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The mapping cone is a special case of the double mapping cylinder.
Mapping cylinders are quite common homotopical tools.
The mapping telescope is a sequence of mapping cylinders, joined end-to-end.
The mapping cylinder of f is by definition the mapping cone of g.
Conversely, the mapping cylinder is the homotopy pushout of the diagram where and .
That is, the mapping cylinder is obtained by gluing one end of x to via the map .
The mapping cylinder may be viewed as a way to replace an arbitrary map by an equivalent cofibration, in the following sense:
One use of mapping cylinders is to apply theorems concerning inclusions of spaces to general maps, which might not be injective.
Mapping cylinder (homological algebra)
The mapping cone is the degenerate case of the double mapping cylinder (also known as the homotopy pushout), in which one space is a single point.
Given a map , the mapping cylinder is a space , together with a cofibration and a surjective homotopy equivalence (indeed, Y is a deformation retract of ), such that the composition equals f.
One can use the mapping cylinder to construct homotopy limits: given a diagram, replace the maps by cofibrations (using the mapping cylinder) and then take the ordinary pointwise limit (one must take a bit more care, but mapping cylinders are a component).
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