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.

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A null-homotopic chain map

Definition

Let $C_{∙}, D_{∙}$ be chain complexes in an abelian category. A chain map $f : C_{∙} \to D_{∙}$ is null-homotopic if it is chain homotopic to the zero chain map.

Equivalently, there is a degree- $1$ graded morphism $s : C_{∙} \to D_{∙}$ such that $f_{n} = d_{n + 1}^{D} s_{n} + s_{n - 1} d_{n}^{C}$ for every $n \in Z$ , so $f$ is a degree- $0$ boundary in $\underset{}{Hom} (C, D)_{∙}$ .

Depends on

Used by

Null-homotopic maps induce zero on homology Corollary
A contractible complex Definition
Null-homotopic maps form a two-sided additive ideal Proposition
Hom in the homotopy category is zero-degree homology of the Hom complex 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)