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.
In an abelian category, monic means zero kernel and epic means zero cokernel
Statement
For a morphism in an abelian category:
- is monic if and only if its kernel is zero;
- is epic if and only if its cokernel is zero.
Facts & Assumptions
Given: An abelian category and a morphism in it.
An abelian category is additive, hence preadditive and equipped with a zero object (Abelian category).
In a preadditive category with a zero object, a morphism is monic exactly when its kernel is zero (In a preadditive category with a zero object, a morphism is monic exactly when its kernel is zero).
In a preadditive category with a zero object, a morphism is epic exactly when its cokernel is zero (In a preadditive category with a zero object, a morphism is epic exactly when its cokernel is zero).
Proof
The monomorphism claim is exactly [L2], because [L1] supplies the preadditive and zero-object hypotheses that [L2] needs.
The epimorphism claim is exactly [L3], for the same reason.