DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

The axioms AB3 and AB3*

Definition

An abelian category satisfies AB3 when it has all small coproducts, and it satisfies AB3* when it has all small products (Finite, small, and large limits and colimits; complete and cocomplete categories).

Because an abelian category already has cokernels and kernels, the criterion A category is complete exactly when it has all small products and equalizers, and cocomplete exactly when it has all small coproducts and coequalizers identifies AB3 with cocompleteness and AB3* with completeness inside the abelian setting.

Depends on

Used by

The axioms AB4 and AB4* Definition
The axioms AB5 and AB5* Definition
A coproduct of projectives is projective and a product of injectives is injective Theorem
The cancellation and epimorphism descriptions of a generator agree Theorem

Dependency tree · two levels

7 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

Charles A. Weibel, An Introduction to Homological Algebra, Appendix A.4 (standard reference, not scraped)
The Stacks Project, Tag 079B (standard reference, not scraped)