TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31

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The homotopy category is additive

Statement

If $A$ is an additive category, then $K (A)$ is an additive category.

Facts & Assumptions

Given: An additive category $A$ .

[L1]

In $K (A)$ , morphisms are homotopy classes of chain maps (The homotopy category of chain complexes).

[L2]

Null-homotopic maps form a two-sided additive ideal (Null-homotopic maps form a two-sided additive ideal).

[L3]

The category $Ch (A)$ is additive (The category of complexes in an additive category is additive).

[L4]

Finite biproducts of complexes are computed degreewise (Finite biproducts of complexes are computed degreewise).

Proof

technique · direct

1.1

By [L3], each hom-set of $Ch (A)$ is an abelian group. Quotienting by the additive subgroup of null-homotopic maps from [L2] therefore gives an abelian group structure on each ${Hom}_{K (A)} (C, D)$ from [L1].

L1L2L3givenalgebra

2.1

The zero complex and the degreewise biproduct complex exist in $Ch (A)$ by [L3] and [L4]. Because [L2] is a two-sided ideal, the usual injections and projections descend to homotopy classes and still satisfy the biproduct identities in the quotient. Therefore $K (A)$ has a zero object and finite biproducts, so it is additive.

L2L3L4step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

12 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)
The Stacks Project, Section 13.8: The homotopy category (standard reference, not scraped)