Re: monic epics

["David Karapetyan" <dkarapetyan@ucdavis.edu>]

> Yes.  Another common example of a morphism that is both a monomorphis and
> an
> epimorphism but not an isomorphism is the inclusion of the rational
> numbers
> into the real numbers in the category of topological spaces.

i understand that such arrows exist and i'm trying to get an intuitive feel
for why they are epic. one way i think of a surjective function is that it
is a map that entirely covers the codomain so any two function that agree
on all of the codomain must be the same. it is not the case with epic
arrows that they cover the entire codomain as set functions but that is i
think because most categories are much more structured than the category of
sets so it is enough to cover certain parts of the codomain and the rest of
the structure can be recovered. i think that is what happens with the
inclusion of the rationals into the reals because the reals are defined as
equivalence classes of sequences of rationals so if two functions agree on
the rationals and they are continuous then they automatically agree on the
reals.