Re: monic epics

[Michael Barr <barr@barrs.org>]

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?
>
>

-- 
Any society that would give up a little liberty to gain a little
security will deserve neither and lose both.

            Benjamin Franklin