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.



Slick verification of the model category axioms for Spaces and SSets with the q-model structure?
The Joyal and Tierney notes contain a combinatorial proof as Dan says. They are available [here](http://www.crm.cat/HigherCategories/tierney.pdf). (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).


