nLab mapping telescope

Idea

Given a sequence

$$X_\bullet = \left( X_0 \overset{f_0}{\longrightarrow} X_1 \overset{f_1}{\longrightarrow} X_2 \overset{f_2}{\longrightarrow} \cdots \right)$$

of (pointed) topological spaces, then its mapping telescope is the result of forming the (reduced) mapping cylinder $Cyl(f_n)$ for each $n$ and then attaching all these cylinders to each other in the canonical way.

At least if all the $f_n$ are inclusions, this is the sequential attachment of ever “larger” cylinders, whence the name “telescope”.

The mapping telescope is a representation for the homotopy colimit over $X_\bullet$. It is used for instance for discussion of lim^1 and Milnor sequences (and that’s maybe the origin of the concept?).

Definition

Definition

For

$$X_\bullet = \left( X_0 \overset{f_0}{\longrightarrow} X_1 \overset{f_1}{\longrightarrow} X_2 \overset{f_2}{\longrightarrow} \cdots \right)$$

a sequence in Top, its mapping telescope is the quotient topological space of the disjoint union of product topological spaces

$$Tel(X_\bullet) \coloneqq \left( \underset{n \in \mathbb{N}}{\sqcup} \left( X_n \times [n,n+1] \right) \right)/_\sim$$

where the equivalence relation quotiented out is

$$(x_n, n) \sim (f(x_n), n+1)$$

for all $n\in \mathbb{N}$ and $x_n \in X_n$.

Analogously for $X_\bullet$ a sequence of pointed topological spaces then use reduced cylinders to set

$$Tel(X_\bullet) \coloneqq \left( \underset{n \in \mathbb{N}}{\sqcup} \left( X_n \wedge [n,n+1]_+ \right) \right)/_\sim \,.$$

Properties

For CW-complexes

Proposition

For $X_\bullet$ the sequence of stages of a (pointed) CW-complex $X = \underset{\longleftarrow}{\lim}_n X_n$, then the canonical map

$$Tel(X_\bullet) \longrightarrow X$$

from the mapping telescope, def. , is a weak homotopy equivalence.

Proof

Write in the following $Tel(X)$ for $Tel(X_\bullet)$ and write $Tel(X_n)$ for the mapping telescop of the substages of the finite stage $X_n$ of $X$. It is intuitively clear that each of the projections at finite stage

$$Tel(X_n) \longrightarrow X_n$$

is a homotopy equivalence, hence a weak homotopy equivalence. A concrete construction of a homotopy inverse is given for instance in (Switzer 75, proof of prop. 7.53).

Moreover, since spheres are compact, so that elements of homotopy groups $\pi_q(Tel(X))$ are represented at some finite stage $\pi_q(Tel(X_n))$ it follows that

$$\underset{\longrightarrow}{\lim}_n \pi_q(Tel(X_n)) \overset{\simeq}{\longrightarrow} \pi_q(Tel(X))$$

are isomorphisms for all $q\in \mathbb{N}$ and all choices of basepoints (not shown).

Together these two facts imply that in the following commuting square, three morphisms are isomorphisms, as shown.

$$\array{ \underset{\longleftarrow}{\lim}_n \pi_q(Tel(X_n)) &\overset{\simeq}{\longrightarrow}& \pi_q(Tel(X)) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow \\ \underset{\longleftarrow}{\lim}_n \pi_q(X_n) &\underset{\simeq}{\longrightarrow}& \pi_q(X) } \,.$$

Therefore also the remaining morphism is an isomorphism (two-out-of-three). Since this holds for all $q$ and all basepoints, it is a weak homotopy equivalence.

Related concepts

• homotopy colimit

examples of universal constructions of topological spaces:

$\phantom{AAAA}$limits	$\phantom{AAAA}$colimits
$\,$ point space$\,$	$\,$ empty space $\,$
$\,$ product topological space $\,$	$\,$ disjoint union topological space $\,$
$\,$ topological subspace $\,$	$\,$ quotient topological space $\,$
$\,$ fiber space $\,$	$\,$ space attachment $\,$
$\,$ mapping cocylinder, mapping cocone $\,$	$\,$ mapping cylinder, mapping cone, mapping telescope $\,$
	$\,$ cell complex, CW-complex $\,$

References

• Robert Switzer, Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.