Localization of $\infty$-categories

Question

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!

Answer 1

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 $\mathrm{Core}(X) \to X$.

Answer 2

While it can be obtain formally using the adjoint functor theorem as mentioned by Marc Hoyois in the comment. There are several explicit constructions.

First, the localization functor which send a small $(\infty,1)$-category $C$ together with $S$, a fullsubcategory of its arrow category, to the $(\infty,1)$-category $C[S^{-1}]$ can be seen as the left adjoint to the functor from the category of small $(\infty,1)$-category to the category of small $(\infty,1)$-category with a full subcategory of their arrow category "marked" which send send any $(\infty,1)$-category $C$ to $C$ with all the invertible arrows of $C$ marked. It shouldn't be too hard to check that this functor is $\omega$-accessible and preserve limits.

There are however explicit construction of this functor that are worth mentioning. Dwyer-Kan localization, mentioned in Francesco Genovese answer is of course the first one historically (even long before $(\infty,1)$-category were a thing). But there is another one which I personally tend to like better, and which is more quasi-categorical:

In Higher topos theory Chap 3. Lurie introduce a model structure of "marked simplicial sets". If you take the "unbased" version (I.e. take "$S$" to be the terminal object everywhere) it is a model structure on marked simplicial sets (a marked simplicial set is a simplicial set with a collection of marked $1$-cells) whose fibrant object are the quasi-category in which the marked cells are exactly the invertible cells. This model structure is shown to be Quillen equivalent to the one for quasi-category, with the functor forgetting the marking being the right Quillen functor

Starting from a small quasi-category $C$ a nice way to construct its localization at a set $S$ of arrow is to take the marked simplicial set $C$ with all the arrows in $S$ marked and take a fibrant replacement in the model structure mentioned above (which is constructed relatively explicitly, using the small object argument).

Of course, without smallness assumptions on $C$ and $S$ there is no guaranty that the localization exists. Or, depending on your framework/philosophical stands, it always exists, simply because assuming some inacessible cardinal, you can apply all this machinery to the huge category large quasi-categories directly, but there is no guaranty that the localization is a locally small category (the hom can become proper classes in the localization process).

Answer 3

A possibility is the Dwyer-Kan "simplicial localization": https://ncatlab.org/nlab/show/simplicial+localization. The three main references are the articles Simplicial localizations of categories, Calculating simplicial localizations and Function complexes in homotopical algebra by Dwyer-Kan. It is worked out in the setting of simplicially enriched categories, which are actually a model of $\infty$-categories. If $C$ is an ordinary category and $S$ is a class of morphisms, then the simplicial localization produces a simplicial category $L(C,S)$ which is an "enhancement" of the ordinary localization, in the precise sense that $\pi_0(L(C,S)) \cong C[S^{-1}]$.

Answer 4

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.