Definition 6.1.14 (Mapping cone). The mapping cone of a chain map \(f\colon C_{\bullet } \to D_{\bullet }\) is the chain complex \(\Cone (f)\) with \[ \Cone (f)_n := D_n \oplus C_{n-1}, \qquad d^{\Cone (f)}(y,x) := (d^D(y)+f(x),-d^C(x)). \] It fits into a natural short exact sequence \(0 \to D_{\bullet } \to \Cone (f) \to C_{\bullet }[1] \to 0\).
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