A split monomorphism in a category$C$ is a morphism$m\colon A \to B$ in $C$ such that there exists a morphism $r\colon B \to A$ such that the composite$r \circ m$ equals the identity morphism$1_A$. Then the morphism $r$, which satisfies the dual condition, is a split epimorphism.

A split monomorphism in $C$ can be equivalently defined as a morphism $m\colon A \to B$ such that for every object$X\colon C$, the function$C(m,X)$ is a surjection in $\mathbf{Set}$; a preimage of $1_A$ under $C(m,A)$ yields a retraction $r$.

Alternatively, it is also possible to define a split monomorphism as an absolute monomorphism: a morphism such that for every functor $F$ out of $C$, $F(m)$ is a monomorphism. From the definition as a morphism having a retraction, it is obvious that any split monomorphism is absolute; conversely, that the image of $m$ under the representable functor$C(1,A)$ is a monomorphism reduces to the characterization above.

In higher category theory, we may still consider the notion of “split monomorphism”, i.e. a morphism $m\colon A \to B$ in $C$ such that there exists a morphism $r\colon B \to A$ with $r \circ m$ being equivalent to the identity of $A$. However, in a higher category, such a morphism $m$ will not necessarily be a “monomorphism”, that is, it need not be $(-1)$-truncated.

In general, we can say that in an $(n,1)$-category, a “split monomorphism” will be $(n-2)$-truncated. Thus: