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

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In a preadditive category with a zero object, the zero morphism is the neutral element of each hom-group

Statement

Let $C$ be a preadditive category with a zero object $0$ . For any objects $A, B$ , the published zero morphism $A \to 0 \to B$ is the additive identity of the abelian group $C (A, B)$ .

Facts & Assumptions

Given: A zero object $0$ in a preadditive category $C$ and objects $A, B$ .

[L1]

A zero object supplies a unique compatible system of zero morphisms (A zero object supplies a unique compatible system of zero morphisms).

[L2]

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

[L3]

Every hom-set in a preadditive category is an abelian group and composition is bilinear (Preadditive category).

Proof

technique · direct

1.1

Let $u : A \to 0$ and $v : 0 \to B$ be the unique arrows from [L1]. Because $0$ is both initial and terminal by [L2], the endomorphism group $C (0, 0)$ has only one element, so $1_{0} = 0_{0, 0}$ .

L1L2L3

2.1

The published zero morphism is $v u$ . Using step 1.1, $v u = v 1_{0} u = v 0_{0, 0} u$ . Bilinearity in [L3] says composing with the zero element of $C (0, 0)$ gives the zero element of $C (A, B)$ , so $v u = 0_{A, B}$ .

L3step 1.1

3.1

Therefore the zero morphism coming from the zero object is exactly the neutral element of the hom-group $C (A, B)$ .

L1step 2.1∎

Depends on

Used by

Dependency tree · two levels

6 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 12.3: Preadditive and additive categories (standard reference, not scraped)