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.
Best Answer
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
http://groupoids.org.uk/nonab-a-t.html
and also this 2016 preprint on Modelling and Computing Homotopy Types: I.
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!