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.

A set-valued skyscraper sheaf and its stalks

Example

Fix a point $x_{0} \in X$ and a set $A$ . Define a presheaf $S_{x_{0}, A}$ by $S_{x_{0}, A} (U) = {\begin{cases} A, & x_{0} \in U, \\ {*}, & x_{0} \notin U, \end{cases}$ with identity restrictions between opens containing $x_{0}$ and the unique map to ${*}$ when the target does not contain $x_{0}$ . Then $S_{x_{0}, A}$ is a sheaf. Its stalk at $x_{0}$ is canonically $A$ , and its stalk at any $y \in X$ with an open neighbourhood $V$ satisfying $x_{0} \notin V$ is the singleton ${*}$ . In particular, this holds for every $y \neq x_{0}$ in a $T_{1}$ space.

Facts & Assumptions

Given: A point $x_{0} \in X$ , a set $A$ , and a point $y \in X$ . For the second stalk computation, also assume $y$ has an open neighbourhood $V$ with $x_{0} \notin V$ .

[L1]

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

[F1]

Stalks are colimits over neighbourhoods, and germs are represented by local sections (The stalk of a presheaf at a point, Germs of sections).

Verification

technique · direct

1.1

If an open set $U$ does not contain $x_{0}$ , then every section of $S_{x_{0}, A} (U)$ is the unique element $*$ , so locality and gluing are trivial. If $x_{0} \in U$ and $U = ⋃_{i} U_{i}$ , then at least one $U_{i}$ contains $x_{0}$ . Compatibility forces all sections on such $U_{i}$ to be the same element of $A$ , and every $U_{j}$ not containing $x_{0}$ contributes only the unique section $*$ . Thus there is a unique glued section on $U$ . Therefore [L1] holds and $S_{x_{0}, A}$ is a sheaf.

L1given

1.2

For the stalk at $x_{0}$ , every neighbourhood of $x_{0}$ has section set $A$ and every transition map is the identity on $A$ . Hence the colimit in [F1] is canonically $A$ .

F1given

2.1

If $y$ has an open neighbourhood $V$ with $x_{0} \notin V$ , then every smaller neighbourhood of $y$ inside $V$ also has section set ${*}$ . Hence the stalk diagram is eventually constant at ${*}$ , so [F1] gives $(S_{x_{0}, A})_{y} ≅ {*}$ .

F1given∎

Depends on

Used by

Nothing in the library uses this result yet.

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, Section 27 (standard reference, not scraped)