Exactness without pullbacks

[Toby Bartels <toby+categories@math.ucr.edu>]

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