 Mapping cone (homological algebra)

For a chain map $f:C\to C^{\prime }$, one mapping-cone convention takes $M(f{)}_i=C_{i-1}\oplus C_i^{\prime }$ with a differential combining $d,d^{\prime }$ and $f$. Its short exact sequence with $C^{\prime }$ and a shift of $C$ produces a long exact sequence in homology.

In homological algebra, a mapping cone is a construction that allows us to define a new complex from a given morphism of chain complexes. It plays a significant role in various contexts, such as in the study of derived categories and in the formulation of the Long Exact Sequence in homology.