TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Additive functors from a small preadditive category to an abelian category form an abelian category

Statement

If $C$ is a small preadditive category and $A$ is abelian, then the category of additive functors $C \to A$ and natural transformations is abelian.

Facts & Assumptions

Given: A small preadditive category $C$ and an abelian category $A$ .

[L1]

Additive functors and natural transformations form a preadditive category (Additive functors and natural transformations form a preadditive category).

[L2]

The smallness of $C$ makes the relevant functor categories 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]

Limits and colimits in functor categories are computed pointwise (For small source and index categories, chosen target limits and colimits compute the corresponding functor-category limits and colimits pointwise).

[L4]

Abelian categories are additive and have pointwise kernels, cokernels, and coimage-image isomorphisms (Abelian category).

Proof

technique · direct

1.1

By [L1], the additive functors already form a preadditive category. The zero functor is additive, and binary biproducts are computed pointwise because [L4] gives biproducts in $A$ and [L3] computes them pointwise. So the additive functor category is additive.

L1L3L4

2.1

Let $η : F \Rightarrow G$ be a natural transformation. By [L3], its kernel and cokernel in the ambient functor category are computed pointwise, and the pointwise constructions lie in $A$ . Because $C$ is preadditive and $F, G$ are additive, the induced structure maps on those pointwise kernels and cokernels are again additive by uniqueness in the kernel and cokernel universal properties. The canonical coimage-to-image map is likewise computed pointwise and is an isomorphism at each object by [L4]. Therefore the additive functor category satisfies the axioms of an abelian category.

L2L3L4step 1.1∎

Depends on

Used by

Representations of the quiver 1 -> 2 in abelian groups form an abelian category Example

Dependency tree · two levels

15 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

Alexandre Grothendieck, Some aspects of homological algebra, §1.6 (standard reference, not scraped)
Gautam Tamme, Algebra II Lecture 9, §9.5 (standard reference, not scraped)
Junhan Tan, The Freyd-Mitchell Embedding Theorem, Theorem 5.1 (standard reference, not scraped)