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 presheaf on a topological space

Definition

Let $X$ be a topological space. A presheaf of sets on $X$ is a contravariant functor $F : Open (X) ⟶ S e t$ (Covariant functor, identity functor, composite functor, and contravariant functor, Sets and functions form the large locally small category $S e t$ ).

Equivalently, a presheaf of sets on $X$ consists of:

a set $F (U)$ for every open set $U \subseteq X$ ;
for every inclusion $V \subseteq U$ , a restriction map $ρ_{V}^{U} : F (U) ⟶ F (V),$

such that $ρ_{U}^{U} = {id}_{F (U)}$ and, whenever $W \subseteq V \subseteq U$ , $ρ_{W}^{U} = ρ_{W}^{V} \circ ρ_{V}^{U} .$

Because the arrows of $Open (X)$ are the inclusions $V \to U$ with $V \subseteq U$ , contravariance is exactly the rule that sections over a larger open set restrict to sections over a smaller one.

A presheaf on a topological space

Definition

Depends on

Dependency tree · 2 levels

Sources