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.
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.
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.