CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27

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In a preadditive category with a zero object, a morphism is epic exactly when its cokernel is zero

Statement

Let $f : A \to B$ be a morphism in a preadditive category with a zero object, and assume its cokernel exists. Then $f$ is epic if and only if its cokernel is zero.

Facts & Assumptions

Given: A morphism $f : A \to B$ in a preadditive category with a zero object.

[L1]

In this setting, monicity is equivalent to zero kernel (In a preadditive category with a zero object, a morphism is monic exactly when its kernel is zero).

[L2]

The opposite of a preadditive category is preadditive (The opposite of a preadditive category is preadditive).

Proof

technique · direct

1.1

In the opposite category, $f$ becomes a morphism $f^{o p} : B \to A$ . By [L2], that opposite category is again preadditive with a zero object, and epicity of $f$ is monicity of $f^{o p}$ .

2.1

Applying [L1] to $f^{o p}$ says that $f^{o p}$ is monic exactly when its kernel is zero. But that kernel is precisely the cokernel of $f$ viewed back in the original category.

L1step 1.1

3.1

Therefore $f$ is epic exactly when its cokernel is zero.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

The Stacks Project, Section 12.3: Preadditive and additive categories (standard reference, not scraped)