Re: monic epics

[Steve Vickers <s.j.vickers@cs.bham.ac.uk>]

Dear David,

There are more complicated examples. Here's one.

Take A to be the semigroup {0,a,b,c} in which x0=0x=0 for all x, aa=a,
cc=c, ab = bc = b, and all other products are 0. (This of this as being
derived from a category with two objects and three morphisms, so a and c
represent the two identities and b is a morphism between the two
objects. 0 takes care of products of non-composable pairs.)

Take B = A u {d}, with cd = da = d, bd=a, db=c and all other binary
products involving d give 0. (Think of adjoining an inverse to b in the
category.)

Now the inclusion A -> B is a semigroup epi. To see this, suppose f: A
-> C is a semigroup homomorphism, and x in C satisfies

    xf(a) = f(c)x = x
    f(b)x = f(a)
    xf(b) = f(c)

Then x is the unique such. For if x' is another then

    x' = x'f(a) = x'f(b)x = f(c)x = x

If g: B -> C is a semigroup homomorphism agreeing with f on A, then g(d)
does satisfy those equations for x, and so any two such g's must be equal.

This semigroup example can be easily turned into monoids by adjoining a
unit element.

There is still the same idea that B is got by adjoining inverses to
elements of A, but they are not inverses in the monoid sense and it is
not clear to me in general how one would formalize the idea that they
are inverses when embedded in some category.

There is a related epi in rings: the inclusion of upper triangular 2x2
matrices (over any ring) into all 2x2 matrices.

Regards,

Steve Vickers.


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?
>
>