TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 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.

A left or right exact functor between abelian categories is automatically additive

Statement

Let $F : A \to B$ be a functor between abelian categories. If $F$ is left exact or right exact, then $F$ is additive.

Facts & Assumptions

Given: A functor $F : A \to B$ between abelian categories.

[L1]

Left exact means preserving finite limits, and right exact means preserving finite colimits (Left exact and right exact functors).

[L2]

Abelian categories are additive (Abelian category).

[L3]

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

Proof

technique · direct

1.1

If $F$ is left exact, then by [L1] it preserves the zero object and binary products. In an additive category, products are biproducts by [L2], so $F$ preserves finite biproducts.

L1L2L3

1.2

If $F$ is right exact, then by [L1] it preserves the zero object and binary coproducts. Again [L2] identifies those with finite biproducts, so $F$ preserves finite biproducts.

L1L2L3

2.1

In either case, [L3] applies and shows that $F$ is additive.

L3step 1.1step 1.2∎

Depends on

Used by

Left exactness, right exactness, and exactness are characterized by short exact sequences Theorem

Dependency tree · two levels

10 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

The Stacks Project, Section 12.7, Lemma 12.7.2(1) (standard reference, not scraped)