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 CW-complexes):

“There is a functor Γ : hU→hU where hU is the homotopy category for unbased spaces , and a natural transformation γ : Γ→Id that asssigns a CW-complex ΓX and a weak equivalence γ_e : ΓX→X to an arbitrary space X, such that the following diagram commutes:

and Γf : ΓX → ΓY is unique up to homotopy equivalence.”

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

Remark 0.1. The CW-complex specified in the approximation theorem for an arbitrary space is constructed as the colimit ΓX of a sequence of cellular inclusions of CW-complexes X₁,...,X_n , so that one obtains X ≡ colim[X_i]. As a consequence of J.H.C. Whitehead’s Theorem, one also has that:

γ∗ : [ΓX, ΓY ]→[ΓX,Y ] is an isomorphism.

Furthermore, the homotopy groups of the CW-complex ΓX are the colimits of the homotopy groups of X_n and γ_n+1 : π_q(X_n+1)π_q(X) is a group epimorphism.