ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck 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.

Locally constant functions form a sheaf and have constant stalks

Example

Fix a set $A$ . For each open set $U \subseteq X$ , let ${\underline{A}}_{l o c} (U) : = {f : U \to A locally constant} .$ With the usual restriction maps, this is a sheaf of sets on $X$ . For every $x \in X$ , evaluation at $x$ induces a canonical bijection ${\underline{A}}_{l o c, x} ≅ A .$

Facts & Assumptions

Given: A set $A$ , an open set $U \subseteq X$ , and a point $x \in X$ .

[L1]

The sheaf condition is locality and unique gluing on open covers (A sheaf on a topological space).

[F1]

The stalk at $x$ is the colimit of sections on neighbourhoods of $x$ (The stalk of a presheaf at a point).

[F2]

The germ of a section is its class in that stalk (Germs of sections).

Verification

technique · direct

1.1

Restriction preserves local constancy. If two locally constant functions on $U$ agree on an open cover, then they agree pointwise on $U$ . If locally constant functions $f_{i} : U_{i} \to A$ are compatible on an open cover of $U$ , the pointwise glued function $f$ is well defined and locally constant because near any point it agrees with one of the local functions $f_{i}$ . Hence [L1] holds and ${\underline{A}}_{l o c}$ is a sheaf.

L1givenconstruct

2.1

Define $ϵ_{x} : {\underline{A}}_{l o c, x} \to A$ by $ϵ_{x} ([U, f]) = f (x)$ . This is well defined because equal germs agree on some neighbourhood of $x$ , hence have the same value at $x$ . Every $a \in A$ is the value at $x$ of the constant function $a$ on any neighbourhood of $x$ , so $ϵ_{x}$ is surjective. If $ϵ_{x} ([U, f]) = ϵ_{x} ([V, g])$ , then $f (x) = g (x)$ . Since both functions are locally constant, there is a neighbourhood $W$ of $x$ on which $f$ and $g$ are both constantly this common value, so $[U, f] = [V, g]$ . Therefore $ϵ_{x}$ is bijective.

F1F2given∎

Depends on

Used by

The constant presheaf need not be a sheaf on a disconnected open set Counterexample

Dependency tree · two levels

7 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, Definition 7.4 (standard reference, not scraped)