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edited Apr 18, 2012 at 19:41

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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"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''structures" that this category is exactly the cofibrant objects in his mixed model structure.

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.

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.

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created Apr 18, 2012 at 18:42

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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.