1
$\begingroup$

Take a morphism $f:X\to X$ in a category with the same domain and codomain. I want to test whether $f$ is a monomorphism. This means, taking arbitrary $g_1,g_2:Y \to X$ with $f\circ g_1=f\circ g_2$ it should follow that $g_1=g_2$.

Does it suffice for a retraction $f:X\to X$ to be a monomorphism that for all $g_1,g_2:X \to X$ with $f\circ g_1=f\circ g_2$ it follows that $g_1=g_2$?

  • 1
    The last sentence doesn't make any sense to me. The condition "for all $g_1,g_2:X\to X$ it follows that $g_1=g_2$" has nothing to do with $f$.2012-03-03
  • 0
    Sorry, typo. Fixed.2012-03-03
  • 0
    If your category is additive (don't actually remember if additive is enough, but if you go with "abelian" then you're sure) it is enough that $\ker f =0$.2012-03-03

1 Answers 1