Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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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.

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Identities and composites of monomorphisms or epimorphisms retain cancellation; split monomorphisms are monic and split epimorphisms are epic

Statement

Identity morphisms are monic and epic; composites of monomorphisms are monic, composites of epimorphisms are epic; every split monomorphism is monic, and every split epimorphism is epic.

Facts & Assumptions

Given: Composable morphisms in a category.

[L1]

Monomorphisms and epimorphisms are defined by left and right cancellation (Monomorphism and epimorphism by left and right cancellation).

[L2]

A split monomorphism has a left inverse and a split epimorphism has a right inverse (Split monomorphism, split epimorphism, retraction, and section); the two halves are dual by Every theorem about categories has a formal dual obtained by reversing morphisms and composition.

Proof

technique · direct
1.1

Identities cancel trivially; if ff and gg are monic and (g∘f)u=(g∘f)v(g\circ f)u=(g\circ f)v, cancellation by gg and then by ff gives u=vu=v, so composites of monomorphisms are monic.

givenL1
2.1

Applying the dual argument of [L2] proves that identities and composites of epimorphisms are epic.

step 1.1L1L2
3.1

If r∘f=1r\circ f=1, then fu=fvfu=fv implies u=rfu=rfv=vu=r fu=r fv=v, so a split monomorphism is monic; dually, f∘s=1f\circ s=1 makes a split epimorphism epic.

step 2.1L1L2∎

Depends on

Used by

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Sources