Michael Barr wrote:
Category theory is too abstract for any such statement to be true (or even make sense). For example, in the category denoted . ---> . (with two objects and one non-identity map), that map is monic and epic for want of any test maps. More concretely, the inclusion of Z into R is both in the category of commtative rings. In fact the following a characterization of monic/epics in commutative rings is this: a subring R \inc S is epic iff every element of of S can be written s = vAw where for some n, v is an n-dimensional row vector, w is an n-dimensional column vector and A is an n x n matrix of elements of S such that the entries of A, vA, and Aw all belong to R. In general, very little can be said about monic/epics.
ok i got it. in all the examples given the subobjects given by the monics are "generators" for the object, where by "generators" i mean the elements of the subobject in some way determine the elements of the bigger object. so how about this then: any time we have the situation described above the monic arrow will also be epic.