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

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The mapping cone of a chain map

Definition

Let $f : C_{∙} \to D_{∙}$ be a chain map in an additive category. The mapping cone of $f$ is the chain complex $Cone (f)_{∙}$ whose degree- $n$ term is $Cone (f)_{n} : = D_{n} \oplus C [1]_{n} = D_{n} \oplus C_{n - 1},$ and whose differential is $d_{n}^{Cone (f)} (y, x) : = (d_{n}^{D} (y) + f_{n - 1} (x), - d_{n - 1}^{C} (x)) .$

Thus $Cone (f)$ is the direct sum of $D$ and the shift $C [1]$ as a graded object, with the off-diagonal term given by $f$ .

Depends on

Used by

The canonical inclusion and projection for a mapping cone Definition
The cone triangle of a chain map Definition
The mapping cylinder of a chain map Definition
The relative homology of a chain map Definition
The cone of multiplication by m on the integers Example
FALSE: the mapping-cone differential needs no minus sign False statement
Homotopic maps have chain-isomorphic mapping cones Lemma
The mapping-cone differential squares to zero Lemma
The three-cone calculation for a composite chain map Lemma
An exact functor carries mapping-cone sequences to mapping-cone sequences Proposition
Cones preserve chain-homotopy equivalences of arrows Proposition
Isomorphic chain maps have isomorphic cones Proposition
The cone construction commutes with shift up to the canonical sign isomorphism Proposition
The cone of the zero map is the direct sum with a shift Proposition
The quotient of the mapping cylinder by its source is the mapping cone Proposition
A chain map is a homotopy equivalence exactly when its cone is contractible Theorem
A chain map is a quasi-isomorphism exactly when its cone is acyclic Theorem
A morphism of chain maps induces a chain map of cones Theorem
The cone of an identity map is contractible Theorem

Dependency tree · two levels

6 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)
The Stacks Project, Section 13.9: Cones and termwise split sequences (standard reference, not scraped)