Monomorphism

A morphism $m:X\to Y$ in a Category $\mathcal{C}$ is a monomorphism if for all $f_{1},f_{2}:A\to X$, $m\circ f_{1}=m\circ f_{2}$ implies $f_{1}=f_{2}$.

Properties

• If $f:{{}(Y,Z)}$ and $g:{{}(X,Y)}$ are monic, then $f\circ g$ is monic.

• If $f\circ g$ is monic, then $g$ is monic.

  Let $h,k$ be a pair of morphisms with $g\circ h=g\circ k$; we need to show $h=k$. $f\circ g$ is monic, so it suffices to show that $f\circ g\circ h=f\circ g\circ k$. However, $g\circ h=g\circ k$ by assumption.

• Postcomposition with a monomorphism is Injective; this basically follows from the definition.