TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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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A small product of abelian categories is abelian

Statement

Every set-indexed product of abelian categories is abelian. In particular, the empty product category is abelian.

Facts & Assumptions

Given: A set-indexed family $(A_{i})_{i \in I}$ of abelian categories.

[L1]

A small product of preadditive categories is preadditive (A small product of preadditive categories is preadditive).

[L2]

Abelian categories are additive and compute kernels, cokernels, and coimage-image comparison maps internally (Abelian category).

Proof

technique · direct

1.1L1L2

By [L1], the product category $\prod_{i \in I} A_{i}$ is preadditive. If $I = \emptyset$ , this product has one object and one morphism, which is simultaneously zero and identity, so it is already an abelian zero category. For nonempty $I$ , the zero object, finite biproducts, kernels, cokernels, and canonical coimage-image maps are all computed coordinatewise from the corresponding structures in each factor from [L2].

2.1L2step 1.1∎

Therefore the product category is additive and satisfies the AB1 and AB2 clauses coordinatewise. So every small product of abelian categories is abelian.

Depends on

Used by

Finite-support families of finite-dimensional vector spaces are locally finite but not finite Lemma

Dependency tree · two levels

7 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

Saunders Mac Lane, Categories for the Working Mathematician, I.2 and VIII.3 (standard reference, not scraped)