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

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A chain homotopy

Definition

Let $C_{∙}, D_{∙}$ be chain complexes in an abelian category $A$ , and let $f, g : C_{∙} \to D_{∙}$ be chain maps. A chain homotopy $s : f ≃ g$ is a graded morphism of degree $1$ , $s_{n} : C_{n} \to D_{n + 1},$ such that for every $n \in Z$ , $f_{n} - g_{n} = d_{n + 1}^{D} s_{n} + s_{n - 1} d_{n}^{C} .$

Equivalently, $f - g$ is the degree- $0$ boundary of the degree- $1$ family $s$ in The Hom complex of chain complexes.

Depends on

Used by

A chain homotopy equivalence Definition
A null-homotopic chain map Definition
Shifted chain maps and shifted chain homotopies Definition
A contracting homotopy for the two-term identity complex Example
Two homotopic maps with different components Example
FALSE: chain-homotopic maps are equal as chain maps False statement
Chain homotopy is compatible with addition and composition Lemma
Chain homotopy is an equivalence relation Proposition
Chain-homotopic maps induce the same map on homology 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)