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

Proposition

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.

Evident but important and in contrast to general epimorphisms:

Proposition

All functors preserve split epimorphisms.

Proposition

A morphism is an isomorphism if and only if it is an monomorphism and a split epimorphism.

For a proof, see Yuan 2012.

Proposition

(relation to the axiom of choice)
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

Example

In Sets the axiom of choice is the statement that every epimorphism (surjective map) is split.

Example

Assuming the axiom of choice and thus the basis theorem, every epimorphism in Vect 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 $\phi$.

For more on this see at split exact sequence the section Of free modules and vector spaces

Related concepts

• epimorphism

• split monomorphism

• idempotent

References

• 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)