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For a category whose objects are CW-complexes (with a chosen cell structure), what is the most natural notion of morphism? Are there choices, and if so, what are the pros and cons?

Types of maps that immediate come to mind would be continuous maps where the direct image a cell in X is a cell Y, or where the inverse image of a cell in Y is a cell in X. There are probably lots of elegant ways to get to maps like this, perhaps involving conditions on chain maps. It seems such an obvious thing to do that surely there are good references on this.

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The class of maps that you pick will surely depend on what you want to use the category for! Unless you tell us what your objective is, deciding what is a pro and what is a con is sort of impossible. – Mariano Suárez-Alvarez Aug 3 '10 at 17:07
The most conceptional definition is, of course, a cellular map. However, cellular approximation tells you that when you are doing homotopy theory, you can relax this condition and just consider continuous maps. In most cases then CW is considered as a full subcategory of Top (or CGHaus). – Martin Brandenburg Aug 3 '10 at 17:10
@Martin: The whole category Top is really poorly behaved with respect to homotopy theory. CGHaus is also not good because it's a bit too strict. The category of topological spaces usually used is CGWH-spaces with the Quillen model structure, since it enjoys a host of very nice properties (cartesian-closed, proper, exhibits the category of CW complexes as the full subcategory of fibrant-cofibrant objects (and hence by the Whitehead theorem that weak equivalences lift to honest homotopy equivalences). – Harry Gindi Aug 3 '10 at 17:18
Top is not cartesian-closed, so it is not suitable for doing homotopy theory... – Harry Gindi Aug 3 '10 at 23:26
@Harry: There are other reasons that Top isn't quite suitable. Just pick up Counterexamples in topology and you will see all kinds of reasons for not doing homotopy theory in top. @Mariano: Perhaps he is asking for just such a list of different choices with the appropriate applications. – Sean Tilson Aug 4 '10 at 0:48

My work on Higher Order Seifert-van Kampen Theorems has led to the conclusion that a very useful category is that of filtered spaces. Of course a CW-complex gives rise to a filtered space with the skeletal filtration.

The reason behind this utility is that we can define (non trivially) a functor $\rho$ from filtered spaces to a form of strict cubical $\omega$-groupoid nicely related to classical invariants (i.e. relative homotopy groups) and which satisfies such a Seifert-van Kampen theorem; this has as a Corollary the Relative Hurewicz Theorem, as well as allowing the computation of the homotopy 2-types of certain pushouts, in nonabelian terms. This then allows some computations of, for example, second homotopy groups, as modules over the fundamental group. The functor $\rho$, being cubical, is also very useful for consideration of tensor products.

For more information, see

Notice also that Grothendieck in Section 5 of `Esquisse d'un programme' makes some interesting comments on the lack of relevance to geometry of the standard notion of topological space, and feels much more structure is needed. This comment needs further discussion in this blog!

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