TITLE: Limits in an $(\infty,1)$-category
QUESTION [11 upvotes]: In ordinary category theory, the notion of limit in a category $C$ is usually formulated with a category (of indices) $J$ and a functor $F:J\to C$ (a diagram in $C$), and a limit of this diagram is something satisfying some universal property.
In the context of quasi-categories (I looked in the nLab and in Higher Topos Theory), the category $J$ is replaced by a simplicial set $K$ and the functor $F$ is replaced by a map of simplicial sets $K\to C$ (a quasi-category is a particular simplicial set). But this definition is specific of quasi-categories, if you take another model for $(\infty,1)$-categories, it does not make sense to talk about a map between a simplicial set and an $(\infty,1)$-category.
Is there a definition of limits for $(\infty,1)$-categories independent of the model?
In particular, can we replace $K$ by an $(\infty,1)$-category and $F$ with an $(\infty,1)$-functor? If we can, why does everybody take a simplicial set instead (for quasi-categories)?

REPLY [3 votes]: One notion which is sometimes convenient, and can be considered to "underlie" all notions of $(\infty,1)$-category, is a derivator.  A prederivator is just a 2-functor $Cat^{op} \to Cat$ (modulo size questions).  And any $(\infty,1)$-category, in any model, will have an underlying "representable" prederivator defined by $X\mapsto Ho(C^X)$, where $C^X$ is the "functor $(\infty,1)$-category" in whatever model you've chosen and $Ho$ denotes the homotopy category of an $(\infty,1)$-category.
A derivator is a prederivator satisfying axioms saying that (among other things) all limits and colimits exist.  But it is also possible to characterize when a particular limit or colimit exists in a prederivator; see for instance here.  One expects, though I don't know to what extent it has been proven, that a limit in any model for $(\infty,1)$-categories will give rise to a limit in the underlying prederivator.  I don't know about the converse, but for many applications of limits and colimits, the structure of a derivator is sufficient.