approximation theorem for an arbitrary space

Theorem 0.1.

(Approximation theorem for an arbitrary topological space in terms of the colimit of a sequence of cellular inclusions of $C\mathrm{}W$-complexes):

“There is a functor $\mathrm{\Gamma }:\text{𝒉𝑼}⟶\text{𝒉𝑼}$ wherehU is the homotopy category for unbased spaces , and a natural transformation $\gamma :\mathrm{\Gamma }⟶Id$ that asssigns a $CW$-complex $\mathrm{\Gamma }X$ and a weak equivalence ${\gamma }_e:\mathrm{\Gamma }X⟶X$ to an arbitrary space $X$, such that the following diagram commutes:

$$\begin{array}{ccc}\hfill \mathrm{\Gamma }X\hfill & \hfill \stackrel{\mathrm{\Gamma }f}{\to }\hfill & \hfill \mathrm{\Gamma }Y\hfill \\ \hfill \gamma (X)\downarrow \hfill & & \hfill \downarrow \gamma (Y)\hfill & & \\ \hfill X>f\gg Y\hfill \end{array}$$

and $\mathrm{\Gamma }f:\mathrm{\Gamma }X\to \mathrm{\Gamma }Y$ is unique up to homotopy equivalence.”

(viz. p. 75 in ref. [1]).

Remark 0.1.

The $CW$-complex specified in theapproximation theorem for an arbitrary space (http://planetmath.org/ApproximationTheoremForAnArbitrarySpace) is constructed as the colimit $\mathrm{\Gamma }X$ of a sequence of cellular inclusions of $CW$-complexes $X_1,\mathrm{\dots },X_n$ , sothat one obtains $X\equiv colim[X_i]$. As a consequence of J.H.C. Whitehead’s Theorem, one also has that:

$\gamma *:[\mathrm{\Gamma }X,\mathrm{\Gamma }Y]⟶[\mathrm{\Gamma }X,Y]$ is an isomorphism.

Furthermore, the homotopy groups of the $CW$-complex $\mathrm{\Gamma }X$ are the colimits of thehomotopy groups of $X_n$ and ${\gamma }_{n+1}:{\pi }_q(X_{n+1})⟼{\pi }_q(X)$ is a group epimorphism.