TITLE: Localization of $\infty$-categories
QUESTION [8 upvotes]: In ordinary category theory, the localization $C[S^{-1}]$ at a class of morphisms $S$ (with possibly some assumptions on $S$) is a category $C[S^{-1}]$ together with a map $L:C \to C[S^{-1}]$ such that any functor $C \to D$ that sends $S$ to isomorphisms factors uniquely through $L$. 
Does a generalization of this notion exist in $\infty$-category theory? Can one simply carry out the construction as one does for ordinary categories? It's unclear to me how one would define the mapping space in the localized category. 
A reference would be sufficient!

REPLY [3 votes]: There are a number of good references for this. Section 5 of Danny Stevenson's paper on simplicial localization and covariant model structures contains an extremely simple explicit construction of the localization which models, in simplicial sets, the construction suggested by Hurkyl. One just pushes out $C$ with a free-living isomorphism for each arrow in $S$, along the canonical map from the disjoint union of the elements of $S$ to $C$. Of course, one then has to take a fibrant replacement to get an $\infty$-category on the nose. I would argue that this gives the most elementary understanding of this fundamental construction, relative to DK localization or the use of marked simplicial sets.
Why "fundamental?" Stevenson also gives a short proof of a theorem known in some form to Dwyer and Kan, and due in this context to Joyal: every $\infty$-category arises as a localization of its ordinary category of simplices. Chapter 7 of Cisinski's recent book is also all about localization, based on the same construction that appears in Stevenson, and in particular its interaction with limits and continuous functors, which imports much of classical homotopical algebra into the $\infty$-context. This was also in part the goal of Mazel-Gee's thesis, which focused on $\infty$-model categories.