Many definitions of regular category have been given. The original one required that every arrow have a kernel pair (it will not do to call it a kernel since that has another, and related, meaning in abelian categories), that the kernel pair of every arrow have a coequalizer and that every coterminous pair of arrows of which one is a regular epi (a coequalizer of a kernel pair) have a pullback in which the arrow opposite the regular epi is regular epi. This definition can be both strengthened and weakened (and has been, including by me), while retaining its essential intent. Strengthenings: You can assume the category has more (finite) limits or more coequalizers. All the way up to all finite limits and finite colimits. Weakenings: You can define a regular epi as I did last fall, in terms of pairs of arrows. You can force it to be stable by requiring that it be a cover in the finest topology for which the representable functors are sheaves. You can then require that every arrow factor as a regular epi followed by a mono. I think that if I were starting from scratch, I would use the latter definition and then strengthen it as needed. The full embedding theorem is certainly true in that context. As for examples, that's harder. I haven't worked out the details, but I think that finite ordinals and order-preserving maps should be an example. If you stick to non-zero ordinals, then this is the standard simplicial category. All the monics and all the epics are split and hence regular in any definition, but I don't think the epis have kernel pairs, although coequalizers probably exist. The point is that there is no reason to constrict the notion, since you can add whatever you need when you need it. In a regular category, it is certainly equivalent to say that every epi is regular and that every monic epi is an isomorphism. Moreover, it works with the weakest definition, since you need only take an epi and factor it as a regular epi followed by a mono. Then the mono is the second factor of an epi and hence also epic and so if every monic epi is an ismorphism, the original is, up to isomorphism, the regular epi part. The converse is even easier. I thought the definition of pretopos implied regular. Michael ==========================================================================