CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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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.

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Sheafification is idempotent

Statement

For every presheaf $F$ on a topological space $X$ , the canonical map $η_{a F} : a F ⟶ a (a F)$ is an isomorphism. Equivalently, sheafification is idempotent: $a (a F) ≅ a F .$

Facts & Assumptions

Given: A presheaf $F$ on $X$ .

[F1]

The object $a F$ is the sheafification of $F$ (Sheafification of a presheaf).

[L1]

Any map from a presheaf to a sheaf factors uniquely through its sheafification (Sheafification is left adjoint to the inclusion of sheaves into presheaves).

Proof

technique · direct

1.1F1L1

Since $a F$ is already a sheaf by [F1], apply [L1] to the identity map $1_{a F} : a F \to a F$ . There is a unique morphism $ε : a (a F) ⟶ a F$ such that $ε \circ η_{a F} = 1_{a F} .$

2.1L1step 1.1

Apply [L1] again to the map $η_{a F} : a F \to a (a F)$ , whose target is also a sheaf. The identity map on $a (a F)$ is one factorization through $η_{a F}$ . The composite $η_{a F} \circ ε$ is another, because step 1.1 gives $(η_{a F} \circ ε) \circ η_{a F} = η_{a F}$ . By uniqueness in [L1], $η_{a F} \circ ε = 1_{a (a F)}$ .

3.1step 1.1step 2.1∎

Steps 1.1 and 2.1 show that $ε$ and $η_{a F}$ are two-sided inverses. Therefore $η_{a F}$ is an isomorphism, so $a (a F) ≅ a F$ .

Depends on

Used by

Nothing in the library uses this result yet.

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, Section 17 (standard reference, not scraped)