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.
Monomorphism and epimorphism by left and right cancellation
Definition
Let be a morphism in a category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection).
- is a monomorphism, or monic, when implies for every parallel pair .
- is an epimorphism, or epic, when implies for every parallel pair .
Thus monomorphisms are left-cancellable and epimorphisms are right-cancellable. The definitions are formally dual.
Depends on
Used by
- A functor need not preserve monomorphisms Counterexample
- Identities and composites of monomorphisms or epimorphisms retain cancellation; split monomorphisms are monic and split epimorphisms are epic Proposition
- In Set, monomorphisms are exactly injections and epimorphisms are exactly surjections Theorem
- 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)