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.

A chain homotopy equivalence

Definition

A chain map $f : C_{∙} \to D_{∙}$ is a chain homotopy equivalence if there exists a chain map $g : D_{∙} \to C_{∙}$ such that $g \circ f ≃ 1_{C}, f \circ g ≃ 1_{D} .$

Such a map $g$ is called a homotopy inverse of $f$ .

Depends on

Used by

A contractible complex Definition
A chain isomorphism is a chain homotopy equivalence Proposition
Homotopy equivalence is an equivalence relation on complexes Proposition
Shift preserves homotopy equivalences, contractibility, and quasi-isomorphisms Proposition
A chain homotopy equivalence is a quasi-isomorphism Theorem

Dependency tree · two levels

5 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

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
Joseph J. Rotman, An Introduction to Homological Algebra, 2nd ed. (standard reference, not scraped)