nLab
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Definition

A section of a morphism f:ABf : A \to B in some category is a right-inverse: a morphism σ:BA\sigma : B \to A such that

fσ:BσAfB f \circ \sigma : B \stackrel{\sigma}{\to} A \stackrel{f}{\to} B

equals the identity morphism on BB.

Split idempotents

In the case that ff has a section σ\sigma, ff may also be called a retraction or cosection of σ\sigma, BB may be called a retract of AA, and the entire situation is said to split the idempotent

AfBσA. A \stackrel{f}{\to} B \stackrel{\sigma}{\to} A \,.

A split epimorphism is a morphism that has a section; a split monomorphism is a morphism that is a section. A split coequalizer is a particular kind of split epimorphism.

Sections of bundles and sheaves

If one thinks of f:ABf : A \to B as a bundle then its sections are sometimes called global sections. This leads to a notion of global sections of sheaves and further of objects in a general topos. See

for more on this.

Revised on August 22, 2013 05:12:40 by Urs Schreiber (150.212.99.10)