Re: monic epics

[David Karapetyan <dkarapetyan@ucdavis.edu>]

Steve Vickers wrote:
> 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.
>
i like the example of the semigroups. i think in some sense the addition
of d does not add enough information to our monoid so if we have two
semigroup homomorphisms that agree on A then by using the properties of
semigroup homomorphisms we are forced to define how they behave on d in
only one way. i'm still trying to incorporate the inclusion of the 2x2
upper triangular matrices into all 2x2 matrices and Michael Barr's
example into this framework.