CorollaryStatement: AI-adaptedProof: 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 cocomplete locally small abelian category with a generator supplies the category-side SAFT hypotheses dually

Statement

Let $A$ be a cocomplete locally small abelian category with a generator. Then $A$ is well-powered and co-well-powered. In the opposite category $A^{o p}$ , the object $G$ is coseparating. If a supplied well-powering of $A$ is given, taking cokernels supplies a co-well-powering of $A$ , equivalently a supplied well-powering of $A^{o p}$ . Thus $A^{o p}$ supplies the category-side data in the supplied-well-powering branch of the objectwise special adjoint functor theorem. The target-category and continuity hypotheses remain hypotheses on the particular functor to which that theorem is applied.

Facts & Assumptions

Given: A cocomplete locally small abelian category $A$ with a generator $G$ .

[L1]

Such a category is well-powered (An AB3 locally small abelian category with a generator is well-powered).

[L2]

The supplied-well-powering branch of objectwise SAFT requires a complete locally small domain with a supplied small coseparating set and a supplied well-powering; the target must be locally small and the functor must preserve all small limits. (Special adjoint functor theorem, objectwise form with explicit intersection smallness or preservation data)

[L3]

In an abelian category, subobjects and quotient objects correspond by kernel and cokernel (Kernel and cokernel are mutually inverse order-preserving correspondences between subobjects and quotient objects).

[L4]

A generator is a separating object (Generator and cogenerator of a category).

[L5]

The opposite of an abelian category is abelian. (The opposite of an abelian category is abelian)

Proof

technique · direct

1.1

By [L1], every object of $A$ has a set of representative monomorphisms for its subobject classes. By [L3], taking cokernels transfers these to representative epimorphisms for all quotient-object classes. Thus $A$ is both well-powered and co-well-powered.

L1L3

1.2

By [L5], $A^{o p}$ is abelian, and cocompleteness of $A$ becomes completeness of $A^{o p}$ . Local smallness is unchanged. The separating property of $G$ from [L4] becomes the coseparating property in the opposite category.

L4L5algebra

2.1

A supplied well-powering of $A$ gives a supplied family of representative monomorphisms. Applying cokernels objectwise using [L3] gives a supplied co-well-powering of $A$ , which is a supplied well-powering of $A^{o p}$ . Hence the domain-side hypotheses in branch 1 of [L2] hold for $A^{o p}$ .

L2L3step 1.1step 1.2construct

3.1

Therefore the category supplies exactly the stated dual SAFT data. As [L2] requires, any application must still provide a locally small target and a functor preserving all small limits; those are not consequences of the present category-level hypotheses.

L2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 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

Peter Freyd, Abelian Categories, Section 3.3 (standard reference, not scraped)