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

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A small product of preadditive categories is preadditive

Statement

Let $(C_{i})_{i \in I}$ be a set-indexed family of preadditive categories. Then their product category $\prod_{i \in I} C_{i}$ , with objects, morphisms, identities, and composition taken coordinatewise, is preadditive.

Facts & Assumptions

Given: A set $I$ and a family $(C_{i})_{i \in I}$ of preadditive categories.

[L1]

A preadditive category has abelian-group hom-sets and bilinear composition (Preadditive category).

[L2]

Product categories compose and take identities componentwise (Product category and its projection functors).

[L3]

The phrase "small product" means the indexing family is set-sized (Small, locally small, and large categories).

Proof

technique · direct

1.1L1L3

For objects $A = (A_{i})$ and $B = (B_{i})$ , define $\prod_{i} C_{i} (A_{i}, B_{i})$ with componentwise addition. Each factor is an abelian group by [L1], so the componentwise law makes the product an abelian group.

1.2L1L2

By [L2], composition in the product category is coordinatewise. Therefore for composable families $(f_{i}), (g_{i}), (h_{i})$ one has $(h_{i}) \circ ((f_{i}) + (g_{i})) = (h_{i} \circ (f_{i} + g_{i})) = (h_{i} \circ f_{i} + h_{i} \circ g_{i})$ coordinatewise, and likewise on the other side. So composition is bilinear.

2.1step 1.1step 1.2L1∎

Steps 1.1 and 1.2 are exactly the preadditive axioms, hence the set-indexed product category is preadditive.

Depends on

Used by

An exact functor need not be faithful Counterexample
A small product of abelian categories is abelian Theorem

Dependency tree · two levels

6 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

Emily Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)