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

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Additive functors and natural transformations form a preadditive category

Statement

If $C$ is small and $D$ is preadditive, then the category whose objects are additive functors $C \to D$ and whose morphisms are natural transformations is preadditive.

Facts & Assumptions

Given: A small category $C$ and a preadditive category $D$ .

[L1]

The functor category has functors as objects and natural transformations as morphisms (Functor category $[C, D]$ ).

[L2]

If the source is small and the target locally small, then the functor category is locally small (If $C$ is small and $D$ is locally small then $[C, D]$ is locally small; if both are small it is small).

[L3]

In a preadditive category each hom-set is an abelian group and composition is bilinear (Preadditive category).

[L4]

A functor is additive exactly when each induced map on hom-sets is a group homomorphism (Additive functor).

Proof

technique · direct

1.1

Since a preadditive category is locally small by definition of its hom-sets, [L2] makes the natural transformations between any two additive functors $F, G : C \to D$ into a set. For $α, β : F \Rightarrow G$ , define $(α + β)_{A} : = α_{A} + β_{A}$ in $D (F A, G A)$ . The naturality square for $α + β$ commutes because composition in $D$ is bilinear by [L3]. Thus each hom-set is an abelian group under pointwise addition.

L1L2L3

2.1

If $β : G \Rightarrow H$ and $α : F \Rightarrow G$ , then vertical composition satisfies $(β α + β^{'} α^{'})_{A} = (β_{A} α_{A} + β_{A}^{'} α_{A}^{'})$ , so fixing either factor and using bilinearity in $D$ shows composition of natural transformations is bilinear.

L1L3step 1.1

3.1

The underlying category structure comes from [L1], the hom-sets are abelian groups by step 1.1, and composition is bilinear by step 2.1. Hence the additive functors and natural transformations form a preadditive category.

L1L4step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Mike Prest, Modules as exact functors, Functor categories (standard reference, not scraped)