nLab split monomorphism



Definitions and terminology

A split monomorphism in a category CC is a morphism m:ABm\colon A \to B in CC such that there exists a morphism r:BAr\colon B \to A such that the composite rmr \circ m equals the identity morphism 1 A1_A. Then the morphism rr, which satisfies the dual condition, is a split epimorphism.

We say that:

A split monomorphism in CC can be equivalently defined as a morphism m:ABm\colon A \to B such that for every object X:CX\colon C, the function C(m,X)C(m,X) is a surjection in Set\mathbf{Set}; a preimage of 1 A1_A under C(m,A)C(m,A) yields a retraction rr.

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



Any split monomorphism is a monomorphism, in fact a regular monomorphism (it is the equalizer of mrm\circ r and 1 B1_B), and therefore also a strong monomorphism, an extremal monomorphism, and (of course) a monomorphism.

Evident but important and in contrast to general monomorphisms:


All functors preserve split monomorphisms.


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

For a proof, see Yuan 2012.


In vector spaces

In the category Vect of finite dimensional vector spaces (over any field) every monomorphism V 1V 2V_1 \hookrightarrow V_2 splits. The corresponding idempotent is the projection onto V 1V_1 in V 2V_2.

In higher category theory

In higher category theory, we may still consider the notion of “split monomorphism”, i.e. a morphism m:ABm\colon A \to B in CC such that there exists a morphism r:BAr\colon B \to A with rmr \circ m being equivalent to the identity of AA. However, in a higher category, such a morphism mm will not necessarily be a “monomorphism”, that is, it need not be (1)(-1)-truncated.

In general, we can say that in an (n,1)(n,1)-category, a “split monomorphism” will be (n2)(n-2)-truncated. Thus:


Last revised on September 15, 2023 at 12:03:24. See the history of this page for a list of all contributions to it.