%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: approximation theorem for an arbitrary space
%%% Primary Category Code: 00.
%%% Filename: ApproximationTheoremForAnArbitrarySpace.tex
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\begin{document}
\begin{theorem}({\em Approximation \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html} for an arbitrary \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} space in terms of the colimit of a sequence of cellular inclusions of $CW$-complexes}):
\begin{quote}
``There is a functor $\Gamma: \textbf{hU} \longrightarrow \textbf{hU}$ where
$\textbf{hU}$ is the homotopy category for unbased spaces , and a natural transformation $\gamma: \Gamma \longrightarrow Id$ that asssigns a $CW$-complex $\Gamma X$ and a weak equivalence $\gamma _e:\Gamma X \longrightarrow
X$ to an arbitrary space $X$, such that the following diagram commutes:
$$
\begin{CD}
\Gamma X @> \Gamma f >> \Gamma Y
\\ @V $20:32, 25 June 2015 (UTC)~~$\gamma (X) VV @VV \gamma (Y) V
\\ X @ > f >> Y
\end{CD}
$$
and $\Gamma f: \Gamma X\rightarrow \Gamma Y$ is unique up to homotopy equivalence.''
\end{quote}
(viz. p. 75 in ref. \cite{MJP1999}).
\end{theorem}
\begin{remark}
The $CW$-complex specified in the
\htmladdnormallink{approximation theorem for an arbitrary space}{http://planetphysics.us/encyclopedia/ApproximationTheoremForAnArbitrarySpace.html} is constructed as the colimit $\Gamma X$ of a sequence of cellular inclusions of $CW$-complexes $X_1, ..., X_n$ , so
that one obtains $X \equiv colim [X_i]$. As a consequence of J.H.C. Whitehead's Theorem, one also has that:
$\gamma* : [\Gamma X,\Gamma Y] \longrightarrow[\Gamma X, Y]$ is an \htmladdnormallink{isomorphism}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html}.
Furthermore, the \htmladdnormallink{homotopy groups}{http://planetphysics.us/encyclopedia/ExtendedHurewiczFundamentalTheorem.html} of the $CW$-complex $\Gamma X$ are the colimits of the
homotopy groups of $X_n$ and $\gamma_{n+1} : \pi_q(X_{n+1})\longmapsto\pi_q (X)$ is a \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} \htmladdnormallink{epimorphism}{http://planetphysics.us/encyclopedia/Epimorphism2.html}.
\end{remark}
\begin{thebibliography} {9}
\bibitem{MJP1999}
May, J.P. 1999, \emph{A Concise Course in Algebraic Topology.}, The University of Chicago Press: Chicago
\end{thebibliography}
\end{document}
