I read several times that $(\infty,1)$-categories (weak Kan complexes, special simplicial sets) are a generalization of the concept of model categories. What does this mean? Can one associate an $(\infty,1)$-category to a model category without losing the information on the co/fibrations? How? Why is the $(\infty,1)$-category viewpoint the better one?

$(\infty,1)$-categories are equivalent to simplicial categories (categories enriched over simplicial sets). This is outlined in Lurie's *higher topoi*. A simplicial model category is in particular a simplicial category. Is this the way the association works? Every model category is Quillen equivalent to a simplicial model category and can thus be enriched over simplicial sets.

It would be nice if somebody could help me to clarify this.

**Edit:** Thank you all for the answers. It seems to me that $(\infty,1)$-categories are *not* a generalization of the concept of model categories. A model category is more than a category $C$ together with a class of maps $W$ such that $C[W^{-1}]$ is a category. A model structure data on a category $C$ contains the information on what a cofibration and what a fibration is. This is important for the structure. There exist *different* model structures with the same homotopy category as for example model structures on functor categories.

This means that if there is a kind of functor
$$
F: \{\mbox{model categories}\} \to \{\mbox{($\infty,1)$-categories}\}
$$
it is at least *not* an embedding. In spite of the answers, I still don't see how this functor (if it is really a functor) works. Where is a model category mapped to?

bookis called higher topos theory. I assume that is what you're referencing? $\endgroup$ – Harry Gindi Feb 23 '10 at 23:19notpreserved under Quillen equivalence of model categories and so it is unfair to try to remember that exact structure. In contrast they will have the same induced $(\infty, 1)$-categories. I think, but am not sure, that if you look at the $(\infty,1)$-category of model categories (what you get by (derived) localizing at Quillen equivalences) then the functor F you describe is an embedding. $\endgroup$ – Chris Schommer-Pries Feb 24 '10 at 12:451more comments