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. On Tue, 6 Mar 2007, David Karapetyan wrote:
Hi, I've been trying to learn some category theory and I came upon the example of a monic, epic in the category of monoids given by the inclusion function of (N,0,+) into (Z,0,+). I know that in monoids every monic arrow is also an injective function but the inclusion function of N into Z provides a counterexample of 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?
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