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!



Localization of $\infty$-categories
An *abstract* construction I find appealing is that a localization should satisfy a natural pullback square of spaces


$$ \require{AMScd} \begin{CD}
\hom(C[S^{-1}], X) @>>> \hom(C, X)
\\ @VVV @VVV
\\ \hom(S, \mathrm{Core}(X)) @>>> \hom(S, X) \end{CD}$$


expressing the universal property that a functor on $C[S^{-1}]$ is a functor on $C$ whose restriction to $S$ factors through the subcategory of equivalences. Core is right adjoint to groupoidification (which is right adjoint to the inclusion of groupoids in categories), so this is equivalent to having a pushout square


$$ \begin{CD} S @>>> C
\\ @VVV @VVV
\\ \mathrm{Gpdify}(S) @>>> C[S^{-1}] \end{CD} $$


Warning: It's important to note that the pullback square above is between hom-spaces, not between functor $\infty$-categories. The full subcategory of $\mathrm{Fun}(S, X)$ spanned by the $S$-inverting functors is not $\mathrm{Fun}(S, \mathrm{Core}(X))$ — the issue is that the natural transformations bewteen two such functors need not be natural isomorphisms.




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Here is another construction I believe to be correct. Let $[1]$ denote the arrow category (i.e. $\bullet \to \bullet$).


There is a functor $R : \mathrm{Cat}\_\infty \to \mathrm{Cat}\_\infty^{[1]} $ that sends a category $X$ to the arrow $\mathrm{Core}(X) \to X$.


This functor is a limit-preserving accessible functor between presentable categories, so it has a left adjoint $L : \mathrm{Cat}\_\infty^{[1]} \to \mathrm{Cat}\_\infty $.


I claim that $L$ is (a generalization of) the localization functor; i.e. given an inclusion of a subcategory, $L(S \to C) \simeq C[S^{-1}]$. This follows from the fact the pullback at the top is *also* the pullback describing the hom-space of commutative squares from $S \to C$ to $\mathrm{Core}(X) \to X$.


