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Definition

A section of a morphism f:A→B in some category is a right-inverse: a morphism σ:B→A such that

f∘σ:B→σA→fBf \circ \sigma : B \stackrel{\sigma}{\to} A \stackrel{f}{\to} B

equals the identity morphism on B.

Split idempotents

In the case that f has a section σ, f may also be called a retraction or cosection of σ, B may be called a retract of A, and the entire situation is said to split the idempotent

A→fB→σA.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:A→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 January 21, 2013 18:54:45 by Urs Schreiber (89.204.139.229)
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