CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-27

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Any adjoint between additive categories is additive

Statement

Every left adjoint or right adjoint between additive categories is additive.

Facts & Assumptions

Given: An adjunction between additive categories.

[L1]

A functor between additive categories preserving finite products is additive (A functor between additive categories preserving finite products is additive).

[L2]

Right adjoints preserve all limits that exist (Right adjoints preserve every limit that exists).

[L3]

Additive categories remain additive after passing to the opposite (Additive categories are closed under passage to the opposite).

Proof

technique · direct

1.1L1L2

A right adjoint preserves finite products by [L2], so [L1] makes it additive.

2.1L3step 1.1

If $F : C \to D$ is a left adjoint, then $F^{o p} : C^{o p} \to D^{o p}$ is a right adjoint. By [L3], the opposite categories are additive, so step 1.1 applied in the opposite categories shows that $F^{o p}$ is additive. But additivity is checked on the same hom-group maps before and after taking opposites, so $F$ is additive as well.

3.1step 1.1step 2.1∎

Therefore any adjoint between additive categories is additive.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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 12.7, Lemma 12.7.2(1) (standard reference, not scraped)