TheoremStatement: Literature-sourcedProof: AI-adaptedSession-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.

Freyd and Mitchell's characterisation of abelian categories

Statement

For a category $A$ , the following are equivalent.

$A$ is abelian in the working sense of Abelian category.
$A$ satisfies Freyd's axioms A0, A1, A1*, A2, A2*, A3, and A3*.
$A$ has a zero object, pullbacks, and pushouts, and every monomorphism is a kernel while every epimorphism is a cokernel.

Facts & Assumptions

Given: A category $A$ .

[L1]

An abelian category is additive, has kernels and cokernels, and has invertible coimage-image comparison maps (Abelian category).

[L2]

In an abelian category every monomorphism is the kernel of its cokernel, and dually every epimorphism is the cokernel of its kernel (Every monomorphism is the kernel of its cokernel, and dually every epimorphism is the cokernel of its kernel).

[L3]

Freyd's axioms imply the working abelian definition (Freyd's axioms force the additive structure and recover the AB2 definition).

[L4]

An additive category with all kernels and cokernels has all finite limits and finite colimits (An additive category with all kernels and cokernels has all finite limits and colimits).

Proof

technique · direct

1.1

If clause 1 holds, then [L1] gives additivity, kernels, and cokernels. The zero object and binary biproducts in [L1] supply the zero-object, product, and coproduct clauses, while [L2] supplies the normality and conormality clauses. So clause 1 implies clause 2.

L1L2

1.2

Clause 2 implies clause 1 by [L3].

2.1

If clause 2 holds, then step 1.2 and [L4] give all finite limits and finite colimits, hence in particular pullbacks and pushouts. So clause 2 implies clause 3. Conversely, if clause 3 holds, then the pullback of $A \to 0 \leftarrow B$ is a product of $A$ and $B$ , the pushout of $A \leftarrow 0 \to B$ is a coproduct, the pullback of $A \overset{f}{\to} B \overset{}{\leftarrow} 0$ is a kernel of $f$ , and the pushout of $0 \overset{}{\leftarrow} A \overset{f}{\to} B$ is a cokernel of $f$ . Together with the stated normal and conormal clauses, that is exactly Freyd's list.

L1L3L4step 1.2

3.1

Steps 1.1, 1.2, and 2.1 prove the three-way equivalence.

step 1.1step 1.2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 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

Barry Mitchell, Theory of Categories, Theorem 20.1 (standard reference, not scraped)
Junhan Tan, The Freyd-Mitchell Embedding Theorem, §2 (standard reference, not scraped)