Let \(f\colon X\to Y\) be a continuous map with mapping cylinder \(M_f=\qty{(X\times I)\disjoint Y}/\qty{(x,1)\sim f(x)}\). The mapping cone of \(f\) is \[ C_f\coloneqq M_f/(X\times\ts{0})=\qty{CX\disjoint Y}/\qty{(x,1)\sim f(x)\text{ for }x\in X}, \] where \(CX=(X\times I)/(X\times\ts{0})\) is the cone on \(X\); that is, \(C_f\) is \(Y\) with \(CX\) attached along \(f\) [@Hat02].
If \((X,A)\) is a CW pair and \(\iota\colon A\injects X\) is the inclusion, then the mapping cone \(C_\iota=X\cup CA\) is homotopy equivalent to \(X/A\) [@Hat02].