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.

Subsheaves

Definition

Let $G$ be a sheaf of sets on a topological space $X$ .

A subsheaf of $G$ is a sheaf $H$ on $X$ together with a morphism of presheaves $ι : H ⟶ G$ (Morphisms of presheaves) such that for every open set $U \subseteq X$ , the component $ι_{U} : H (U) ⟶ G (U)$ is injective.

Equivalently, one may identify $H (U)$ with a subset of $G (U)$ for every open $U$ , with restriction maps inherited from $G$ , and then require that these subsets define a sheaf. A subpresheaf of a sheaf need not be a subsheaf, because it may fail gluing.

Depends on

Used by

The objectwise image of a sheaf morphism need not be a sheaf Counterexample
The image sheaf is the sheafification of the presheaf image Lemma

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 29 (standard reference, not scraped)