CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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.

An exact functor need not be faithful

Statement refuted

Every exact functor between abelian categories is faithful.

Facts & Assumptions

Given: Two nonzero abelian categories $A$ and $B$ , and an object $Y$ of $B$ with $1_{Y} \neq 0_{Y, Y}$ .

[L1]

A finite product of preadditive categories is preadditive (A small product of preadditive categories is preadditive).

[L2]

Abelian categories are additive and have kernels, cokernels, and coimage-image isomorphisms (Abelian category).

[L3]

Exact means additive, left exact, and right exact (Exact functor between abelian categories).

Counterexample

1.1

The product category $A \times B$ is preadditive by [L1], and its zero object, kernels, cokernels, finite biproducts, and canonical coimage-image maps are all computed componentwise from the corresponding structures in the two factors. Since both factors satisfy [L2], the product does too. The projection $π_{1} : A \times B \to A$ preserves those componentwise constructions, so it is exact in the sense of [L3].

L1L2L3

2.1

The endomorphism $(0, 1_{Y}) : (0, Y) \to (0, Y)$ is nonzero in $A \times B$ , but $π_{1} (0, 1_{Y}) = 0$ . So $π_{1}$ is not faithful. Therefore the refuted claim is false.

givenstep 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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 (standard reference, not scraped)