$(\infty,1)$-categories and model categories

Question

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?

Answer 1

Mostly I refer you to my answer here and also this question.

To answer the question about (co)fibrations: No, there is no notion corresponding to (co)fibration in the (∞,1)-category associated to a model category. After all, being a (co)fibration has no homotopical information: every map is equivalent to a (co)fibration. For the sorts of things you need the (co)fibrations to define in model categories, such as homotopy (co)limits, you can give direct definitions in terms of mapping spaces in the (∞,1)-category.

There are two sensible notions of "sameness" of model categories: categorical equivalence, by which I mean an equivalence of categories which preserves each of the three classes of arrows, and Quillen equivalence. This is a lot like the difference between two objects in a model category being isomorphic or merely weakly equivalent, though I don't think anyone has a framework in which to make this idea precise. When you consider, say, the projective and injective model structures on a diagram category, these model structures are Quillen equivalent but not categorically equivalent. They have different 1-categorical properties (it's easy to describe left Qullen functors out of the projective model structure and left Quillen functors into the injective model structure) but they model the same homotopy theory. The passage to associated (∞,1)-categories eliminates the distinction between categorical equivalence and Quillen equivalence: two model categories are Quillen equivalent if and only if their associated (∞,1)-categories are equivalent. (Actually, I am not sure whether there are some technical conditions needed for the last assertion, but if so they are satisfied in practice.)

Answer 2

You have the basic sense correct. In a model category any pair of objects has a function space of maps between them. However, it is possible to define this function space in far more generality.

Given a category C with a subcategory W < C with the same objects, consisting of a collection of maps that one wants to make into isomorphisms, Dwyer and Kan defined under certain circumstances a "simplicial localization" of C which is a functor from C to a category LC enriched in simplicial sets. If C has a model category structure, then this recovers the spaces of maps that you describe as coming from the model category structure.

The model category could then be viewed as a tool to recover the mapping spaces and the associated $(\infty,1)$-category in a more constructive manner, as the category LC is generally quite large.

Answer 3

The functor $F$ you are looking for can be described as follows. Given a model category $M$, with class of weak equivalences $W$, one may associate to it an $\infty$-category $M_\infty$, equipped with a map $M \to M_\infty$, which is characterized by the following universal property: For every $\infty$-category $D$, the natural map $$Fun(M_\infty,D) \to Fun(M,D)$$ is fully faithful, and its essential image is spanned by those functors $M \to D$ which send $W$ to equivalences.

The $\infty$-category $M_\infty$ may be constructed follows: Construct the Hammock localization $L^H(M,W)$ of $M$ with respect to $W$, which is a simplicial category. The $\infty$-category $M_\infty$ can then be obtained by taking the coherent nerve of any fibrant model of $L^H(M,W)$ (with respect to the Bergner model structure). We refer to

http://arxiv.org/abs/1311.4128

for the above universal property.

Note that the above construction works for any relative category, that is, a category equipped with a subcategory of weak equivalences containing all the objects. Let us call this procedure $\infty$-localization.

Morphisms of model categories are given by Quillen pairs: $$F:M\rightleftarrows N:R.$$ According to Proposition 1.5.1 in the paper above, every Quillen pair as above induces an adjoint pair of $\infty$-categories: $$F_\infty:M_\infty\rightleftarrows N_\infty:R_\infty.$$ It can be shown that a Quillen equivalence goes to an equivalence of $\infty$-categories by this construction.

So consider the category of model categories and left Quillen functors between them. This is actually a relative category, where the weak equivalences are the Quillen equivalences. The above construction defines a relative functor from this relative category to the relative category of $\infty$-categories (weak Kan-complexes and simplicial maps between them) with the weak equivalences being the equivalences of $\infty$-categories. This relative functor should be a derived fully faithful embedding of the $\infty$-category of model categories (in the sense of the $\infty$-localization above) into the $\infty$-category of $\infty$-categories and left adjoints between them. (Perhaps there are some delicate technical issues here that I am suppressing.)