The notion of epimorphism can be quite different from surjection, e.g. in Rings.
Though I recently learned epimorphisms can be characterized in terms of Isbell's zig-zags: https://en.wikipedia.org/wiki/Isbell%27s_zigzag_theorem.
Whereas monic seems to capture the notion of "injective" quite well in a categorical def. And indeed the two agree on any variety of algebras in the sense of universal algebra.
But! TIL there's a categorical definition that supposedly agrees w/ "surjection" on any variety of algebras:
h is "categorically surjective" (a term I just made up) if for any factorization h=fg with f monic, f must be an iso.
(h/t Knoebel's book https://doi.org/10.1007/978-0-8176-4642-4)
Are there categorical definitions that agree w/ injective (resp. surjective) on all concrete categories?
@JordiGH when you do that you precisely get the def of epimorphism.
The issue is that epimorphism and surjective agree a lot less often than monic and injective do. Eg the inclusion of Z into Q is epi in the category of rings, but not surjective.
Joshua Grochow
JordiGH