TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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A functor between additive categories is additive exactly when it preserves finite biproducts

Statement

For a functor between additive categories, the following are equivalent:

the functor is additive;
the functor preserves finite biproducts.

Facts & Assumptions

Given: A functor $F : C \to D$ between additive categories.

[L1]

Additive functors preserve finite biproducts (An additive functor preserves finite biproducts).

[L2]

Additive categories are preadditive with finite biproducts (Additive category).

[L3]

In a preadditive category, finite products are automatically biproducts (In a preadditive category, a finite product is automatically a biproduct).

[L4]

In an additive category, the preadditive hom-group law is a bilinear commutative-monoid enrichment compatible with the finite biproduct diagrams. Indeed, product uniqueness gives $⟨ f, g ⟩ = i_{1} f + i_{2} g$ , and bilinearity then gives $\nabla ⟨ f, g ⟩ = f + g$ . By uniqueness this is the canonical biproduct addition (The commutative-monoid enrichment of a category with finite biproducts is unique).

Proof

technique · direct

1.1L1

The implication from 1 to 2 is exactly [L1].

1.2L2L3

Assume $F$ preserves finite biproducts. In an additive category, for parallel morphisms $f, g : A \to B$ , the pairing into $B \oplus B$ is the unique morphism $⟨ f, g ⟩$ with projections $f$ and $g$ , and the codiagonal is the unique morphism $\nabla_{B} : B \oplus B \to B$ with both composites equal to $1_{B}$ . Since $F$ preserves the relevant biproducts, it preserves those pairings and codiagonals.

2.1L2L4step 1.2

By [L4], the hom-group addition in an additive category is the canonical biproduct addition. [L2, L4, step 1.2] $f + g = \nabla_{B} ⟨ f, g ⟩$ . Step 1.2 therefore gives $F (f + g) = F (\nabla_{B}) F (⟨ f, g ⟩) = \nabla_{F B} ⟨ F f, F g ⟩ = F f + F g$ . So $F$ is additive.

A functor between additive categories is additive exactly when it preserves finite biproducts

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources