DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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.

The arrow category of an abelian category

Definition

Let $[1]$ denote the category with two objects and one nonidentity arrow. For an abelian category $A$ , the arrow category $A^{\to}$ is the functor category $[[1], A]$ .

Thus an object of $A^{\to}$ is a morphism $x : X_{0} \to X_{1}$ in $A$ , and a morphism in $A^{\to}$ is a commutative square between such arrows.

Because $[1]$ is small, this is an honest functor category by Functor category $[C, D]$ and If $C$ is small and $D$ is locally small then $[C, D]$ is locally small; if both are small it is small. Because limits and colimits in a functor category are computed pointwise, an abelian category gives an abelian arrow category as well.

The arrow category of an abelian category

Definition

Depends on

Dependency tree · 2 levels

Sources