DefinitionDefinition: Literature-sourcedProof: 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.

Cycle and boundary subobjects of a complex

Let $C_{∙}$ be a chain complex in an abelian category and fix $n \in Z$ .

The $n$ th cycle subobject of $C$ is $Z_{n} (C) : = \ker (d_{n}) ↪ C_{n} .$

The $n$ th boundary subobject of $C$ is $B_{n} (C) : = im (d_{n + 1}) ↪ C_{n} .$

Thus both are subobjects of $C_{n}$ , one defined by the kernel of $d_{n}$ and the other by the image of $d_{n + 1}$ .

Depends on