CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

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

1.1L1L3

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.

1.2L4L5algebra

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.

2.1L2L3step 1.1step 1.2construct

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 $A^{o p}$ .

3.1L2step 2.1∎

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.

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)