Re: Exactness without pullbacks

[Michael Barr <barr@barrs.org>]

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
>
>