Edit & Note: I'm declaring a convention here because I don't feel like trying to fix this in a bunch of spots: If I said model category and it doesn't make sense, I meant a model-category "model" of an (infinity,1)-category. Also, "model" in quotes means the English word model, whereas without quotes it has do do with model categories.
At the very beginning of Lurie's higher topos theory, he mentions that there is a theory of $(\infty,1)$-categories that can be directly constructed by using model categories.
What I'd like to know is:
Where can I find related papers (Lurie mentions two books that are not available for download)?
How dependent on quasicategories is the theory developed in HTT? Can the important results be proven for these $(\infty,1)$-model-categories by proving some sort of equivalence (not equivalence of categories, but some weaker kind of equivalence) to the theory of quasicategories?
When would we want to use quasicategories rather than these more abstract model categories?
And also, conversely, when would we want to look at model categories rather than quasicategories?
Does one subsume the other? Are there disadvantages to the model category construction just because it requires you to have all of the machinery of model categories? Are quasicategories better in every way?
The only "models" of infinity categories that I'm familiar with are the ones presented in HTT.
Best Answer
You can't produce every ($\infty$,1)-category from a model category. The slogan is that every presentable ($\infty$,1)-category comes from a model category, and every adjoint pair between such comes from a Quillen pair of functors between model categories. The paper by Dugger on Universal model categories works out this formalism from the point of view of model categories. (A companion paper shows that a large class of model categories (the combinatorial ones) are "presentable" in this sense.)
(I say it's a slogan, but I'm sure it's a theorem; I just don't have a reference for you.)
Presentable ($\infty$,1)-categories are special among all ($\infty$,1)-categories; in particular, they are complete and cocomplete.
For instance, you can define the notion of ($\infty$,1)-topos in terms of model categories, since ($\infty$,1)-topoi are presentable, and morphisms between such are certain kinds of adjoint functor pairs.