Re: Exactness without pullbacks

Re: Exactness without pullbacks

[Michael Barr]

My recollection is that in the original definition only pullbacks of
regular epis as well as kernel pairs were assumed to exist.  Although you
could just assume that when the pullback of a regular epi exists it is a
regular epic, I think that would vitiate the definition.  However, one
possibility that I have known of for a long time but not written about is
to suppose that when A --> B is regular epic and B' --> B is arbitrary and
you look at all pairs A' --> A, A' --> B' that make the evident square
commute, then the family of all those A' --> B' is an effective epic
family.  In that category, a pullback, if it exists, is terminal.

On Thu, 18 Jan 2007, Toby Bartels wrote:

> Has anybody considered (and are there any references with standard results)
> categories that do *not* have *all* pullbacks
> but nevertheless have some nice exactness properties?
>
> For example, instead of saying that regular epis are stable under pullback
> (so that the pullback of a regular epi along any map is also regular-epic),
> I might say that any pullback of a regular epi is regular-epic *if* it exists.
> (I might instead use a weaker variant, requiring this only in the case
> that *all* pullbacks of the regular epi in question exist;
> or else requiring that all pullbacks of *all* regular epis exist,
> yielding a stronger variant).
>
> For a more specific example, the category of smooth manifolds
> misses many pullbacks but has the property above (at least the weaker form;
> as I recall, the surjective submersions are precisely those regular epis
> that have all pullbacks, but I forget if any other regular epis exist;
> in any case, the pullback of a surjective submersion along any smooth map
> exists and is also surjective-submersive).
>
>
> --Toby
>
>

Re: Exactness without pullbacks

[Eduardo Dubuc]

M. Barr wrote (in part, concerning the question of defining the stability
of a regular epi under pull-backs without pull-backs)

>
> However, one
> possibility that I have known of for a long time but not written about
> is
> to suppose that when A --> B is regular epic and B' --> B is
> arbitrary and
> you look at all pairs A' --> A, A' --> B' that make the evident square
> commute, then the family of all those A' --> B' is an effective epic
> family.  In that category, a pullback, if it exists, is terminal.
>
refer to this property as (*)

Well, (*) is the same of what I wrote in my posting in the subject:

(*):
> you can say that a strict epi is "stable under pullbacks" also in
> the
> absence of pullbacks:
>
>                    Z_i -------> X
>                     |           |
>                     |f_i        |f
>                    \/     h     \/
>                     Z --------> Y
>
> a strict epi  f  is  universal  if  given any  h  there exists a strict
> epi family f_i as indicated in the diagram.
>
> this exactness property is as good as stability under pullbacks
> see the links
>
> http://arXiv.org/abs/math/0611701
>
> http://arXiv.org/abs/math/0612727
>

Of course, it is the same if we are talking of the same thing. That we
are.

When I say "strict", I mean it in the sense of SGA4 Expose I, 10.2 10.3,
and we should assume that it coincides with what M. Barr calls
"effective". Contrary to M. Barr terminology, "effective" is also utilizad
in SGA4, presicely, when the kernel pair exists !

Concerning the above notion (*) of "stability under pull-backs without
pull-backs" (an instance of "universality"), it is also defined in
SGA4 Expose II 2.5, and it is simply the following:

an arrow F: X ---> Y (singleton family) is a strict universal epimorphism
if it is a cover for the canonical topology.

In Proposition 2.6 it is stablished the characterization of strict
universal epimorphisms by the property (*) above.

e.d.

Re: Exactness without pullbacks

[Michael Barr]

Of course I meant the definition in LNM#236.  However, I don't have the
original source at home anyway, so I would have to wait to check it.

Re: Exactness without pullbacks

[Eduardo Dubuc]

>
> Has anybody considered (and are there any references with standard results)
> categories that do *not* have *all* pullbacks
> but nevertheless have some nice exactness properties?
>
> For example, instead of saying that regular epis are stable under pullback
> (so that the pullback of a regular epi along any map is also regular-epic),


grothendieck notion of strict epi (SGA4) is equivalent to the notion of
regular epi in the presence of the kernel-pair, but it makes
sense in the absence of pull-backs.

you can say that a strict epi is "stable under pullbacks" also in the
absence of pullbacks:

                     Z_i -------> X
                      |           |
                      |f_i        |f
                     \/     h     \/
                      Z --------> Y

a strict epi  f  is  universal  if  given any  h  there exists a strict
epi family f_i as indicated in the diagram.

this exactness property is as good as stability under pullbacks

see the links

http://arXiv.org/abs/math/0611701

http://arXiv.org/abs/math/0612727

i am afraid thought that you have different examples in mind.

eduardo j. dubuc

Re: Exactness without pullbacks

[Toby Bartels]

Michael Barr wrote:

>Toby Bartels wrote:

>>Has anybody considered (and are there any references with standard results)
>>categories that do *not* have *all* pullbacks
>>but nevertheless have some nice exactness properties?

>My recollection is that in the original definition only pullbacks of
>regular epis as well as kernel pairs were assumed to exist.

By "the original definition", you mean the definitions here?:
 Michael Barr, Exact categories,
 in Exact Categories and Categories of Sheaves,
 Lecture Notes in Mathematics 236, Springer-Verlag, 1971.
I've never read this, since you-exact categories are now standard,
but I guess that one should always go back to the source!


--Toby

Exactness without pullbacks

[Toby Bartels]

Has anybody considered (and are there any references with standard results)
categories that do *not* have *all* pullbacks
but nevertheless have some nice exactness properties?

For example, instead of saying that regular epis are stable under pullback
(so that the pullback of a regular epi along any map is also regular-epic),
I might say that any pullback of a regular epi is regular-epic *if* it exists.
(I might instead use a weaker variant, requiring this only in the case
that *all* pullbacks of the regular epi in question exist;
or else requiring that all pullbacks of *all* regular epis exist,
yielding a stronger variant).

For a more specific example, the category of smooth manifolds
misses many pullbacks but has the property above (at least the weaker form;
as I recall, the surjective submersions are precisely those regular epis
that have all pullbacks, but I forget if any other regular epis exist;
in any case, the pullback of a surjective submersion along any smooth map
exists and is also surjective-submersive).


--Toby