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TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28
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The quotient by the kernel followed by the image inclusion is the canonical epi-mono factorization

Statement

For a morphism f:A→B in an abelian category, the factorization

A↠A/ker⁡(f)⟶∼im⁡(f)↣B

is the canonical epimorphism-monomorphism factorization of f.

Facts & Assumptions

Given: An abelian category and a morphism f:A→B.

[L1]

The first isomorphism theorem gives a canonical isomorphism A/ker⁡(f)≅im⁡(f) (First isomorphism theorem in an abelian category).

[L2]

Epic-monic factorizations exist and are unique up to unique isomorphism (Every morphism factors as an epimorphism followed by a monomorphism, uniquely up to unique isomorphism).

Proof

technique · direct
1.1

By [L1], there is an isomorphism ϕ:A/ker⁡(f)→im⁡(f) such that the composite of the quotient map q:A→A/ker⁡(f), the isomorphism ϕ, and the image inclusion if:im⁡(f)→B equals f.

L1
2.1

The quotient map q is epic and the image inclusion if is monic, so step 1.1 is an epic-monic factorization of f. By the uniqueness clause in [L2], it is the canonical one.

L2step 1.1∎

Depends on

Used by

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Dependency tree · two levels

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