nLab local section

Idea

For $E \to X$ a bundle, (often taken to be a fiber bundle or at least typically taken to be a regular epimorphic map) a local section is a section of the pullback of the bundle along some $U \to X$, typically required to be an element of a covering family from some coverage.

The assignment of local sections of some $E \to X$ to all admissible $U \to X$ is the (pre-)sheaf of local sections assigned to a bundle.

In a finitely complete site $(S,J)$ the assignment $X \mapsto \{p\colon Y \to X | p$ admits local sections over a $J$-cover $\}$ is a singleton pretopology on $S$.

Examples

• coset space coprojections admitting local sections

• logarithm

Related concepts

• global section

• flat section

• partial section

• slice of presheaves is presheaves on slice