TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge 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.

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

1.1L1given

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

2.1L1F2step 1.1

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.

3.1F1step 1.1step 2.1∎

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 $η_{F}$ .

Sheafification preserves stalks

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources