TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-28

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The opposite of an abelian category is abelian

Statement

If $A$ is an abelian category, then the opposite category $A^{o p}$ is also abelian.

Facts & Assumptions

Given: An abelian category $A$ .

[L1]

An abelian category is additive and every morphism in it has a kernel and a cokernel (Abelian category).

[L2]

The opposite of an additive category is additive (Additive categories are closed under passage to the opposite).

[L3]

Passing to the opposite reverses every morphism while keeping the same objects (Opposite category $C^{o p}$ ).

Proof

technique · direct

1.1L1L2L3

By [L1] and [L2], the opposite category is additive. Under [L3], a kernel in $A$ becomes a cokernel in $A^{o p}$ , and a cokernel becomes a kernel, so every morphism of $A^{o p}$ also has both.

2.1L1L3step 1.1∎

The image of $f^{o p}$ in the opposite category is the opposite of the coimage of $f$ , and the coimage of $f^{o p}$ is the opposite of the image of $f$ . Therefore the canonical comparison for $f^{o p}$ is the opposite of the canonical comparison for $f$ , which is an isomorphism by [L1]. So $A^{o p}$ satisfies the same AB2 clause and is abelian.

Depends on

Dependency tree · 2 levels

Sources

The Stacks Project, Section 12.5, Lemma 12.5.2