Re: Exactness without pullbacks

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

>
> 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