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.
We say that:
$r$ is a retraction of $m$,
$m$ is a section of $r$,
$A$ is a retract of $B$,
the pair $(r,m)$ is a splitting of the idempotent $m \circ r\colon B \to B$.
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.
Any split monomorphism is a monomorphism, in fact a regular monomorphism (it is the equalizer of $m\circ r$ and $1_B$), and therefore also a strong monomorphism, an extremal monomorphism, and (of course) a monomorphism.
Evident but important and in contrast to general monomorphisms:
All functors preserve split monomorphisms.
A morphism is an isomorphism if and only if it is an epimorphism and a split monomorphism.
For a proof, see Yuan 2012.
In the category Vect of finite dimensional vector spaces (over any field) every monomorphism $V_1 \hookrightarrow V_2$ splits. The corresponding idempotent is the projection onto $V_1$ in $V_2$.
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:
in a (0,1)-category (a poset), a split mono is $(-2)$-truncated, i.e. an isomorphism;
in a 1-category, a split mono is $(-1)$-truncated, i.e. a monomorphism;
in a (2,1)-category, a split mono is $0$-truncated, i.e. a discrete morphism;
in an (∞,1)-category, a split mono is not necessarily truncated at any finite level.
Saunders MacLane, §I.5 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]
Qiaochu Yuan, Split epimorphisms and split monomorphisms, blog post (2012),
Last revised on September 15, 2023 at 12:03:24. See the history of this page for a list of all contributions to it.