 ##  [Mapping Cone](/mapping-cone) 

  ##  [Mapping Cone](https://puremath.quantumdictionary.io/mapping-cone-0) 

  
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**Pure Mathematics Dictionary**

 
Definition

The mapping cone Cf of a map f: X→Y is the quotient of the mapping cylinder Mf obtained by collapsing X×{0} to a point; equivalently Cf = Mf / (X×{0} ∼ *). The mapping cone models the homotopy cofiber of f and measures the failure of f to be surjective on homotopy.