If $J$ is a class of maps in a category, the $J$-cellular maps are by definition transfinite compositions of pushouts of coproducts of maps in $J$.

Now if $J$ denotes the family of inclusions $S^{n-1}\hookrightarrow D^n$ for all $n$, then CW complexes are elements of Cell($J$).

However, not every map of Cell($J$) is a relative CW complex since for the latter we are only allowed to use the $n$-th inclusion $S^{n-1}\hookrightarrow D^n$ at the $n$-th step of a transfinite composition (we can only glue $n$-cells at step $n$).

The retracts of maps in Cell($J$) are exactly the cofibrations in a $(J, J_{ac})$-cofibrantly generated model structure. The Quillen model structure on Top (with Serre fibrations) is cofibrantly generated with generating cofibrations the family of inclusions $S^n\hookrightarrow D^n$.

Also, the CW complexes are exactly the cofibrant objects in this model structure (read on nlab https://ncatlab.org/nlab/show/CW+complex).

That must mean that CW complexes are exactly the retracts of cell complexes. This is false since there are cell complexes which are not CW complexes.

Where is the mistake ??? are there more cofibrant objects in Top than just CW complexes ?

  • $\begingroup$ Yes, there are more cofibrant objects in Top than just CW complexes, for the reason you've pointed out. $\endgroup$
    – JHF
    Apr 12 '19 at 15:34

They are not all the cofibrant objects in the Quillen model structure on spaces, they are "among" the cofibrant objects- as is stated in the nlab article you linked in the question.

They are all of the cofibrant objects in the mixed model structure, see here, at least up to actual homotopy equivalence.


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