TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Sheafification preserves stalks

Statement

Let $F$ be a presheaf on a topological space $X$ . For every $x \in X$ , the sheafification map induces a bijection $η_{F, x} : F_{x} \overset{≅}{\to} (a F)_{x} .$

Facts & Assumptions

Given: A presheaf $F$ on $X$ and a point $x \in X$ .

[F1]

Sheafification is $a F = (F^{+})^{+}$ with unit $η_{F} : F \to a F$ (Sheafification of a presheaf).

[L1]

For any presheaf, the first plus construction preserves stalks (The first plus construction is separated and preserves stalks).

[F2]

Equality in a filtered-colimit stalk is eventual on a smaller neighbourhood (Two representatives in a filtered colimit of sets are equal exactly when they become equal at one common later stage).

Proof

technique · direct

1.1

Apply [L1] to the presheaf $F$ . This gives a bijection $F_{x} \overset{≅}{\to} (F^{+})_{x} .$

L1given

2.1

Apply [L1] again, now to the presheaf $F^{+}$ . Since [L1] holds for every presheaf, it yields a bijection $(F^{+})_{x} \overset{≅}{\to} ((F^{+})^{+})_{x} = (a F)_{x} .$ The reference to [F2] is the same eventual-equality argument used inside [L1] to prove injectivity on stalks.

L1F2step 1.1

3.1

Composing the bijections of steps 1.1 and 2.1 gives the required isomorphism $F_{x} \overset{≅}{\to} (a F)_{x} .$ By [F1], this composite is exactly the map induced by $η_{F}$ .

F1step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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 17 (standard reference, not scraped)
Ravi Vakil, Foundations of Algebraic Geometry, Class 3, Section 4.8 (standard reference, not scraped)