The comments are getting a bit too tightly nested up above, so I’ll go down here to tackle Urs’ questions about model categories versus $(\infty,1)$categories.
Urs wrote:
Every model category can be turned into an $(\infty,1)$category. Right?
Right. The classic version of this result uses the oldest version of $(\infty,1)$categories: simplicially enriched categories. These are categories enriched over the category of simplicial sets — or roughly speaking, categories where you’ve got a simplicial set of morphisms between any two objects.
(Simplicially enriched categories were the first and in some ways the simplest approach to $(\infty,1)$categories.)
In 1980, Dwyer and Kan found two different ways to turn a model category into a simplicially enriched category. I think these are usually called the ‘Dwyer–Kan simplicial localization’ and the ‘hammock localization’. But, they proved these two methods give weakly equivalent simplicially enriched categories.
Here are the relevant papers. As usual for Dwyer and Kan, they’re frighteningly elegant:

William G. Dwyer and Daniel M. Kan,
Simplicial localizations of categories,
J. Pure Appl. Algebra 17 (1980), 267284.

William G. Dwyer and Daniel M. Kan,
Calculating simplicial localizations,
J. Pure Appl. Algebra 18 (1980), 1735.

William G. Dwyer and Daniel M. Kan,
Function complexes in homotopical algebra,
Topology 19 (1980), 427440.
Urs wrote:
And the converse should be easy and well known, I guess.
Umm, that doesn’t sound right. I think there’s more information in a model category than just its underlying simplicially enriched category.
The reason is that to build the simplicially enriched category we only need the ‘most important portion’ of our model category. We need the very nice objects (those that are both fibrant and cofibrant), we need the morphisms between these, and we need to know which of those morphisms are weak equivalences.
So, don’t shoot me if I’m wrong, but I’m pretty sure you can cook up Quillen inequivalent model categories that give weakly equivalent simplicially enriched categories.
Also, I’ve never heard of a way to take an arbitrary simplicially enriched category and promote it to a model category! That sounds tough.
So, I think we should view model categories as a convenient trick for constructing and working with $(\infty,1)$categories — but not ‘the same thing’.
I have to remind myself how this construction of an $(\infty,1)$category from a model category works.
Again: there are two famous constructions, which give weakly equivalent answers.
Is it really the same $(\infty,1)$category on the nose for two Quillenequivalent model categories?
If by ‘the same on the nose’ you mean equal or isomorphic, the answer is no.
Instead, Quillen equivalent model categories give weakly equivalent simplicially enriched categories.
Re: The Volume of a Differentiable Stack
This is great! Now: roll on the applications!