In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function between topological spaces and is the quotient where the denotes the disjoint union, and ∼ is the equivalence relation generated by That is, the mapping cylinder is obtained by gluing one end of to via the map. Notice that the "top" of the cylinder is homeomorphic to, while the "bottom" is the space. It is common to write for, and to use the notation or for the mapping cylinder construction. That is, one writes with the subscripted cup symbol denoting the equivalence. The mapping cylinder is commonly used to construct the mapping cone, obtained by collapsing one end of the cylinder to a point. Mapping cylinders are central to the definition of cofibrations.
For a fiber bundle with fiber the mapping cylinder has the equivalence relation for. Then, there is a canonical map sending a point to the point, giving a fiber bundle whose fiber is the cone. To see this, notice the fiber over a point is the quotient space where every point in is equivalent.
Interpretation
The mapping cylinder may be viewed as a way to replace an arbitrary map by an equivalent cofibration, in the following sense: Given a map, the mapping cylinder is a space, together with a cofibration and a surjective homotopy equivalence , such that the composition equals f. Thus the space Y gets replaced with a homotopy equivalent space, and the map f with a lifted map. Equivalently, the diagram gets replaced with a diagram together with a homotopy equivalence between them. The construction serves to replace any map of topological spaces by a homotopy equivalent cofibration. Note that pointwise, a cofibration is a closed inclusion.
Applications
Mapping cylinders are quite common homotopical tools. One use of mapping cylinders is to apply theorems concerning inclusions of spaces to general maps, which might not be injective. Consequently, theorems or techniques which are only dependent on the homotopy class of spaces and maps involved may be applied to with the assumption that and that is actually the inclusion of a subspace. Another, more intuitive appeal of the construction is that it accords with the usual mental image of a function as "sending" points of to points of and hence of embedding within despite the fact that the function need not be one-to-one.
Categorical application and interpretation
One can use the mapping cylinder to construct homotopy colimits: this follows from the general statement that any category with all pushouts and coequalizers has all colimits. That is, given a diagram, replace the maps by cofibrations and then take the ordinary pointwise limit. Conversely, the mapping cylinder is the homotopy pushout of the diagram where and.
Mapping telescope
Given a sequence of maps the mapping telescope is the homotopical direct limit. If the maps are all already cofibrations, then the direct limit is the union, but in general one must use the mapping telescope. The mapping telescope is a sequence of mapping cylinders, joined end-to-end. The picture of the construction looks like a stack of increasingly large cylinders, like a telescope. Formally, one defines it as
