[Math] Advantages of working with CW complexes/spaces over Kan complexes/simplicial sets

Question

Many topologists express a clear preference for working with CW complexes instead of simplicial sets.

One of the reasons is that the cellular chain complex of a CW complex is often easier to work with than a simplicial chain complex. However, simplicial sets have many nice features that spaces do not. The category of simplicial sets has a proper and combinatorial (in the sense of Jeff Smith) model structure and is a presheaf topos, which makes the objects behave very much like sets. Surely these make up for the problems with specifying combinatorial data?

The question: Why do many topologists and homotopy theorists prefer to work with spaces and CW complexes over simplicial sets and Kan complexes? What are some other advantages that CW complexes enjoy over Kan complexes?

Answer 1

I think there are many times that simplicial sets are preferable (e.g for classifying spaces the simplicial construction is often advantageous), but to answer the stated question:

• CW complexes connect more immediately to manifold theory (Morse functions give CW structures; a finite CW complex is homotopy equivalent to a manifold by embedding it in some Euclidean space and "fattening it up").
• CW structures can be simpler and more explicit in "small" cases. For example, I do not know an explicit simplicial set whose realization is $CP^2$ (though perhaps I could work one out using a simplicial model for the Hopf map.)
• CW complexes can be analyzed using manifold theory. For example, maps from manifolds to $n$-dimensional CW complexes such as attaching maps can be understood in part by taking a "smooth" approximation and looking at preimages of points in each cell (Goodwillie uses this kind of technique to generalize the Blakers-Massey theorem).

But why should one have to choose "once and for all" between building things from sets vs. from vector spaces, anyways?