Alphabeta MathDefinitionDefinition: 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 is a chain homotopy equivalence if there exists a chain map such that
Such a map is called a homotopy inverse of .
Depends on
Used by
- A contractible complex Definition
- Whitehead torsion of a finite CW homotopy equivalence Definition
- A lifted finite CW equivalence has a contractible group-ring mapping cone Lemma
- Basis-change, direct-sum and based exact-sequence formulas Lemma
- The handle complex of an h-cobordism is contractible over the group ring, with an explicit contraction Lemma
- The three-cone calculation for a composite chain map Lemma
- A chain isomorphism is a chain homotopy equivalence Proposition
- Cones preserve chain-homotopy equivalences of arrows 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
- A chain map is a homotopy equivalence exactly when its cone is contractible Theorem
- The mapping cylinder factors a chain map 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)