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The category of simplicial sets has a standard model structure, where the weak equivalences are those maps whose geometric realization is a weak homotopy equivalence, the cofibrations are monomorphisms, and the fibrations are Kan fibrations.

Simplicial sets are combinatorial objects, so morally their model structure should not be dependent on topological spaces. Are there any approaches to this model structure which do not use the geometric realization functor, and do not use topological spaces?

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I should mention: Any time you deal with material relating to generating model structures, you (somewhat unfortunately) have to deal with annoying set-theoretic technicalities about size. In particular, there's a very important argument of Quillen, called the small object argument, which allows us to build functorial factorizations, but you cannot apply it unless you have small generating sets for your classes of morphisms. To be able to do a lot of stuff in homotopy theory, it's pretty much a given that you'll want to take functorial holims and hocolims if you want to work with anything... –  Harry Gindi Nov 15 '10 at 1:14
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Harry: Could you expand on your comment in terms of how it is relevant? that is, for someone not well versed in set-theoretic problems and nuances of model categories. –  Sean Tilson Nov 15 '10 at 3:40
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To prove that functorial factorizations exist in the diagram category, you want to apply Quillen's small object argument, but this won't work if you can't find small generating sets for your cofibrations and trivial cofibrations. What Jeff Smith's theorem allows you to do is produce a small generating set for the trivial cofibrations given only an accessibly embedded accessible class of weak equivalences and a generating set for the cofibrations. Accessibility of object-wise weak equivalences can easily be determined from the accessibility of weak equivalences in the original model category –  Harry Gindi Nov 15 '10 at 4:00
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And a small generating set for the cofibrations is easy to find in the injective model structure (again, since these are exactly cofibrations object-by-object and our diagram is small). For the projective model structure, (fibrations are exactly the ones object-by-object), the argument is a bit harder, but you can again determine a small generating set for the cofibrations. –  Harry Gindi Nov 15 '10 at 4:04
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WRT Vopenka's principle, I think that the idea is that it lets you show that a model category is combinatorial iff it is cofibrantly generated. –  Harry Gindi Nov 15 '10 at 4:07
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up vote 15 down vote accepted

Quillen's original proof (in Homotopical Algebra, LNM 43, Springer, 1967) is purely combinatorial (i.e. does not use topological spaces): he uses the theory of minimal Kan fibrations, the fact that the latter are fiber bundles, as well as the fact that the classifying space of a simplicial group is a Kan complex. This proof has been rewritten several times in the literature: at the end of

S.I. Gelfand and Yu. I. Manin, Methods of Homological Algebra, Springer, 1996

as well as in

A. Joyal and M. Tierney An introduction to simplicial homotopy theory

(I like Joyal and Tierney's reformulation a lot). However, Quillen wrote in his seminal Lecture Notes that he knew another proof of the existence of the model structure on simplicial sets, using Kan's $Ex^\infty$ functor (but does not give any more hints).

A proof (in fact two variants of it) using Kan's $Ex^\infty$ functor is given in my Astérisque 308: the fun part is not that much about the existence of model structure, but to prove that the fibrations are precisely the Kan fibrations (and also to prove all the good properties of $Ex^\infty$ without using topological spaces); for two different proofs of this fact using $Ex^\infty$, see Prop. 2.1.41 as well as Scholium 2.3.21 for an alternative). For the rest, everything was already in the book of Gabriel and Zisman, for instance.

Finally, I would even add that, in Quillen's original paper, the model structure on topological spaces in obtained by transfer from the model structure on simplicial sets. And that is indeed a rather natural way to proceed.

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The man himself! –  Harry Gindi Nov 15 '10 at 17:39
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In the introduction to Chapter 3 of Mark Hovey's book on Model Categories, he states that he "does not know if it is possible to give a proof that simplicial sets form a model category without ever referring to topological spaces." This was in 1998. I seem to recall a comment somewhere in the book saying roughly Hovey couldn't find simple proof of some statement made by Quillen and repeated by others; this might be where Hovey had to use topological spaces (but now I can't find the quote). So it's good that people are writing down new versions of the proof. –  Dan Ramras Nov 15 '10 at 23:46
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@ Dan: Quillen's proof is really complete, as well as its reformulations by Gelfand, Manin, Joyal and Tierney (who did not repeat Quillen's proof without understanding it fully!), no matter what M. Hovey might have written and/or believed. –  Denis-Charles Cisinski Nov 16 '10 at 1:48
    
I believe you. It's just evidence of a difficult proof, maybe. I've never gone through all the details, but if I ever do I'm sure I'll be glad to have multiple references to consult. –  Dan Ramras Nov 16 '10 at 3:47
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@ Dan: By any chance, are you not referring to the following remarks in 2.4 "top.spaces"? "Our proof differs from the proofs in [Qui67] and [DS95] mostly in the level of detail. We give full proofs of the required smallness results, and we provide a careful proof that trivial fibrations have the right lifting property with respect to relative cell complexes. Both of these issues are completely avoided in both [Qui67] and [DS95]. We also briefly discuss the model categories of pointed topological spaces and compactly generated topological spaces." And one before Lemma 2.4.11,p.54,(2007 ed): –  mmm Mar 7 '11 at 17:16
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Denis-Charles Cisinski has a beautiful book called Les Préfaisceaux commes modèles des Types d'Homotopie, which gives a very very powerful framework for building model structures on presheaf categories (and more generally Grothendieck toposes), and after building up this framework, the model structure for simplicial sets drops out literally for free.

Here's a link to it from his website: link.

He also proves some nontrivial conjectures of Grothendieck that are important for derivator theory, among other things. Rick Jardine published a summary paper of this book, which is also worth reading.

Note: The framework is built up entirely in chapter 1, so even if you don't want to read the whole book, the first chapter is what you need.

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Which builds on work of Grothendieck from Pursuing Stacks, not that I advise you to read it in depth, because it is very rambling and has backtracks. It is however in English, but for the proofs you need to look in Cisinski. –  David Roberts Nov 15 '10 at 0:11
    
I mean, read Pursuing Stacks in depth. –  David Roberts Nov 15 '10 at 0:11
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He means, don't read pursuing stacks in depth. –  Harry Gindi Nov 15 '10 at 0:14
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That's right. I was impatient, and wouldn't wait for the published version (which is in development limbo, say no more). Killed my eyes... –  David Roberts Nov 15 '10 at 1:33
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Yes. The only one I know of is in a book by Joyal and Tierny. I heard some time ago that the book was going to be published, but I don't know if that has happened. There's a version on the Hopf topology archive:

http://hopf.math.purdue.edu/cgi-bin/generate?/Joyal-Tierney/JT-chap-01

If you look at the first page, they state what you're looking for as their main goal.

If anyone knows of a more recent version, maybe with more chapters, let us know!

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There are many ways to define weak equivalences of simplicial sets without referring to topological spaces.

A morphism f is a weak equivalence of simplicial sets if one of the following equivalent conditions is satisfied:

  • Ex^∞(f) is a simplicial homotopy equivalence.
  • Ex^∞(f) induces isomorphism on simplicial homotopy groups.
  • Hom(f, A) is an isomorphism for every Kan complex (fibrant simplicial set) A.
  • The morphism f is a composition of a trivial cofibration and a trivial fibration.

Note that fibrations and trivial fibrations, as well as cofibrations and trivial cofibrations can be defined using left and right lifting properties.

A book on homological algebra by Gelfand and Manin contains a sketchy construction of the standard model structure on simplicial sets without referring to topological spaces.

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What does the notation Hom(f,A) mean, by the way? –  user332 Nov 15 '10 at 11:52
    
@RC: If f: X→Y is a morphism, then Hom(f,A): Hom(Y,A)→Hom(X,A) is the morphism given by composition with f. –  Dmitri Pavlov Nov 15 '10 at 17:46
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