Mapping cone

Summary

The mapping cone of a chain map $f \colon C_\bullet \to D_\bullet$ is a chain complex $\mathrm{cone}(f)$. It is built from $D$ and a shifted copy of $C$, with a differential that uses $f$. It measures the failure of $f$ to be a quasi-isomorphism: $f$ is one exactly when the cone has zero homology. The cone fits into a short exact sequence of complexes, and in the homotopy category and the derived category it gives the distinguished triangles.