CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 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.

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

1.1L1L2L3

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].

2.1givenstep 1.1∎

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.

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)