Mapping cylinder
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In mathematics, specifically algebraic topology, the mapping cylinder of a function f between topological spaces X and Y is the quotient
![M_f = (X\times [0,1]) \cup Y\,/\,\sim](http://upload.wikimedia.org/math/7/6/2/762e7c180f6a236e6a6c04b6034bf22a.png)
where the union is disjoint, and ∼ is the equivalence relation
That is, the mapping cylinder Mf is obtained by gluing one end of X × [0,1] to Y via the map f. Notice that the "top" of the cylinder X × {0} is homeomorphic to X, while the "bottom" is the space Y.
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[edit] Basic properties
The bottom Y is a deformation retract of Mf. The projection 
splits (via 
), and the deformation retraction (with time parameterized by s) is given by:
![\begin{cases}(x,t) \mapsto (x,t+s) \in X \times [0,1] & t+s \leq 1\\
(x,t) \mapsto f(x) \in Y& t+s \geq 1
\end{cases}](http://upload.wikimedia.org/math/9/9/e/99e7b52f6d7fe80aeff1be8c3e62c35f.png)
(where all points in Y are fixed, as this is a deformation retraction).
[edit] 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 Mf, together with a cofibration 
and a surjective homotopy equivalence (indeed, Y is a deformation retract of Mf), such that the composition 
equals f.
Thus the space Y gets replaced with a homotopy equivalent space Mf, 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.
[edit] Applications
The use of mapping cylinders is to apply theorems concerning subspaces or inclusions of spaces to general maps which may not be injective.
Consequently, theorems or techniques (such as homology, cohomology, or homotopy theory itself) which are independent of the homotopy class of the spaces and maps involved may be applied to X,Y,f with the assumption that 
and that f 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 X to points of Y, and hence of embedding X within Y, despite the fact that the function need not be one-to-one. That the construction yields a picture which is homotopy equivalent to the intuitive one indicates that intuition is a correct picture so long as deformation of Y is not an obstacle.
[edit] Categorical application and interpretation
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).
Conversely, the mapping cylinder is the homotopy pushout of the diagram where and 
.
[edit] Mapping telescope
Given a sequence of maps
the mapping telescope is the homotopical direct limit. If the maps are all already cofibrations (such as for the orthogonal groups 
), 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, and is so called because the picture of the construction looks like a stack of increasingly large cylinders, like a telescope.
Formally, one defines it as
![\coprod_i X_i \times [0,1] / (x_i,1) \sim (f(x_i),0)](http://upload.wikimedia.org/math/6/a/b/6ab1a137dae3f5c4308ec9c611b0fe0a.png)
