# Advantages of working with CW complexes/spaces over Kan complexes/simplicial sets?

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?

-
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

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?

-
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

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.

-
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
There's a difference between saying you think of simplicial sets geometrically and saying that you think of them as spaces. At the very least, I'm pretty sure that simplicial sets aren't the same kinds of spaces as manifolds, complex varieties, etc. I'd actually say that whether simplicial complexes are spaces depends on how you think of simplicial complexes. I think of them as a bunch of simplices glued together, so to me they are spaces. But I have the feeling that some people think of them as baby versions of simplicial sets... – Andy Putman Jul 13 '10 at 6:19
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

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.

-

It is always good to know both CW complexes and simplicial sets. Let me give a few examples.

1. To define cup products in some multiplicative cohomology theory, one needs the diagonal map $X\to X\times X$. It does not respect a CW structure in general, so one has to approximate it. One knows that such an approximation always exists. If you need a concrete one, you have to work. For simplicial sets on the other hand, the diagonal map is simplicial. But then it is harder to relate $h^\bullet(X\times X)$ with $h^\bullet(X)\otimes h^\bullet(X)$. One of the possible solutions for ordinary cohomology leads to the well-known cup-product formula in singular cohomology. It is interesting to notice that this cup-product formula looks as if it came from an approximation of the diagonal in the CW-product of the geometric realisations.

2. Every topological space has an approximation by a weakly homotopy equivalent CW complex. If you are lucky, you find one with very few cells, for example a single point suffices for the polish circle. But there always is a natural choice, the realisation of the singular complex $\left|S_\bullet(-)\right|$, which is awfully large in most cases. However, in the case of the polish circle, you may argue that a weakly homotopy equivalent approximation loses too much information, and prefer to use some entirely different theory.

3. Starting from a smooth manifold $M$, a Morse function together with a gradient-like vector field gives a CW complex, which is again far from natural. But you immediately recover the dimension of $M$ from the cell structure (which you can then use to prove a cup-length estimate, for example). The singular simplicial complex has nondegenerate simplices in all dimensions, so you really have to work until you recover $\dim M$.

4. Yet another point. For any topological group $G$, Milnor's join construction gives a model for the classifying space $BG$ that is a simplicial space. It is "made" to classify $G$-bundles gives by $G$-cocycles. On the other hand, if $G$ is a classical Lie group, you can approximate $BG$ through Grassmannians. These classify vector bundles that are given as subbundles of trivial bundles (which is how you usually view vector bundles in noncommutative geometry). The construction is again less universal, but it connects better with some analytic methods. And one has Schubert cells to work with.

There may even be situations where one wants to combine the strength of both approaches in some hybrid object.

-
point 1 sounds strange to me; the diagonal is beautifully simplicial; it's the isomorphism H*(X x X) ~~ H*(X) o H*(X) that runs simplicially-weird --- and which is beautifully CW. – Jesse C. McKeown Dec 15 '15 at 16:55
@JesseC.McKeown You are right - I was thinking CW-ish. The composition of the diagonal and the Alexander-Whitney map looks like an approximation off the diagonal in the geometric realisation. – Sebastian Goette Dec 16 '15 at 7:58

I feel topologists should be eclectic and able to move across a number of models, to see which better aids understanding. Here is a part quotation from Einstein, which I feel also reflects the concerns of the questioner:

"What is essential and what is based only on the accidents of development?... Concepts which have proved useful for ordering things easily assume so great an authority over us, that we forget their terrestrial origin and accept them as unalterable facts. ....It is therefore not just an idle game to exercise our ability to analyse familiar concepts, and to demonstrate the conditions on which their justification and usefulness depend, and the way in which these developed, little by little..."

Simplicial sets have a very well developed theory, are "convenient" in many ways, but they have limitations. The results of the book Nonabelian Algebraic Topology:filtered spaces, crossed complexes, cubical omega-groupoids (NAT) (pdf available) would not even have been conjectured simplicially, but the notion of multiple compositions of cubes led to their conjecture and proof. A search on "cubical" in mathoverflow gives more relevant information. To go back to the question, the above book gives a new outlook on structures related to CW-filtrations, and the border between homotopy and homology, using cubical sets (with connections).

See also this brief 2015 presentation on "A philosophy of modelling and computing homotopy types": aveiro.

January 5, 2016 Since products are referred to in other answers, I mention that the isomorphism $$C_*(X_*) \otimes C_*(Y_*) \cong C_*(X_* \otimes Y_*)$$ in the cellular case is extended in this case to an isomorphism $$\Pi(X_*) \otimes \Pi(Y_*) \cong \Pi(X_* \otimes Y_*)$$ in the above NAT book. Here $\Pi$ is a homotopically defined functor on filtered spaces with values in crossed complexes: this functor contains information on relative homotopy groups $$\pi_n(X_n, X_{n-1},x), x \in X_0, n \geqslant 2,$$ and on the operation of fundamental group(oid)s on these. Proofs use cubical methods, relying on the isomorphism $I^m_* \otimes I^n_8 \cong I^{m+n}_*$.

-