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

1.1L1F1

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 $d_{0} e = d_{1} e$ .

1.2L1L2F1

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.

2.1L1L2F1∎

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.

Depends on

Used by

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, Lemma 4.2 (standard reference, not scraped)
Ravi Vakil, Foundations of Algebraic Geometry, Class 3 (standard reference, not scraped)