Re: Exactness without pullbacks

[Eduardo Dubuc <edubuc@dm.uba.ar>]

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.