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

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 $0\leq i \leq n$.

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 '10 at 17:22
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 '10 at 17:23
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 '10 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 '10 at 1:45
@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 '10 at 4:25