TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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An abelian category is balanced

Statement

If $f : A \to B$ in an abelian category is both monic and epic, then $f$ is an isomorphism.

Facts & Assumptions

Given: An abelian category and a morphism $f : A \to B$ that is both monic and epic.

[L1]

The kernel of a monomorphism is zero, and the cokernel of an epimorphism is zero (The kernel of a monomorphism is zero and the cokernel of an epimorphism is zero).

[L2]

The cokernel of $0 \to A$ is $A$ , and the kernel of $B \to 0$ is $B$ (The cokernel of the zero map out of the zero object is the target, and dually for kernels).

[L3]

Every morphism has a canonical factorization $A \to coim (f) \to im (f) \to B$ (The canonical morphism from the coimage to the image exists and is unique).

[L4]

In an abelian category the canonical map $coim (f) \to im (f)$ is an isomorphism (Abelian category).

Proof

technique · direct

1.1

Because $f$ is monic and epic, [L1] identifies $\ker (f)$ and $coker (f)$ with zero objects. Hence [L2] gives isomorphisms $q_{f} : A \tilde{⟶} coim (f)$ and $i_{f} : im (f) \tilde{⟶} B$ for the coimage projection and image inclusion of $f$ .

L1L2L3

2.1

By [L4], the middle map $\overline{f} : coim (f) \to im (f)$ is an isomorphism. Since $f = i_{f} \overline{f} q_{f}$ , step 1.1 shows that $f$ is a composite of three isomorphisms, so $f$ itself is an isomorphism.

L3L4step 1.1∎

Depends on

Used by

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Junhan Tan, The Freyd-Mitchell Embedding Theorem, Theorem 2.4 (standard reference, not scraped)
The Stacks Project, Section 12.5, Lemma 12.5.2 (standard reference, not scraped)