Knowlpedia

Mapping cone

A complex combining the source and target of a chain map and measuring its failure to be an equivalence.

For a f:X∙→Y∙f:X^\bullet\to Y^\bullet, its mapping cone is the complex with graded object

Cone⁡(f)n=Yn⊕Xn+1\operatorname{Cone}(f)^n=Y^n\oplus X^{n+1}

and differential

dCone⁡(f)n(y,x)=(dYny+fn+1x, −dXn+1x).d_{\operatorname{Cone}(f)}^n(y,x) =\bigl(d_Y^n y+f^{n+1}x,\,-d_X^{n+1}x\bigr).

This formula uses cochain complexes; other grading conventions change the displayed signs. The cone fits into a canonical triangle

X→fY→Cone⁡(f)→X[1].X\xrightarrow{f}Y\to\operatorname{Cone}(f)\to X[1].

Mapping-cone triangles motivate the and the of triangulated categories.