TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-30

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The category of complexes in an additive category is additive

Statement

Let $Ch (A)$ denote the category whose objects are \mathbb Z-graded objects of $A$ equipped with differentials $d_{n} : C_{n} \to C_{n - 1}$ satisfying $d_{n - 1} d_{n} = 0$ , and whose morphisms are chain maps. If $A$ is an additive category, then $Ch (A)$ is an additive category.

Facts & Assumptions

Given: An additive category $A$ .

[L1]

An additive category has a zero object and finite biproducts (Additive category).

[L2]

A biproduct is a common product-coproduct object (Biproduct).

[L3]

A chain map is a degreewise family commuting with the differentials (Chain map).

Proof

technique · direct

1.1

Since $A$ is additive, [L1] includes the preadditive structure that supplies zero morphisms, so the displayed chain-complex condition is meaningful in $A$ . The zero object of $A$ from [L1] therefore gives the zero complex, which is a zero object of $Ch (A)$ because every chain map to or from it is forced degreewise.

L1L3

1.2

For chain complexes $C$ and $D$ , let $(C \oplus D)_{n} : = C_{n} \oplus D_{n}$ using the biproducts from [L1]. Define the differential by $d_{n}^{C \oplus D} : = d_{n}^{C} \oplus d_{n}^{D}$ . Then $(d_{n - 1}^{C} \oplus d_{n - 1}^{D}) (d_{n}^{C} \oplus d_{n}^{D}) = 0$ , so this is a chain complex, and the degreewise injections and projections are chain maps. By [L2], they make $C \oplus D$ a biproduct in $Ch (A)$ .

L1L2L3algebra

2.1

Addition of chain maps is defined degreewise on the additive hom-groups of $A$ , and the chain-map equation is preserved because composition is bilinear. Together with steps 1.1 and 1.2, [L1] shows that $Ch (A)$ is additive.

L1L3step 1.1step 1.2algebra∎

Depends on

Used by

Finite biproducts of complexes are computed degreewise Proposition
Homology is an additive functor Theorem
The category of complexes in an abelian category is abelian Theorem

Dependency tree · two levels

9 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

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
Romyar Sharifi, Homological Algebra, Proposition 2.7.5 (standard reference, not scraped)