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.

Homology object of a chain complex

Definition

Let $C_{∙}$ be a chain complex and let $β_{n} : B_{n} (C) \to Z_{n} (C)$ be the canonical morphism supplied by The boundary subobject factors through the cycle subobject.

The $n$ th homology object of $C$ is the cokernel $H_{n} (C) : = coker (β_{n}) .$

Equivalently, one writes $H_{n} (C) = Z_{n} (C) / B_{n} (C),$ with the understanding that this quotient notation abbreviates the cokernel of the canonical monomorphism from boundaries to cycles.

Homology object of a chain complex

Definition

Depends on

Dependency tree · 2 levels

Sources