When is the model structure on functors correct, i.e. when does localization commute with taking functor categories?

Let $$C$$ be a small category and $$M$$ a model category. Then there are various "global" model structures (projective, injective, Reedy) on the category $$Fun(C,M)$$ of functors from $$C$$ to $$M$$, all with the same (levelwise) weak equivalences. The whole point of having such a model structure is that it should present the $$\infty$$-category $$Fun(C,\tilde M)$$, where $$\tilde M$$ is the $$\infty$$-category presented by $$M$$. But I'm not sure when this is actually the case.

Of course, by "the $$\infty$$-category presented by $$M$$", I mean $$\tilde M = M[W^{-1}]$$ is $$M$$ localized at the weak equivalences in the $$\infty$$-categorical sense, and similarly "the $$\infty$$-category presented by $$Fun(C,M)$$" is the $$\infty$$-categorical localization $$Fun(C,M)[Fun(C,W)^{-1}]$$.

Questions:

1. If $$C$$ is a small category and $$M$$ is a model category, then under what conditions do the standard model structures on $$Fun(C,M)$$ present the $$\infty$$-category $$Fun(C,\tilde M)$$, where $$\tilde M = M[W^{-1}]$$ is the $$\infty$$-category presented by $$M$$?

2. More generally, if $$C$$ and $$M$$ are relative categories, then under what conditions does the mapping relative category $$\widetilde{Fun(C,M)} = Fun(\tilde C, \tilde M)$$ where $$\tilde{(-)}$$ denotes taking the associated quasicategory?

3. In a more model-independent direction, when does localization of $$\infty$$-categories commute with taking functor categories? That is, when does $$Fun(C,M[W^{-1}]) = Fun(C,M)[Fun(C,W)^{-1}]$$ where $$C,M$$ are $$\infty$$-categories and $$W \subseteq M$$ is a subcategory?

• Here's a possible way of setting up the problem. You have a functor from the localization of the functor category to the category of functors into the localization. You want to know when it's essentially surjective and fully faithful. The former is like a `rectification' question while the latter can be tackled using the end formula for natural transformations together with Lemma I.3.4 of Nikolaus-Scholze. I think this setup makes it clearer why the model category structures help give the equivalence, and I think you ought to get it for all the favorite model structures on functors. – Dylan Wilson Feb 10 at 17:52
• I remember there are general results in Cisinski's work for the special case in which $M$ is a Brown category of Fibrant objects. I think it can be found in his book on higher categories, in the part on localization and categories with fibrations. – Simon Henry Feb 10 at 18:45
• There is an article by Lenz, which formulates a positive answer nicely in the language of derivators: arxiv.org/abs/1712.07845 – Lennart Meier Feb 11 at 4:15

If your model structures are assumed to have small limits or colimits, the answer to the question of the title is: always. For any model category $$M$$ and any small category $$C$$, inverting levelwise weak equivalences in $$Fun(C,M)$$ is equivalent to considering the $$\infty$$-category of functors from $$C$$ to $$M[W^{-1}]$$. This is a special case of Theorem 7.9.8 (and Remark 7.9.7) in my book on higher categories. It is even possible to take for $$M$$ a model $$\infty$$-category in the sense of Mazel-Gee. In fact, Theorem 7.5.8 gives sufficient conditions on $$M$$ which are much more general: essentially, the mere existence of a class of well behaved fibrations is good enough (this includes Brown's categories of fibrant objects, but, more generally, a version where we do not assume all objects to be fibrant; in particular, all the variations on semi-model structures are OK) if we assume further properties: we mainly need this extra structure to exist on functor categories $$Fun(C,M)$$ as well (which is automatic in practice, as explained in Example 7.9.6 and Remark 7.9.7 of loc. cit.). If we restrict ourselves to those $$C$$ whose nerve is a finite simplicial set (e.g. finite partially ordered sets), this kind of properties is true in a much greater level of generality; see Theorem 7.6.17.
Observe that, if $$M$$ is good enough in the sense that $$\widetilde{Fun(C,M)}\cong Fun(C,\tilde M)$$ for any small category $$C$$, then, it is automatic that, for any subcategory $$S\subset C$$, the localization of the full subcategory of $$Fun(C,M)$$ whose objects are those functors sending the maps of $$S$$ to weak equivalences will autmatically be a model of the $$\infty$$-category of functors from $$C[S^{-1}]$$ to $$\tilde M$$.