Section and retraction

Remarks

Every split monomorphism is a monomorphism: if $f \circ g_1 = f \circ g_2$, then $g_1 = r \circ f \circ g_1 = r \circ f \circ g_2 = g_2$. Dually, every split epimorphism is an epimorphism. Example: in the category of sets, the inclusion $f \colon \set{0} \to \set{0, 1}$ and the constant map $r \colon \set{0, 1} \to \set{0}$ satisfy $r \circ f = 1_{\set{0}}$. Non-example: in the category of abelian groups, the map $f \colon \Z \to \Z$, $f(x) = 2x$, is a monomorphism but not a split monomorphism. A retraction $r$ would satisfy $r(2) = 1$, so $2 r(1) = 1$ in $\Z$, which is impossible. A morphism is an isomorphism exactly when it is both a split monomorphism and an epimorphism. In the category of sets, the statement that every surjection is a split epimorphism is equivalent to the axiom of choice.