nLab split epimorphism

Definitions and terminology

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

We say that:

• $s$ is a section of $e$,

• $e$ is a retraction of $s$,

• $B$ is a retract of $A$,

• the pair $(e,s)$ is a splitting of the idempotent $s \circ e\colon A \to A$.

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

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

Properties

• Any split epimorphism is automatically a regular epimorphism (it is the coequalizer of $s\circ e$ and $1_A$), and therefore also a strong epimorphism, an extremal epimorphism, and (of course) an epimorphism.

• The axiom of choice internal to a category $C$ can be phrased as “all epimorphisms are split.” In Set this is equivalent to the usual axiom of choice; in many other categories it may be true without assuming the axiom of choice (in $Set$), or it may be false regardless of the axiom of choice.

Applications

The notion of split epimorphism arises often as a condition on fibrations in categories of chain complexes. See there for details.

Examples

• In Vect, every epimorphism is split. For $\phi\colon V \to W$ a surjective linear map, we can find an isomorphism $V \simeq ker(\phi) \oplus V'$. Then $\phi|_{V'}$ is an isomorphism, and its inverse $W \to V' \hookrightarrow ker(\phi) \oplus V'$ is a section of $\phi$.