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