ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-28

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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.

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Representations of the quiver 1 -> 2 in abelian groups form an abelian category

Example

A representation of the quiver $1 \to 2$ in abelian groups is just a homomorphism $u : A_{1} \to A_{2}$ , and a morphism of such representations is a commutative square. These representations form an abelian category.

Facts & Assumptions

Given: The free preadditive category on the quiver $1 \to 2$ and the target category $A b$ .

[L1]

Abelian groups form an abelian category (Abelian groups form an abelian category).

[L2]

Additive functors from a small preadditive category to an abelian category form an abelian category (Additive functors from a small preadditive category to an abelian category form an abelian category).

Verification

technique · direct

1.1L1L2

The free preadditive category on the quiver $1 \to 2$ is small, and an additive functor out of it is exactly the data of two abelian groups and one homomorphism between them.

2.1L1L2step 1.1∎

Therefore the category of quiver representations is a special case of [L2] with target $A b$ from [L1]. So it is abelian.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Alexandre Grothendieck, Some aspects of homological algebra, §1.6 (standard reference, not scraped)