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: * rr is a retraction of mm, * mm is a section of rr, * AA is a retract of BB, * the pair (r,m)(r,m) is a splitting of the idempotent mr:BBm \circ r\colon B \to B.

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}; the 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 automatically 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.

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: * in a (0,1)-category (a poset), a split mono is (2)(-2)-truncated, i.e. an isomorphism; * in a 1-category, a split mono is (1)(-1)-truncated, i.e. a monomorphism; * in a (2,1)-category, a split mono is 00-truncated, i.e. a discrete morphism; * in an (∞,1)-category, a split mono is not necessarily truncated at any finite level.

Revised on May 14, 2012 07:26:43 by Toby Bartels (