Paths and cylinders
For every topological space there is a CW complex and a weak homotopy equivalence . Such a map is called a CW approximation to .
Such CW-approximation may be constructed case-by-case by iteratively attaching (starting from the empty space) cells for each representative of a homotopy group of and further cells to kill off spurious homotopy groups introduced this way (e.g. Hatcher, p. 352-353).
In the classical model structure on topological spaces , the cofibrant objects are the retracts of cell complexes, and hence CW approximations are in particular cofibrant replacements in this model structure.
The Quillen equivalence to the classical model structure on simplicial sets (“homotopy hypothesis”) yields a functorial CW approximation (by this proposition) via
(geometric realization of the singular simplicial complex of ) with the adjunction counit
a weak homotopy equivalence.
For topological spaces
Let be a continuous function between topological spaces. Then there exists for each a relative CW-complex together with an extension , i.e.
such that is n-connected.
if itself is k-connected, then the relative CW-complex may be chosen to have cells only of dimension .
if is already a CW-complex, then may be chosen to be a subcomplex inclusion.
(tomDieck 08, theorem 8.6.1)
For every continuous function out of a CW-complex , there exists a relative CW-complex that factors followed by a weak homotopy equivalence
Apply prop. 1 iteratively for all to produce a sequence with cocone of the form
where each is a relative CW-complex and where in n-connected.
Let then be the colimit over the sequence (its transfinite composition) and the induced component map. By definition of relative CW-complexes, this is itself a relative CW-complex.
By the universal property of the colimit this factors as
Finally to see that is a weak homotopy equivalence: since n-spheres are compact topological spaces, then every map factors through a finite stage as (by this lemma). By possibly including further into higher stages, we may choose . But then the above says that further mapping along is the same as mapping along , which is -connected and hence an isomorphism on the homotopy class of .
For sequential topological spectra
First let be a CW-approximation of the component space in degree 0, via prop. 2. Then proceed by induction: suppose that for a CW-approximation has been found such that all the structure maps are respected. Consider then the continuous function
Applying prop. 2 to this function factors it as
Hence we have obtained the next stage of the CW-approximation. The respect for the structure maps is just this factorization property: