## Slick verification of the model category axioms for Spaces and SSets with the q-model structure?

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We choose our category of spaces to be compactly generated weak Hausdorff spaces for convenience, denoted $CGWH$.

Questions:

1.) Is there any sort of slick argument to verify that CGWH with the Quillen model structure is a right-proper (closed) model category?

2.) If we give the following presentation of the model structure on SSet:

Cofibrations are monomorphisms

Fibrations have the RLP with respect to all horn inclusions $\Lambda^n_i \subseteq \Delta^n$ for $0\leq i \leq n$.

Or instead of the characterization of cofibrations, we could instead give:

Trivial fibrations have the RLP with respect to all inclusions of the boundary $\partial^n \subseteq \Delta^n$.

(The point of picking a nice presentation is that the (morally) right choice of definition often simplifies a proof.)

Is there any way to verify the model category axioms more easily? The proofs I've seen appeal to all of the hard work done in question 1. It seems like one should be able to verify the axioms for SSet more easily than the case of CGWH spaces.

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Why does it seem like SSet should be easier? SSet has all sorts of problems that CGWH doesn't, like the fact that you can't compose homotopies in general. – Mike Shulman Jun 1 2010 at 17:22

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There are proofs for SSet that don't go via CGWH, though; there's one in some notes that I can't remember where to find, and another one in the forthcoming sequel to "A concise course in algebraic topology." – Mike Shulman Jun 1 2010 at 17:23

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I believe that there's also a forthcoming book by Joyal and Tierney that contains a combinatorial verification of the model category axioms for SSet. Tom Fiore once told me he saw a draft online, but I don't know where, and I can't find it. (These might be the notes Mike mentions?) – Dan Ramras Jun 1 2010 at 20:26
Your second characterization in 2. is not sufficient to determine the q-model structure in sSet---it simply determines the cofibrations, but there are numerous model structures with fewer weak equivalences than the q-model structure but the same cofibrations. One example is the Joyal model structure. What steps of the usual proofs do you think are "hard"? Simplicial sets are convenient for many arguments, but Top has many great properties that sSet does not. – Sam Isaacson Jun 2 2010 at 1:45

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@Harry, sorry I misread your question. By Cisinski's work, you can easily construct a "minimal" model structure on sSet (one with the fewest weak equivalences given that Cof = Mono) and then left Bousfield localize at the horn inclusions. But then checking that weak homotopy equivalences $X \to Y$ are the same as those maps inducing isos $Ho(Y, Z) \to Ho(X, Z)$ for all Kan $Z$ requires some combinatorics. Incidentally, the fact that you detect all acyclic cofs just with inner horn inclusions is fairly special. I think Cisinski's proof is beautiful, but not concise. – Sam Isaacson Jun 2 2010 at 4:25
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## 1 Answer

The Joyal and Tierney notes contain a combinatorial proof as Dan says. They are available here. (Wanted to post this in the comments, but it seems impossible to do without sufficient reputation.) I should also mention that it is possible to give a reasonably slick proof of the model category axioms for simplicial sets by using the Cisinski machinery (see his monograph: Ast'erisque vol. 308).

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Is there any way to get a copy of Cisinki's book without paying 90-some Euros? To be honest, I'd rather do something like pay him directly for an electronic edition instead of shelling out the 20 USD for shipping from France. – Harry Gindi Jun 1 2010 at 21:50

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I'd have thought that most university libraries should have a copy of the Ast'erisque volumes. That said, a fair amount of the same material (as far as the model structure on simplicial sets is concerned) can be found in his paper "Th'eories homotopiques dans les topos" which is available on his website. – Michael A Warren Jun 1 2010 at 22:09
What do you know, they did have it, and it looks brand new! – Harry Gindi Jun 1 2010 at 23:07

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If you're in North America and you want to own your own copy, you can buy Astérisque volumes directly from the AMS (but Cisinski's book is 107 USD). You can download a PDF from his website: www-math.univ-paris13.fr/~cisinski/ast.pdf – Sam Isaacson Jun 2 2010 at 0:41
Thanks for the link to the Joyal-Tierney notes! – Dan Ramras Jun 3 2010 at 4:14
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