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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?

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I'm pretty surprised to hear that you've found a substantial number of homotopy theorists willing to express a clear preference for CW complexes over simplicial sets. Both are very useful, for different purposes. I see no reason to prefer one to the other in general, although certainly there are specific situations in which one is easy to work with and the other would be very difficult or annoying to use. –  Dan Ramras Jul 12 '10 at 23:24
It suffices to specify the degree of the attaching maps (only) if you are interested in computing homology groups, but a CW structure needs the full (homotopy class) of attaching maps. For example, CP^2 and S^2 \vee S^4 have isomorphic cellular chain complexes but are not homotopy equivalent as can be seen through their cohomology rings. The attaching map for CP^2 is the Hopf map S^3 --> S^2, which does not have a sensible degree. Indeed, one fact which "everyone should know" is that the cellular chain complex loses information needed to compute cohomology ring structure. –  Dev Sinha Jul 12 '10 at 23:31
I've never worked with CW complexes. All of my experience with homotopy theory is with simplicial sets and model categories. That's why I'm asking this question =). –  Harry Gindi Jul 12 '10 at 23:46
It's hard to talk about manifolds, classifying spaces, the Pontryagin-Thom construction, $G$-equivariant homotopy theory where $G$ is compact Lie, etc. without making reference to topological spaces. –  Sam Isaacson Jul 12 '10 at 23:56
By the way, there's an amazing theorem by Mike Mandell that roughly says that as an $E_\infty$-algebra, $C^\ast(X; F_p)$ retains all the homotopical information about $X$ if $X$ is $p$-complete, nilpotent, connected, and of finite type. The rational version of this statement is due to Quillen. But as Dev pointed out, there's no obvious way to get this multiplicative structure when you work with cellular chains. –  Sam Isaacson Jul 13 '10 at 1:09
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3 Answers

up vote 9 down vote accepted

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?

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Good points! Just to register a counterpoint, I think it's at least as easy associating a simplicial set to a manifold as associating a CW-complex to one: choose a Riemannian metric, then take the Cech nerve of a covering by geodesically convex open sets. This isn't a Kan complex, though, which reminds us of another convenience of CW-complexes: more constructions are automatically homotopy-invariant. –  Dustin Clausen Jul 13 '10 at 16:10
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My gut reaction is always to work with CW complexes because, being a topologist, I like to work with spaces. Simplicial sets, as nice as they may be, are definitely not spaces.

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Why the negative vote? –  Mariano Suárez-Alvarez Jul 13 '10 at 0:57
I don't know why this was downvoted. There are many types of topologists in the world. While the more algebraically-minded sometimes prefer simplicial objects, the more geometrically-minded (I could myself among this group) feel more comfortable working with actual spaces, and thus often prefer CW complexes. Sometimes it is a matter of technology (eg if your space is a manifold it would be perverse to replace it w/ a simplicial set, as you would lose the ability to talk about the tangent/stable normal bundles, embedded submanifolds, etc), but often it is just a matter of taste. –  Andy Putman Jul 13 '10 at 0:58
I'm not sure I agree with the statement that simplicial sets are definitely not spaces. Of course, set theoretically, that's true. But would you say that simplicial complexes are not spaces? One can describe simplicial complexes as sets equipped with a "downward closed" collection of subsets, and yet topologists certainly think of simplicial complexes as spaces. Simplicial sets are just another, more flexible, way to describe spaces combinatorially. Some homotopy theorists may prefer not to think of simplicial sets as geometric objects, but that's hardly a universal point of view. –  Dan Ramras Jul 13 '10 at 3:23
CW complexes aren't exactly spaces either for that matter - they're spaces with a decomposition... Sometimes we might confuse the notions of "CW complexes" and "spaces homotopy equivalent to CW complexes," and thus not as readily recognize their respective drawbacks. –  Dev Sinha Jul 13 '10 at 6:26
Dan, as a topologist who's only recently learned not to run screaming from the room when simplicial sets come up, I'd argue (from a naive point of view) that simplicial complexes are spaces more than simplicial sets are. This is probably because I tend to (happily) confuse complexes with their realizations. But back to the original question, CW complexes come ready-made with topologies that simplicial sets need an extra step to get to. I'd also argue (honestly) that the literature is better for a student to learn about CW complexes than simplicial sets. –  Greg Friedman Jul 13 '10 at 10:07
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Both languages are very important. Working with cell complexes you can use geometry, approximations to proof things looking irrational on pure simplicial level. On simlicial language universal constructions looks much better.

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