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

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The sheaf axiom is the equalizer condition on a cover

Statement

Let $F$ be a presheaf of sets on a topological space $X$ . Then $F$ is a sheaf if and only if, for every open set $U \subseteq X$ and every open cover $U = ⋃_{i \in I} U_{i}$ , the restriction map $e : F (U) ⟶ \prod_{i \in I} F (U_{i}), e (s) = (s ∣_{U_{i}})_{i \in I},$ is an equalizer of the two maps $d_{0}, d_{1} : \prod_{i \in I} F (U_{i}) ⇉ \prod_{(i, j) \in I \times I} F (U_{i} \cap U_{j}),$ defined by $d_{0} ((s_{i})) = (s_{i} ∣_{U_{i} \cap U_{j}})_{i, j}, d_{1} ((s_{i})) = (s_{j} ∣_{U_{i} \cap U_{j}})_{i, j} .$

Facts & Assumptions

Given: A presheaf $F$ , an open set $U$ , and an open cover $U = ⋃_{i \in I} U_{i}$ .

[L1]

A sheaf is exactly a presheaf satisfying locality and unique gluing for every open cover (A sheaf on a topological space).

[F1]

The notation $s ∣_{U_{i}}$ and $s_{i} ∣_{U_{i} \cap U_{j}}$ is the presheaf restriction notation (Sections, restrictions, and global sections of a presheaf).

[L2]

An equalizer of parallel maps $f, g : A ⇉ B$ is a morphism $e : E \to A$ with $f e = g e$ such that any $h : T \to A$ with $f h = g h$ factors uniquely through $e$ (Equalizers and coequalizers as limits and colimits of a parallel pair).

Proof

technique · direct

1.1

Assume $F$ is a sheaf. For any $s \in F (U)$ , the two families $d_{0} (e (s))$ and $d_{1} (e (s))$ are equal because both entries on $U_{i} \cap U_{j}$ are the common restriction $s ∣_{U_{i} \cap U_{j}}$ by [F1]. Thus $d_{0} e = d_{1} e$ .

L1F1

1.2

Let $(s_{i})_{i \in I} \in \prod_{i} F (U_{i})$ satisfy $d_{0} ((s_{i})) = d_{1} ((s_{i}))$ . Unwinding the definitions, this says exactly that $s_{i} ∣_{U_{i} \cap U_{j}} = s_{j} ∣_{U_{i} \cap U_{j}}$ for all $i, j$ . By the gluing clause of [L1], there exists a unique $s \in F (U)$ with $s ∣_{U_{i}} = s_{i}$ for all $i$ . Hence every equalizing family factors uniquely through $e$ , so [L2] shows that $e$ is an equalizer.

L1L2F1

2.1

Conversely, assume $e$ is an equalizer for every open set and cover. If $s, t \in F (U)$ satisfy $s ∣_{U_{i}} = t ∣_{U_{i}}$ for all $i$ , then $e (s) = e (t)$ . Let $*$ be a singleton and define $h : {*} \to \prod_{i} F (U_{i})$ by $h (*) = e (s) = e (t)$ . Then $d_{0} h = d_{1} h$ , so [L2] gives a unique $u : {*} \to F (U)$ with $e u = h$ . The two maps sending $*$ to $s$ and to $t$ both satisfy this condition, hence are equal and therefore $s = t$ . So locality holds. If $(s_{i})$ is a compatible family on the cover, then $d_{0} ((s_{i})) = d_{1} ((s_{i}))$ , so [L2] yields a unique $s \in F (U)$ with $e (s) = (s_{i})$ . That is exactly unique gluing. Therefore [L1] implies that $F$ is a sheaf.

L1L2F1∎

Depends on

Used by

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Dependency tree · two levels

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Sources

The Stacks Project, Sheaves on Spaces, Lemma 4.2 (standard reference, not scraped)
Ravi Vakil, Foundations of Algebraic Geometry, Class 3 (standard reference, not scraped)