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.

Morphisms of presheaves

Definition

Let $F$ and $G$ be presheaves on a topological space $X$ . A morphism of presheaves $φ : F ⟶ G$ is a natural transformation (Natural transformation and its components) between the two contravariant functors on $Open (X)$ .

Equivalently, it is a family of maps $φ_{U} : F (U) ⟶ G (U)$ for every open $U \subseteq X$ , such that for every inclusion $V \subseteq U$ the square $\begin{aligned} F (U) & \overset{φ_{U}}{\to} G (U) \\ ↓ & ↓ \\ F (V) & \overset{φ_{V}}{\to} G (V) \end{aligned}$ commutes, that is, $φ_{V} (s ∣_{V}) = φ_{U} (s) ∣_{V} for all s \in F (U) .$

Composition and identities are taken componentwise.

Depends on

Used by

Dependency tree · two levels

5 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, Definition 3.1 (standard reference, not scraped)
Ravi Vakil, Foundations of Algebraic Geometry, Class 3 (standard reference, not scraped)