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.
Depends on
Used by
- Faithful, full, fully faithful, essentially surjective, and split essentially surjective functors Definition
- Natural isomorphism Definition
- Skeletal category and skeleton Definition
- Split monomorphism, split epimorphism, retraction, and section Definition
- An action groupoid has the acted-on set as objects, orbits as connected components, and stabilizers as automorphism groups Example
- The fundamental groupoid of a topological space Example
- A morphism is an isomorphism exactly when postcomposition, equivalently precomposition, induces bijections on every hom-collection Lemma
- Every fully faithful functor reflects isomorphisms Proposition
- Every functor preserves isomorphisms Proposition
- The isomorphisms in a category form its maximal subgroupoid Proposition
- Under the Axiom of Choice, a connected small groupoid is equivalent to the automorphism group of any one of its objects Proposition
- The inclusion ℤ↪ℚ is monic and epic but neither surjective nor an isomorphism in Ring Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 11 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)