Stable homotopy theory
A section of a morphism in some category is a right-inverse: a morphism such that
equals the identity morphism on .
Typically is thought of as a bundle and then one speaks of sections of bundles. For topological bundles one considers continuous sections, for smooth bundles smooth sections, etc.
In terms of dependent product
In a locally cartesian closed category , regard the morphism as an object in the slice category over . Then there is the dependent product
This is the space of sections of . A single section is a global element in here
See at dependent product – In terms of spaces of sections for more on this.
In the case that has a section , may also be called a retraction or cosection of , may be called a retract of , and the entire situation is said to split the idempotent
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 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 November 30, 2016 17:24:19
by Urs Schreiber