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The homotopy category is additive
Statement
If is an additive category, then is an additive category.
Facts & Assumptions
Given: An additive category .
In , morphisms are homotopy classes of chain maps (The homotopy category of chain complexes).
Null-homotopic maps form a two-sided additive ideal (Null-homotopic maps form a two-sided additive ideal).
The category is additive (The category of complexes in an additive category is additive).
Finite biproducts of complexes are computed degreewise (Finite biproducts of complexes are computed degreewise).
Proof
By [L3], each hom-set of is an abelian group. Quotienting by the additive subgroup of null-homotopic maps from [L2] therefore gives an abelian group structure on each from [L1].
The zero complex and the degreewise biproduct complex exist in 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 has a zero object and finite biproducts, so it is additive.