Weakened notion of extremal epimorphism? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T15:59:04Z http://mathoverflow.net/feeds/question/81645 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81645/weakened-notion-of-extremal-epimorphism Weakened notion of extremal epimorphism? Sergei Akbarov 2011-11-22T19:47:35Z 2011-11-23T19:30:54Z <p>An epimorphism $f$ is said to be <em>extremal</em>, 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.) </p> <p>Let us say that $f$ is <em>weakly extremal</em>, if for any decomposition $f=i\circ p$ with $i$ a monomorphism <em>and $p$ an epimorphism</em>, the morphism $i$ is automatically an isomorphism.</p> <p>Are these definitions equivalent?</p> http://mathoverflow.net/questions/81645/weakened-notion-of-extremal-epimorphism/81712#81712 Answer by Boris Novikov for Weakened notion of extremal epimorphism? Boris Novikov 2011-11-23T14:40:40Z 2011-11-23T19:30:54Z <p>A counterexample:</p> <p>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.</p>