# Technical question about cell complexes

Hello,

I have a technical question. My terminology:

I - set of standard inclusions $\partial I^n \to I^n$.

I-cell (Relative Cell Complexes) - transfinite compositions of pushouts of maps in $I$.

CW (CW complexes) - the usual definition (like I-cell but with "cells attached by order of dimension).

I-cof - retracts of maps in I-cell, the same as maps having Left Lifting Property w.r.t. maps having Right lifting property w.r.t. I (by Quillen small object argument).

My first question is to confirm that CW in I-cell in I-cof are all different

My second question is: In this page: http://ncatlab.org/nlab/show/model+structure+on+topological+spaces it is written under "Mixed model structure" (the model structure on $Top$ for which equivalences are weak equivalences and fibrations are Hurewicz fibrations) that cofibrant objects are spaces homotopy equivalent to CW complexes. Is this exactly true (or is it for example spaces homotopy equivalent to cell complexes?). I also read somewhere around nlab that Milnor advocated that spaces homotopy equivalent to CW complexes are nice, so I wanted to know if the cofibrant objects in this mixed model structure are exactly such, or a bit more general.

Thank you, Sasha

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They are exactly such. One point is that one can use classical cell complexes, stop at $\omega$ with no transfinite nonsense, in setting up the Quillen model structure: it is a compactly generated (as well as a cofibrantly generated) model category. The distinction and full details are in May and Ponto, "More concise algebraic topology", published Feb. 1 this year. Another is that classical cell complexes are homotopy equivalent to CW complexes, as one can see by approximating attaching maps by maps that land in the n-skeleton. Formally, one has two filtrations on cell complexes, one given by the order of construction, the other given by dimensions of cells. The distinction is familiar and essential when one goes stable and works with spectra rather than spaces. Milnor wrote a classical paper "On spaces of the homotopy type of CW complexes" not just advocating but proving the niceness of the category of CW homotopy types. It is a result of Cole "Mixing model structures" that this category is exactly the cofibrant objects in his mixed model structure.