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.

Morphisms of presheaves

Definition

Depends on

Dependency tree · 2 levels

Sources