For David <dkarapetyan@ucdavis.edu> Karapetyan, who asked,
... the inclusion function of N into Z provides a counterexample [to] every epic arrow being a surjective function. I noticed that N is just a "folded" version of Z, where by "folded" I mean take Z and throw away all the inverses of the natural numbers. So does every monic, epic arrow determine such a "folding" or are there monic, epics that can't be characterized in such a way?
let me offer two further examples of monic epic arrows, not surjective (and both pretty standard): 1) in Hausdorff topological spaces, the inclusion of the rationals in the reals; 2) in boolean rngs (i.e., units not required, and not necessarily preserved when present) with countable intersections, and boolean homomorphisms preserving those intersections, the inclusion of the boolean rng of finite subsets of N in the whole power-set of N (this is epic because boolean homomorphisms (between such boolean rngs) that preserve countable intersections will also preserve whatever countable joins may be available, and every subset of N is the join of all its finite subsets). Does your "folding" insight still stand up? Or must it be modified? -- Fred Linton