Definitions and terminology
A split monomorphism 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 epimorphism.
We say that:
A split monomorphism 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 retraction .
Alternatively, it is also possible to define a split monomorphism as an absolute monomorphism: a morphism such that for every functor out of , 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 under the representable functor is a monomorphism reduces to the characterization above.
Any split monomorphism is automatically a regular monomorphism (it is the equalizer of and ), and therefore also a strong monomorphism, an extremal monomorphism, and (of course) a monomorphism.
In vector spaces
In the category Vect of finite dimensional vector spaces (over any field) every monomorphism splits. The corresponding idempotent is the projection onto in .
In higher category theory
In higher category theory, we may still consider the notion of “split monomorphism”, i.e. a morphism in such that there exists a morphism with being equivalent to the identity of . However, in a higher category, such a morphism will not necessarily be a “monomorphism”, that is, it need not be -truncated.
In general, we can say that in an -category, a “split monomorphism” will be -truncated. Thus:
in a (0,1)-category (a poset), a split mono is -truncated, i.e. an isomorphism;
in a 1-category, a split mono is -truncated, i.e. a monomorphism;
in a (2,1)-category, a split mono is -truncated, i.e. a discrete morphism;
in an (∞,1)-category, a split mono is not necessarily truncated at any finite level.