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.
Isomorphism, groupoid, and connected category
Definition
In a category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection), a morphism is an isomorphism if there is a morphism with and . Such a is unique and is denoted .
A groupoid is a category in which every morphism is an isomorphism. A category is connected when it is nonempty and any two objects can be joined by a finite zigzag of morphisms, with successive arrows allowed to point in either direction. In a groupoid this is equivalent to being nonempty and having an isomorphism between every ordered pair of objects. Nonemptiness cannot be dropped: the empty groupoid has an isomorphism between every ordered pair of its objects, vacuously, and is not connected.