DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-08-31

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The cone triangle of a chain map

Definition

Let $f : C_{∙} \to D_{∙}$ be a chain map. The associated cone triangle is the chain-level sequence $C_{∙} \overset{f}{\to} D_{∙} \overset{j}{\to} Cone (f)_{∙} \overset{q}{\to} C [1]_{∙},$ where $j$ and $q$ are the canonical maps of The canonical inclusion and projection for a mapping cone.

On this page the term "cone triangle" refers only to this explicit construction; the later triangulated-category page supplies the abstract distinguished-triangle language.

Depends on

Used by

Cones preserve chain-homotopy equivalences of arrows Proposition
The cone triangle of a null-homotopic map splits in the homotopy category Proposition

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

The Stacks Project, Section 13.9: Cones and termwise split sequences (standard reference, not scraped)
Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)