The mapping cone of a continuous map 
is the space
formed by gluing the base of the cone on 
to 
using 
. For a pointed map between
pointed CW-complexes
whose basepoints are 0-cells, the reduced mapping cone
also collapses the cone on the basepoint. It models
the cofiber of 
in homotopy theory. For
the inclusion map of a nonempty subcomplex 
in a CW-complex 
, the mapping cone is homotopy
equivalent to the quotient space 
(Hatcher 2002, p. 15). Mapping cones provide the space-level
model for cofiber sequences in stable homotopy theory.
In homological algebra, the mapping cone of a morphism 
of cochain complexes
in an additive category is the cochain
complex 
with biproducts
 |
![d_(C(f))^n=[d_B^n f^(n+1); 0 -d_A^(n+1)].](/images/equations/MappingCone/NumberedEquation2.svg) |
The identity 
implies 
(Stacks Project, Definition 13.9.1). These mapping cones give distinguished triangles
in the homotopy category and, for an Abelian
category, in the derived category.
