DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Sections, restrictions, and global sections of a presheaf

Definition

Let $F$ be a presheaf on a topological space $X$ .

An element of $F (U)$ is called a section of $F$ over $U$ . We also write $Γ (U, F) : = F (U) .$

If $V \subseteq U$ and $s \in F (U)$ , its restriction to $V$ is $s ∣_{V} : = ρ_{V}^{U} (s) \in F (V) .$

A global section of $F$ is a section over the whole space: $Γ (X, F) = F (X) .$

The presheaf identities imply $s ∣_{U} = s, (s ∣_{V}) ∣_{W} = s ∣_{W} for W \subseteq V \subseteq U .$

Depends on

A presheaf on a topological space

Used by

Germs of sections Definition
The sheaf axiom is the equalizer condition on a cover Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Sheaves on Spaces, Section 3 (standard reference, not scraped)