ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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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.

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A preadditive category with two objects and a nonzero hom-group

Example

Let $C$ have two objects $X, Y$ with

$C (X, X) = Z, C (Y, Y) = Z, C (X, Y) = Z, C (Y, X) = 0,$

and let composition be the obvious integer multiplication where defined and the zero map otherwise. Then $C$ is preadditive and has a nonzero mixed hom-group $C (X, Y)$ .

Facts & Assumptions

Given: The category $C$ described in the Example.

[L1]

A preadditive category has abelian-group hom-sets and bilinear composition (Preadditive category).

Verification

technique · direct

1.1

Each displayed hom-set is an abelian group under ordinary integer addition, except $C (Y, X) = 0$ , which is the trivial abelian group. The chosen identity morphisms are the integers $1 \in Z$ on $X$ and $Y$ .

2.1

Composition is bilinear because wherever integer multiplication is defined it distributes over integer addition, and any composite involving $C (Y, X) = 0$ is automatically zero. So the data satisfy the definition [L1], while $C (X, Y) = Z \neq 0$ gives the promised nonzero mixed hom-group.

L1step 1.1∎

Depends on

Preadditive category

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 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

Kiran S. Kedlaya, Solid modules over an ordinary ring, Section 1.2 (standard reference, not scraped)