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

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FALSE: a preadditive category with a zero object must have binary biproducts

Statement

False claim: every preadditive category with a zero object has binary biproducts.

Facts & Assumptions

Given: The full subcategory $C$ of $A b$ on the two objects $0$ and $Z$ .

[L1]

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

[L2]

In a preadditive category, initial and terminal objects coincide (In a preadditive category, an object is initial exactly when it is terminal).

Refutation

technique · direct

1.1

The subcategory $C$ is preadditive because each of its hom-sets is either $0$ or $Z$ , with composition inherited from $A b$ . The object $0$ is both initial and terminal, so $C$ has a zero object by [L2].

L1L2

2.1

If $Z$ had a product with itself inside $C$ , that product would have to be one of the existing objects $0$ or $Z$ . But $C (Z, 0) = 0$ and $C (Z, Z) = Z$ , whereas the universal property of a product $Z \times Z$ would require $C (Z, Z \times Z) ≅ Z \times Z$ . Neither $0$ nor $Z$ has that hom-set.

step 1.1

3.1

So $C$ is preadditive with a zero object and still lacks the binary product of $Z$ with itself, hence lacks a binary biproduct.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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, Exercise 2A (standard reference, not scraped)