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.

A sheaf on a topological space

Definition

Let $F$ be a presheaf on a topological space $X$ . We say that $F$ is a sheaf if for every open set $U \subseteq X$ and every open cover $U = ⋃_{i \in I} U_{i}$ , including the empty cover of $\emptyset$ , the following two conditions hold.

Locality. If $s, t \in F (U)$ satisfy $s ∣_{U_{i}} = t ∣_{U_{i}} for all i \in I,$ then $s = t$ .

Gluing. If sections $s_{i} \in F (U_{i})$ satisfy $s_{i} ∣_{U_{i} \cap U_{j}} = s_{j} ∣_{U_{i} \cap U_{j}} for all i, j \in I,$ then there exists $s \in F (U)$ such that $s ∣_{U_{i}} = s_{i} for all i \in I .$

By locality, such a section $s$ is automatically unique. Thus a sheaf is exactly a presheaf whose compatible local sections glue uniquely.

Depends on

A presheaf on a topological space

A sheaf on a topological space

Definition

Depends on

Dependency tree · 2 levels

Sources