We demonstrate categories with maps that are epic but not surjective:
Theorem: In the category of monoids , the inclusion is epic
Proof: Suppose we have maps and . This essentially means that and are identical on positive integers. We show that this implies that they’re also identical on negative integers. Indeed, let . Then,
then, since , we have :
as desired. Since and are clearly identical on , we have for all , so .
Theorem: In the category of rings , the inclusion is epic.
Proof: Suppose we have maps for some ring , and . This means that and are identical on the integers. We show this implies they are identical on all rationals. Indeed, let be some fully reduced rational number. Then, we have:
and , so
as desired. So for all rational , and .