nLab split epimorphism



Definitions and terminology

A split epimorphism in a category CC is a morphism e:ABe\colon A \to B in CC such that there exists a morphism s:BAs\colon B \to A such that the composite ese \circ s equals the identity morphism 1 B1_B. Then the morphism ss, which satisfies the dual condition, is a split monomorphism.

We say that:

A split epimorphism in CC can be equivalently defined as a morphism e:ABe\colon A \to B such that for every object X:CX\colon C, the function C(X,e)C(X,e) is a surjection in Set\mathbf{Set}; the preimage of 1 B1_B under C(B,e)C(B,e) yields a section ss.

Alternatively, it is also possible to define a split epimorphism as an absolute epimorphism: a morphism such that for every functor FF out of CC, F(e)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 ee under the representable functor C(B,1)C(B,1) is an epimorphism reduces to the characterization above.



Any split epimorphism is automatically a regular epimorphism (it is the coequalizer of ses\circ e and 1 A1_A), and therefore also a strong epimorphism, an extremal epimorphism, and (of course) an epimorphism.

Evident but important and in contrast to general epimorphisms:


All functors preserve split epimorphisms.


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

For a proof, see Yuan 2012.


(relation to the axiom of choice)
The axiom of choice internal to a category CC 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 SetSet), or it may be false regardless of the axiom of choice.


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



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


Assuming the axiom of choice and thus the basis theorem, every epimorphism in Vect is split:

For ϕ:VW\phi\colon V \to W a surjective linear map, we can find an isomorphism Vker(ϕ)VV \simeq ker(\phi) \oplus V'. Then ϕ| V\phi|_{V'} is an isomorphism, and its inverse WVker(ϕ)VW \to V' \hookrightarrow ker(\phi) \oplus V' is a section of ϕ\phi.

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


Last revised on September 15, 2023 at 12:04:12. See the history of this page for a list of all contributions to it.