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I apologize if this is to basic for MO, but it just seems a bit to advanced for SE.

Let $\text{Top}$ be the category of topological spaces and $\text{sSet}$ the category of simplicial sets. There is an adjunction $$ |\cdot|:\text{sSet} \to \text{Top}:Sing $$ which is a Quillen equivalence with respect to the standard model category structures. One of the fundamental results of the general theory is that this adjunction induces an equivalence of the homotopy categories (the localisations with respect to weak equivalences). It is often described in textbooks as an application of model category theory.

It seems to me that in order to prove the equivalence of homotopy categories, by abstract nonsense, one needs only to prove that the counit of the adjunction, (i.e. $|Sing(X)|\to X$) is a weak equivalence. My first question is

Is there a direct/slick proof that $|Sing(X)|\to X$ is a weak equivalence without using model category theory?

Remark: It is easy to see that it induces isomorphism on homology, since the simplicial chain complex of the first is exactly the singular complex of the second, but this is of course not enough.

Frankly, I would be very surprised if the easiest proof of this fact indeed requires model category whose machinery is rather heavy to set up. If my intuition is correct, it seems that mentioning it as an application of model category theory is a bit misleading. I understand completely (at least on the philosophical level) that model structure gives you much more than just equivalence of the homotopy categories, but I would be glad to have a pedagogical answer (to myself) to the following question:

What nice and easy to describe applications model category theory has?

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You can find a "direct proof" in the paper by Curtis Simplicial Homotopy Theory (Advances in Mathematics, Volume 6, Issue 2, April 1971, Pages 107–209). The main ingredient is the use of barycentric subdivision.

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  • $\begingroup$ Beware that Curtis' 1971 proof of the equivalence (his Theorem 1.35) contains a serious mistake. Diagram (ii) on page 201 does not commute, so Barratt's unpublished 1956 construction of a homeomorphism |Sd(X)| \cong |X| fails. A different homeomorphism was found by Fritsch and Puppe in 1967. The rest of Curtis' argument can probably be repaired using sections 2.3 and 2.5 of my book with Waldhausen and Jahren (AoMS 186) -- I should check someday. $\endgroup$ – John Rognes May 6 '14 at 15:30
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It misses the point to think of model category as a tool for proving that homotopy categories are equivalent. In the case of simplicial sets and topological spaces, that equivalence long preceded the introduction of model category theory and therefore, of course, is in no way dependent on it.* Rather the use of model category theory gives a lot more point set level structure to this and other mere equivalences of homotopy categories. There is a great deal of mathematics that cannot be thought, at least not readily, without the use of model category theory. Voevodsky's importation of the methodology of modern algebraic topology in algebraic geometry, which allowed him to prove the Milnor conjecture, is an extreme example. Examples within algebraic topology are abundant. I don't have the time or inclination to pinpoint a small elementary example, since that is not really the point. Use of model category theory in algebraic topology and in homotopical algebra (Quillen's apt term) in general is analogous to the use of category theory in mathematics.

*An unoriginal direct proof is in my 1967 book "Simplicial objects in algebraic topology".

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  • $\begingroup$ I understand that it is not the point. Thank you for the explanation though. $\endgroup$ – KotelKanim Feb 9 '14 at 16:00
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    $\begingroup$ Actually, I first became truly convinced of the force of model category theory while writing EKMM. I had long wanted to prove that the periodic K-theory spectra are E_{\infty} ring spectra, knowing by infinite loop space theory that their connective covers are such. Thinking model theoretically, this became so easy it was like a joke (in fact, I burst out laughing in the shower when I noticed it). $\endgroup$ – Peter May Feb 9 '14 at 19:24
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    $\begingroup$ @KotelKanim - perhaps a simpler example of the application of model categories would be what Quillen first used it for. Namely he answered: what is the cohomology of commutative rings? Concretely, he wanted to extend an exact sequence of abelian groups involving 3 rings (highly non-abelian objects) into a long exact sequence reminiscent of the usual one you get from derived functors cohomology in the sense of abelian categories. In this sense, model categories is a way to make sense of "abelian" ideas (e.g. derived functors, cohomology) in non-abelian settings! $\endgroup$ – Elden Elmanto Mar 26 '14 at 15:38

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