Alphabeta MathHow 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.
A sheaf on a topological space
Definition
Let be a presheaf on a topological space . We say that is a sheaf if for every open set and every open cover , including the empty cover of , the following two conditions hold.
Locality. If satisfy then .
Gluing. If sections satisfy then there exists such that
By locality, such a section is automatically unique. Thus a sheaf is exactly a presheaf whose compatible local sections glue uniquely.
Depends on
Used by
- Bounded continuous functions need not form a sheaf Counterexample
- Subsheaves Definition
- The etale space of a sheaf of sets Definition
- A set-valued skyscraper sheaf and its stalks Example
- Continuous real-valued functions form a sheaf Example
- Locally constant functions form a sheaf and have constant stalks Example
- Sections on an open subset extended by the empty set outside it Example
- The empty space has a unique sheaf section over the empty open set Example
- A section of a sheaf of groups is zero exactly when all of its germs are zero Lemma
- A set-valued sheaf has a unique section over the empty open set Lemma
- Morphisms of sheaves are determined by their maps on stalks Lemma
- Sheafhood of algebraic-structure valued presheaves is detected on underlying sets Lemma
- The second plus construction is a sheaf Lemma
- The sheaf condition can be checked on a basis with basis-refinable intersections Lemma
- A morphism of sheaves is an isomorphism exactly when it is an isomorphism on every stalk Theorem
- Sheaves of sets are equivalent to local homeomorphisms over the base space Theorem
- The sheaf axiom is the equalizer condition on a cover Theorem
Dependency tree · two levels
4 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 7.1 (standard reference, not scraped)
- Ravi Vakil, Foundations of Algebraic Geometry, Class 3 (standard reference, not scraped)