DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Restriction of a sheaf to an open subspace

Definition

Let $j : U ↪ X$ be the inclusion of an open subspace, and let $F$ be a sheaf on $X$ . The restriction of $F$ to $U$ is the inverse image sheaf

F ∣_{U} : = j^{- 1} F .

For an open set $W \subseteq U$ , one may identify $(F ∣_{U}) (W)$ with $F (W)$ : because $U$ is open in $X$ , the set $W$ is also open in $X$ , and in the neighbourhood-colimit defining $j^{- 1} F$ the open set $W$ itself is a neighbourhood of $j (W) = W$ .

Depends on

Inverse image presheaf and inverse image sheaf

Restriction of a sheaf to an open subspace

Definition

Depends on

Dependency tree · 2 levels

Sources