nLab section





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.

Typically AfBA \stackrel{f}{\to} B 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 𝒞\mathcal{C}, regard the morphism f:ABf\colon A \to B as an object [f]𝒞 /B[f] \in \mathcal{C}_{/B} in the slice category over BB. Then there is the dependent product

B[f]𝒞. \underset{B}{\prod} [f] \in \mathcal{C} \,.

This is the space of sections of ff. A single section σ\sigma is a global element in here

σ:*B[f]. \sigma \colon \ast \to \underset{B}{\prod} [f] \,.

See at dependent product – In terms of spaces of sections for more on this.

Split idempotents

In the case that ff has a section σ\sigma, ff may also be called a retraction or cosection of σ\sigma, AA may be called a retract of BB, 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.

Last revised on October 1, 2022 at 19:15:39. See the history of this page for a list of all contributions to it.