3
$\begingroup$

An epimorphism $f$ is said to be extremal, if for any decomposition $f=i\circ p$ with $i$ a monomorphism, the morphism $i$ is automatically an isomorphism. (This is from the textbook by F.Borceux.)

Let us say that $f$ is weakly extremal, if for any decomposition $f=i\circ p$ with $i$ a monomorphism and $p$ an epimorphism, the morphism $i$ is automatically an isomorphism.

Are these definitions equivalent?

$\endgroup$
3
  • $\begingroup$ Which categories have you tried so far? $\endgroup$ Nov 22, 2011 at 21:33
  • $\begingroup$ Of course, this must be the same in some categories. For example, in Abelian categories, or, more generally, in categories where every morphism $f$ can be represented as a composition $f=m\circ g\circ e$, with $m$ a strong monomorphism, $e$ a strong epimorphism, and $g$ a bimorphism (see details in arxiv.org/abs/1110.2013, Theorem 1.3; unfortunately, in Russian). But I do not know counterexamples at all. Perhaps, these definitions are equivalent in any category? $\endgroup$ Nov 22, 2011 at 22:37
  • $\begingroup$ Thanks for the background. I strongly believe that there are counterexamples, but I haven't found one. After all, it may happen that every epi is an iso? What about considering categories given by generators and relations, perhaps even finite ones? $\endgroup$ Nov 23, 2011 at 8:46

1 Answer 1

4
$\begingroup$

A counterexample:

Consider the monoid $\langle a,b,c\mid ac=bc\rangle$ as a category with one object. Then $a,b$ are monics, $bc$ is an epic and $c$ is not an epic. So $bc$ is not an extremal epic, but it is easily to see weakly extremal.

$\endgroup$
7
  • $\begingroup$ Any extremal epi is weakly extremal. Do you mean the opposite? $\endgroup$ Nov 23, 2011 at 17:51
  • $\begingroup$ I don't see why this works. Clearly bc is not an extremal epi since b is not an iso (only 1 is an iso). Why is it weakly extremal? $\endgroup$ Nov 23, 2011 at 17:53
  • $\begingroup$ Sorry, I meant a is not an iso. $\endgroup$ Nov 23, 2011 at 17:55
  • $\begingroup$ Oh, I see it is weakly extremal because it cannot be written as a monic times an epi. Very nice. $\endgroup$ Nov 23, 2011 at 17:57
  • $\begingroup$ @ Benjamin: Thank you very much. I mixed and correct the answer now. $\endgroup$ Nov 23, 2011 at 19:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.