Mapping Cone
Mapping cone may refer to one of the following two different but related concepts in mathematics: * Mapping cone (topology) * Mapping cone (homological algebra) In homological algebra, the mapping cone is a construction on a map of chain complexes inspired by the analogous construction in topology. In the theory of triangulated categories it is a kind of combined kernel and cokernel: if the chain comp ...

Mapping Cone (topology)
In mathematics, especially homotopy theory, the mapping cone is a construction C_f of topology, analogous to a quotient space. It is also called the homotopy cofiber, and also notated Cf. Its dual, a fibration, is called the mapping fibre. The mapping cone can be understood to be a mapping cylinder Mf, with one end of the cylinder collapsed to a point. Thus, mapping cones are frequently applied in the homotopy theory of pointed spaces. Definition Given a map f\colon X \to Y, the mapping cone C_f is defined to be the quotient space of the mapping cylinder (X \times I) \sqcup_f Y with respect to the equivalence relation \forall x,x' \in X, (x, 0) \sim \left(x', 0\right)\,, (x, 1) \sim f(x). Here I denotes the unit interval , 1with its standard topology. Note that some authors (like J. Peter May) use the opposite convention, switching 0 and 1. Visually, one takes the cone on ''X'' (the cylinder X \times I with one end (the 0 end) identified to a point), and glues the ot ...
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