some more essentially algebraic structure so that
we can talk about power-objects using universal Horn sentences and
obtain again the finite axiomatizability and the fact that the the
only models are those categories that are a product of an abelian
category and a pre-topos. This will be the "third task".
Note that each of axioms below is true for both abelian categories and
topoi. Until we reach axiom VAE, all can be turned into universal
Horn sentences if the completeness conditions listed above are turned
into an essentially algebraic theory.
V1) Effective regular category. (Yes this can be stated as universal
Horn conditions on pullbacks and the special pushouts mentioned
above.)
V2) 0 -> 1 is monic.
Note that it follows that all maps from 0 are monic.
In the diagrams below all non-horizontal lines should have arrows
added in downwards direction.
V3) If A -> C is monic and
A
/ \
B C
\ /
D
is a pushout then it's a pullback,
V4) and so is
0xA
/ \
0xB 0xC
\ /
0xD.
Note that binary coproducts therefore exist and are disjoint.
V5) If B -> 0xC is epic and
A
/ \
0 B
\ /
0xC
is a pullback then it's a pushout.
Two definitions: let
0 x X
/ \
0 X
be a product diagram. If the map to 0 is an iso, I'll say
that X is type-T; if the map to X is an iso, I'll say that
it's type-A.
Note that (using V2) an object is type-A iff it has a map to 0. The
type-A objects are closed under binary products and coproducts,
subobjects and quotient objects. The full subcategory of A objects is
coreflective, with AX = 0xX as coreflector. Most important, it is an
abelian category.
V5) If the rhombi in
A B
|\ /|
D C E
\ | /
F
are pullbacks and if
D E
\ /
F
is a coproduct diagram, and if all the objects are type-T
then
A B
\ /
C
is also a coproduct diagram.
Note that an object is type-T iff every type-A object has a unique map
thereto. Type-T objects are closed under binary products and
coproducts, subobjects and quotient objects. (For coproducts use V3.)
The full subcategory of type-T objects forms a reflective subcategory:
the reflector, T, is defined via the pushout diagram:
0xX
/ \
0 X
\ /
TX
One now has what's needed to show that the full subcategory of type-T
objects is a pre-topos.
Note that the functor A preserves pullbacks and pushouts
of pairs of maps one of which is monic (V4). Add
V6) T preserves pullbacks.
(It clearly preserves pushouts.)
Only one more axiom is needed for the first task:
V7) If f is a map such that Af and Tf are both isos then
f is an iso.
For the second task add:
VAE) For every X there's a map TX -> X such that
AX TX
\ /
X
is a coproduct diagram.
Note that TX -> X is a coreflector (making V6 redundant). One can
force the map to be unique by further requiring that TX -> X -> TX
is the identity map. And note that the type-A objects can not be
reflective: if 1 has a map to any type-A object the entire category
collapses.
For the third task, I'll add the following structure: functions
on objects P,E,l, r such that lX:EX -> PX and rX:EX -> X and
V8) :EX -> PXxX is monic and PX, EX are type-T.
I need one more operation, denoted with an upcase lambda, here
written as /\. Given a pair of maps f:R -> Y, g:R -> X
where R is of type-T, then /\(f,g): Y -> PX is defined and
V9> The composition of relations
/\(f,g) (lX)' rX
Y ----------> PX -------> EX ---------> X
is equal to
f' g
Y -------> R --------> X
(where ' is being used to denote the reciprocation operator on
relations).
V10) If
R
f / \ g
Y EX
h \ /lX
TX
is a pullback and R of type-T, then /\(f,(rX)g) = h.
Note that in the full category of type-T objects, P yields
power-objects with E, l, r naming the universal relations. (V10
provides the uniqueness condition.)
In any abelian category the only type-T object is the zero
object, which forces P=E=0 for abelian categories.
It's routine that both A and T preserve the new structure. We can
now remove the existential from VAE. Define TX -> X as the image of
rX. The third task is finished with:
V11) For every X
AX TX
\ /
X
is a coproduct diagram.
These axioms will, no doubt, be much simplified.
Date: Fri, 31 Oct 1997 14:36:29 -0400 (AST)
Subject: Re: Topos-abelian categories that are neither
[Note from moderator: this is two postings from Peter...an error on my
part resulted in the first not being circulated until now.]
Date: Tue, 28 Oct 1997 08:50:20 -0500 (EST)
From: Peter Freyd
Vaughan points out the many remarkable similarities between topoi and
abelian categories. I've always thought that the most remarkable is
the fact that if one forms the pushout of a pair of maps one of which
is monic then the result is a pullback. The known proofs for the two
cases are remarkably different. When both maps are monic this is known
as the amalgamation property (at least it's so known in the category
of belian groups, that is, groups abelian or not).
It's also right here that a big difference shows up. For abelian cats
the lemma remains true when one relaxes the hypothesis from "a pair of
maps one of which is monic" to "a pair of maps that are jointly
monic." Such a lemma is very wrong for topoi. (For a counterexample in
sets pushout any jointly monic pair of maps from a 3-element set. The
result is a pullback iff one of the given maps is already monic. Such
a counterexample sits, in fact, in any non-degenerate topos.)
Vaughan asked:
What can be said of the class of those categories having all the
elementary properties common to toposes and abelian categories? In
particular does it contain anything other than toposes and abelian
categories? And if so, does this outcome change when the language
is extended to say second order logic?
The answer to the second question is no (hence so is the answer to
the third sentence).
There is a complete answer to the first question. You won't like it.
For a 1-sentence axiom of pratt categories first take a 1-sentence
elementary axiom, T, for elementary topoi and a 1-sentence elementary
axiom, A, for abelian categories
Then take the sentence:
T or A.
Sorry.
Vaughan goes on to say:
To the extent that both toposes and abelian categories share much
pleasant structure, the models of the intersection of their
theories, for a suitable choice of language, would seem to be a nice
class in its own right.
This strikes me as an interesting topic (and perhaps we should use the
phrase "pratt categories" for what emerges as the right choice of this
class). My first choice for the suitable choice of language would be
the set of universal Horn sentences in the predicates of finite limits
and colimits (where it is understood that we already have the axioms
for finite bicompleteness). Can one prove the expected representation
theorem: does every pratt category have a faithful limit- and colimit-
preserving functor into a product of an abelian category and a topos?
Date: Fri, 31 Oct 1997 05:35:00 -0500 (EST)
From: Peter Freyd
To: cat-dist@mta.ca
Somehow my first answer to Vaughan's question went astray. He
had asked:
What can be said of the class of those categories having all the
elementary properties common to toposes and abelian categories? In
particular does it contain anything other than toposes and abelian
categories? And if so, does this outcome change when the language
is extended to say second order logic?
The answer to the second question is no, hence so is the answer to
the third. And you won't like the answer to the first.
Given any two elementary theories, *A* and *T*, let *AoT* be the
set of all sentences of the form "A or T", where "A" is a sentence
in *A* and "T" is a sentence in *T*. Clearly every model of *A*
is a model of *AoT* and so is every model of *T*.
Almost as clearly: every model of *AoT* is either a model of *A*
or of *T*.
Since the elementary theories of abelian cats and topoi can each be
finitely axiomatized, there's a single elementary sentence common to
topoi and abelian cats whose only models are either topoi or abelian
cats.
* * *
A few misstatements from my previous post. My attempt to abbreviate
V4 didn't work. I want to say that the diagram is a pushout (of
course it's a pullback). So it should be:
V4) If A -> C is monic and
A
/ \
B C
\ /
D
is a pushout then and so is
0xA
/ \
0xB 0xC
\ /
0xD.
I wrote:
And note that the type-A objects can not be reflective: if 1 has a
map to any type-A object the entire category collapses.
I should have written:
And note that the type-A objects can not be reflective unless all
objects are type-A: if 1 has a map to any type-A object then
0 = 1.
Finally (I must be kidding), as it stands the P-E-l-r-/\ structure
is not properly fixed for abelian categories. Adjust as follows: /\
is defined for pairs f:R -> Y, g:R -> X only when both R and Y
are of type-T. Note that the equation in V9 is a directed equality: if
/\(f,g) is defined then the equality holds. V10 must be modified by
strengthening the hypothesis to include the condition that Y is
type-T (at which point the condition on R becomes redundant).
Date: Fri, 31 Oct 1997 15:40:18 -0400 (AST)
Subject: Re: pratt cat
Date: Fri, 31 Oct 1997 01:03:22 -0800
From: Vaughan R. Pratt
Peter's response to my question about categories having all the common
properties of toposes and abelian categories was a very pleasant
surprise.
First I was not at all expecting such a definite answer.
Second I did not expect a representation theorem, especially not such a
nice one as "every such category is a product of an abelian category
and a topos". This is the converse of the automatic (by Horn) fact
that such a product is such a category, where (for this converse at
least) the language can be taken to be as large as the full Horn theory
common to abelian categories and toposes.
Third I did not expect at all that the class would be finitely
axiomatizable (but on the other hand neither did I have any reason to
suppose not).
Fourth and very nice, unless I'm overlooking something, Peter's
representation theorem implies that my original question (does the
common *elementary* theory allow anything other than abelian categories
and toposes?) can now be answered in the negative. The elementary
sentence "either all the objects are A-type or all the objects are
T-type" is obviously true in both abelian categories and (pre)toposes,
and rules out hybrids like Set x Ab. I'd been wondering what
elementary sentence might fail for that example, and Peter's A-type and
T-type constructs lead straight to it.
Minor remarks:
The trick of multiplying by 0 to tease out both the A- and T-components
at once is beautiful, not just intrinsically but also because it
reveals a certain duality between A and T that is not at all apparent
(to me anyway) from their separate definitions. The actual
multiplication projects out the T part without disturbing the A part,
and then the pushout
AX (where AX = 0xX)
/ \
0 X
\ /
TX
neatly quotients out that surviving A part by forcing the left side
through the zero object while on the right AX->X communicates only the
A part which is therefore the only part that the pushout quotients out
from 0 + X (if that's not too clumsy a way of putting it).
The upshot is then that the AX's form (as an abelian category) a
coreflective subcategory of the whole while the TX's form (as a
pretopos) a reflective subcategory. That's really beautiful!
The following is presumably the key to proving completeness of Peter's
axiomatization...
>VAE) For every X there's a map TX -> X such that
>
> AX TX
> \ /
> X
...by permitting any model to be constructed directly as a
product of an abelian category and a topos (not even "subcategory"
seems to be needed here, so this is an even stronger representation
theorem than the representation of a Boolean algebra as a subalgebra of
a power set).
>And note that the type-A objects can not be
>reflective: if 1 has a map to any type-A object the entire category
>collapses.
Whoops, a bug here if I'm understanding things. Only the T part
collapses, if the category were abelian to begin with then there'd be
no collapse.
>Define TX -> X as the image of rX.
This was lovely (getting rid of the E in VAE above).
Some questions.
1. Can this representation theorem be extended to the full Horn
theory, i.e. any number of alternations of quantifiers? (Note that my
disjunction---every object is A-type or every object is T-type---is not
a Horn sentence.) (Now that I look again, I guess the answer must be
yes, since products preserve *all* Horn sentences.)
2. In a text that treated both toposes and abelian categories, would
your axiomatization be a good starting point prior to doing either?
You could then quickly define an abelian category to be a category of
this general type with the additional property of being strongly
connected. I don't see right off how to define a pretopos as slickly.
3. What about bringing in the closed structure? Might that permit
some simplification of your axiomatization? For example it looks as
though you could define a T-type object to be one having a unique map
to the tensor unit. Then you could answer the last sentence of the
previous paragraph simply with "the tensor unit is terminal."
Vaughan Pratt
Date: Sun, 2 Nov 1997 14:54:55 -0400 (AST)
Subject: Abelian-topos (AT) categories
Date: Sun, 02 Nov 1997 06:52:02 -0800
From: Vaughan R. Pratt
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
Date: Mon, 3 Nov 1997 15:44:52 -0400 (AST)
Subject: Re: Abelian-topos (AT) categories
Date: Mon, 3 Nov 1997 10:00:59 -0500 (EST)
From: Peter Freyd
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.
Date: Tue, 4 Nov 1997 08:17:51 -0400 (AST)
Subject: Re: Abelian-topos (AT) categories
Date: Mon, 03 Nov 1997 21:57:20 -0800
From: Vaughan R. Pratt
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
Date: Wed, 5 Nov 1997 17:35:52 -0400 (AST)
Subject: Re: Abelian-topos (AT) categories
Date: Wed, 5 Nov 1997 11:11:52 +0000
From: Steven Vickers
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.
Date: Thu, 6 Nov 1997 16:36:32 -0400 (AST)
Subject: Re: Abelian-topos (AT) categories
Date: Thu, 6 Nov 1997 10:39:35 GMT
From: Michael Barr
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
Date: Fri, 7 Nov 1997 14:46:40 -0400 (AST)
Subject: Re: Abelian-topos (AT) categories
Date: Thu, 6 Nov 1997 16:38:20 -0500 (EST)
From: Peter Freyd
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.
Date: Tue, 11 Nov 1997 17:19:27 -0400 (AST)
Subject: On a remark of Barr
Date: Tue, 11 Nov 1997 11:35:10 -0500 (EST)
From: Peter Freyd
Mike Barr on the question: "is every small topos fully representable
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?)
One can go further. The embedding will necessarily preserve all finite
colimits and all infinite coproducts that happen to exist.
Extend the Makkai observation as follows. For any pair of objects A
and B note that the lattice of partial maps with complemented
domains can be constructed as the set of maps from A to 1+B (give
1+B the "flat ordering" of CS: start with B with the trivial
partial ordering and adjoin a bottom). The fact that it's a complete
lattice is enough to show that arbitrary disjoint unions are
coproducts.
There is an argument for the case that the topos does not have a
natural numbers object but I'll assume here that it does have such.
The standard points of the natural numbers object are complemented,
hence their union must exist; that union is a coproduct; it must
therefore be the entire natural numbers object. It is preserved by
the embedding.
In Aspects of Topoi there's a lemma entitled "one coequalizer for
all" that now suffices to show that the representation preserves
all coequalizers.