Mapping cylinder

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In mathematics, the mapping cylinder of a function $f$ between topological spaces $X$ and $Y$ is a way to replace an arbitrary map by an equivalent cofibration, in the following sense:

Given a map $f\colon X \to Y$ , the mapping cylinder is a space $M f$ , together with a cofibration $\tilde f\colon X \to M_f$ and a surjective homotopy equivalence $M_f \to Y$ (indeed, Y is a deformation retract of $M f$ ), such that the composition $X \to M_f \to Y$ equals f.

I.e., such that the following diagram commutes:

(fixme: diagram)

You've replaced the space Y with an equivalent space $M f$ , and the map f with a lifted map $\tilde f$ . Alternately, you've replaced the diagram

$f\colon X \to Y$

with the diagram

$\tilde f\colon X \to M_f$

together with a homotopy equivalence between them.

"Homotopically, every map is equivalent to a cofibration."

Note that pointwise, a cofibration is a closed inclusion.

1 Construction
2 Applications
- 2.1 Categorical application and interpretation
- 2.2 Mapping telescope

[edit] Construction

You construct the mapping cylinder $M f$ by gluing one end of $X \times I$ to Y by f.

You include X into $M f$ at the other end, and you map $M f$ to Y by squashing the cylinder onto Y.

Formally, define:

$M_f = ((X \times I) \coprod Y)/(x,1) \sim f(x),$

where $I$ is the unit interval, $\coprod$ denotes the disjoint union of two topological spaces, and $\sim$ is an equivalence relation that identifies $(x,1) \in X \times I$ with $f(x) \in Y$ (glue one end of the cylinder $X \times I$ to Y by f).

Define $X \to M_f$ by $x \mapsto (x,0) \in X \times I$ (include X at the other end).

Define $M_f \to Y$ by $(x,t) \in X \times I \mapsto f(x) \in Y$ (squash the cylinder onto Y).

Note that Y is a deformation retract of $M f$ . The projection $M_f \to Y$ splits (via $y \in Y \mapsto Y \in Y \subset M_f$ ), and the deformation retraction (with time parameterized by s) is given by:

$\begin{cases}(x,t) \mapsto (x,t+s) \in X \times I & t+s \leq 1\\ (x,t) \mapsto f(x) \in Y& t+s \geq 1 \end{cases}$

(where all points in Y are fixed, as this is a deformation retraction).

[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 $X \subset Y$ 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 $f\colon X \to Y$ and $\text{id}_X\colon X \to X$ .

[edit] Mapping telescope

Given a sequence of maps

$X_1 \to_{f_1} X_2 \to_{f_2} X_3 \cdots$

the mapping telescope is the homotopical direct limit. If the maps are all already cofibrations (such as for the orthogonal groups $O(n) \subset O(n+1)$ ), 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.