A split epimorphism in a category is a morphism in such that there exists a morphism such that the composite equals the identity morphism . Then the morphism , which satisfies the dual condition, is a split monomorphism.
We say that:
is a section of ,
is a retraction of ,
is a retract of ,
the pair is a splitting of the idempotent .
A split epimorphism in can be equivalently defined as a morphism such that for every object , the function is a surjection in ; the preimage of under yields a section .
Alternatively, it is also possible to define a split epimorphism as an absolute epimorphism: a morphism such that for every functor out of , 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 under the representable functor is an epimorphism reduces to the characterization above.
Any split epimorphism is automatically a regular epimorphism (it is the coequalizer of and ), and therefore also a strong epimorphism, an extremal epimorphism, and (of course) an epimorphism.
The axiom of choice internal to a category 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 ), 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.
Last revised on June 3, 2021 at 10:28:19. See the history of this page for a list of all contributions to it.