Generators and Colimit Closures

The only proof of this I know is kind of fiddly and not enlightening. But the point is that in most categories that appear ‘in nature,’ regular, strong, and extremal epics are about the same, and all are the correct notion of quotient, while ordinary epimorphisms may not be. For example, N→ Q is epic in Rings, but not extremal epic. In Top the epics are the surjective maps, while the regular = strong = extremal epics are the quotient maps. If you were at Emily’s talk on factorization systems, it is natural to wonder: since strong epics are left orthogonal to monics, i.e. StrEpi = ⊥Mono, when is (StrEpi ,Mono) an orthogonal factorization system? It would suffice to show that any map factors as an extremal epic followed by a monic. Here’s a natural way to do that: given f : X → Y , take its kernel pair, which is the pullback