PropositionStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31

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Homotopy equivalence is an equivalence relation on complexes

Statement

Chain homotopy equivalence is an equivalence relation on chain complexes.

Facts & Assumptions

Given: Chain complexes $C_{∙}, D_{∙}, E_{∙}$ .

[L1]

A homotopy equivalence is a chain map with a homotopy inverse (A chain homotopy equivalence).

[L2]

Chain homotopy is compatible with composition (Chain homotopy is compatible with addition and composition).

Proof

technique · direct

1.1

Every complex is homotopy equivalent to itself: the identity map is its own homotopy inverse, with the zero homotopies witnessing $1_{C} 1_{C} ≃ 1_{C}, 1_{C} 1_{C} ≃ 1_{C} .$ Symmetry is immediate by swapping a map with its chosen homotopy inverse.

L1givenalgebra

2.1

If $f : C \to D$ has homotopy inverse $g$ and $u : D \to E$ has homotopy inverse $v$ , then $(g v) (u f) = g (v u) f ≃ g 1_{D} f = g f ≃ 1_{C}$ and similarly $(u f) (g v) ≃ 1_{E}$ by [L2]. Hence $u f$ is again a homotopy equivalence. Together with step 1.1, this proves reflexivity, symmetry, and transitivity.

L1L2step 1.1algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)