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\begin{document}

%-------------------------------------------------------------------


\section*{mapping telescope}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{homotopy}{}\paragraph*{{Homotopy}}\label{homotopy}

[[!include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{for_cwcomplexes}{For CW-complexes}\dotfill \pageref*{for_cwcomplexes} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Given a sequence

\begin{displaymath}
X_\bullet
  =
  \left(
  X_0 
   \overset{f_0}{\longrightarrow}
  X_1
   \overset{f_1}{\longrightarrow}
  X_2
   \overset{f_2}{\longrightarrow}
  \cdots
  \right)
\end{displaymath}
of ([[pointed topological space|pointed]]) [[topological spaces]], then its \emph{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{\tt \symbol{94}}1 and Milnor sequences]] (and that's maybe the origin of the concept?).

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{MappingTelescope}\hypertarget{MappingTelescope}{}
For

\begin{displaymath}
X_\bullet
  =
  \left(
  X_0 
   \overset{f_0}{\longrightarrow}
  X_1
   \overset{f_1}{\longrightarrow}
  X_2
   \overset{f_2}{\longrightarrow}
  \cdots
  \right)
\end{displaymath}
a sequence in [[Top]], its \textbf{mapping telescope} is the [[quotient topological space]] of the [[disjoint union]] of [[product topological spaces]]

\begin{displaymath}
Tel(X_\bullet)
  \coloneqq
  \left(
  \underset{n \in \mathbb{N}}{\sqcup}  
  \left(
    X_n \times [n,n+1]
  \right)
  \right)/_\sim
\end{displaymath}
where the [[equivalence relation]] quotiented out is

\begin{displaymath}
(x_n, n) \sim (f(x_n), n+1)
\end{displaymath}
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

\begin{displaymath}
Tel(X_\bullet)
  \coloneqq
  \left(
  \underset{n \in \mathbb{N}}{\sqcup}  
  \left(
    X_n \wedge [n,n+1]_+
  \right)
  \right)/_\sim
  \,.
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{for_cwcomplexes}{}\subsubsection*{{For CW-complexes}}\label{for_cwcomplexes}

\begin{prop}
\label{TelescopeOfCWComplexEquivalentToTheOriginal}\hypertarget{TelescopeOfCWComplexEquivalentToTheOriginal}{}
For $X_\bullet$ the sequence of stages of a ([[pointed topological space|pointed]]) [[CW-complex]] $X = \underset{\longleftarrow}{\lim}_n X_n$, then the canonical map

\begin{displaymath}
Tel(X_\bullet)
  \longrightarrow
  X
\end{displaymath}
from the [[mapping telescope]], def. \ref{MappingTelescope}, is a [[weak homotopy equivalence]].

\end{prop}
\begin{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

\begin{displaymath}
Tel(X_n) \longrightarrow X_n
\end{displaymath}
is a [[homotopy equivalence]], hence a [[weak homotopy equivalence]]. A concrete construction of a homotopy inverse is given for instance in (\hyperlink{Switzer75}{Switzer 75, proof of prop. 7.53}).

Moreover, since spheres are [[compact object|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

\begin{displaymath}
\underset{\longrightarrow}{\lim}_n \pi_q(Tel(X_n)) 
    \overset{\simeq}{\longrightarrow} 
  \pi_q(Tel(X))
\end{displaymath}
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.

\begin{displaymath}
\itexarray{
    \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)    
  }
  \,.
\end{displaymath}
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.

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[homotopy colimit]]

\end{itemize}
[[!include universal constructions of topological spaces -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Robert Switzer]], \emph{Algebraic Topology - Homotopy and Homology}, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

\end{itemize}


\end{document}