DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-08-31

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.

Homotopy classes of chain maps

Definition

Let $C_{∙}$ and $D_{∙}$ be chain complexes in an additive category $A$ . Write $N (C, D)$ for the subgroup of ${Hom}_{Ch (A)} (C, D)$ consisting of null-homotopic chain maps.

The homotopy class of a chain map $f : C_{∙} \to D_{∙}$ is its coset modulo $N (C, D)$ , written $[f]$ . The abelian group of all homotopy classes is $[C, D]_{K} : = {Hom}_{Ch (A)} (C, D) / N (C, D) .$

Depends on

Null-homotopic maps form a two-sided additive ideal

Homotopy classes of chain maps

Definition

Depends on

Dependency tree · 2 levels

Sources