Split monomorphism

A split monomorphism $f:A\to B$ has a left inverse $r:B\to A$ with $rf=1_A$. Every functor preserves split monomorphisms.

An absolute monomorphism is a morphism sent to a monomorphism by every functor out of its category. The absolute monomorphisms are exactly the split monomorphisms.

An object $A$ is a retract of $B$ when there are morphisms $s:A\to B$ and $r:B\to A$ with $rs=1_A$. The map $s$ is a split monomorphism and $r$ is a split epimorphism.