nLab
split epimorphism

Contents

Definitions and terminology

A split epimorphism in a category CC is a morphism e:ABe\colon A \to B in CC such that there exists a morphism s:BAs\colon B \to A such that the composite ese \circ s equals the identity morphism 1 B1_B. Then the morphism ss, which satisfies the dual condition, is a split monomorphism.

We say that: * ss is a section of ee, * ee is a retraction of ss, * BB is a retract of AA, * the pair (e,s)(e,s) is a splitting of the idempotent se:AAs \circ e\colon A \to A.

A split epimorphism in CC can be equivalently defined as a morphism e:ABe\colon A \to B such that for every object X:CX\colon C, the function C(X,e)C(X,e) is a surjection in Set\mathbf{Set}; the preimage of 1 B1_B under C(B,e)C(B,e) yields a section ss.

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

Properties

Applications

The notion of split epimorphism arises often as a condition on fibrations in categories of chain complexes. See there for details.

Examples

  • In Vect, every epimorphism is split. For ϕ:VW\phi\colon V \to W a surjective linear map, we can find an isomorphism Vker(ϕ)VV \simeq ker(\phi) \oplus V'. Then ϕ| V\phi|_{V'} is an isomorphism, and its inverse WVker(ϕ)VW \to V' \hookrightarrow ker(\phi) \oplus V' is a section of ϕ\phi.

Revised on March 6, 2012 22:33:44 by Toby Bartels (64.89.59.67)