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