When are homotopy categories of model categories closed modules over the homotopy category of $(\infty, 1)$-categories?

Question

Let $\mathrm{Quillen}$ be the model category of simplicial sets with the Quillen model structure, and $\mathrm{Joyal}$ the model category of simplicial sets with the Joyal model structure.

As is well-known, given an arbitrary model category $\mathcal C$, its homotopy category $h\mathcal C$ is naturally a closed module over the category of spaces, the homotopy category $h\mathrm{Quillen}$ of $\mathrm{Quillen}$.

On the other hand, the category $h\mathrm{Joyal}$, the homotopy category of $\mathrm{Joyal}$, that is the category of (small) $(\infty, 1)$-categories, is naturally a closed module not only over $h\mathrm{Quillen}$ but also over itself, $h\mathrm{Joyal}$.

My question is the following: What nice property makes a model category $\mathcal C$ into one such that its homotopy category is naturally a closed model over $h\mathrm{Joyal}$, so that, in particular, its mapping spaces can naturally be viewed as $(\infty, 1)$-categories?

Of course, the construction should give back the closed module structure over $h\mathrm{Joyal}$ in case of the category of $(\infty, 1)$-categories.

Answer 1

I am not sure if this will answer your question, but it may at least point you in the right direction (or at least some direction).

Let me start with some classical background. Let $C$ be a category with a class of weak equivalences $W$. Dwyer and Kan showed that this data presents an (∞,1)-category $C[W^{-1}]$, called the hammock localization. Like the classical Gabriel-Zisman localization, its 1-morphisms are equivalence classes of zig-zags of morphisms of $C$, but it also encodes the data of homotopies between these morphisms. From this perspective, the data of a model structure on $C$, with class of weak equivalences $W$, can be viewed as a computational tool whose purpose is to ensure that the mapping spaces of the hammock localization have a much more tractable description via taking resolutions in the Reedy model structure on simplicial objects in $C$ (at least under combinatorial and properness assumptions).

Barwick and Kan have built on the work of Dwyer-Kan to show that the (∞,1)-category of pairs $(C,W)$, called relative categories, is in fact equivalent to the (∞,1)-category of (∞,1)-categories. Further, they have developed a model for (∞,n)-categories called relative n-categories. In the case n=2, if I understand correctly, a 2-relative category is the data of a tuple $(C, W, V_1, V_2)$, where $W$, $V_1$ and $V_2$ are subcategories of the category $Arr(C)$ of morphisms of $C$, subject to various axioms. This data should be thought of as two relative categories $(V_1, W)$, $(V_2, W)$, with an ambient category $Arr(C)$ encoding relations between them. See [C. Barwick, D. M. Kan, n-relative categories: a model for the homotopy theory of n-fold homotopy theories, pdf].

Now to your question. Since 2-relative categories present (∞,2)-categories, there is a mapping (∞,1)-category (instead of just a mapping (∞,0)-category = space) between any two objects. The story of relative categories and model categories suggests that there should be a notion of 2-model category, which is some additional structure on a 2-relative category, giving a simpler description of these mapping (∞,1)-categories. Presumably, this structure would be something like compatible model structures on the relative categories $(C,W)$, $(V_1,W)$ and $(V_2,W)$. In other words, I think it is reasonable that, in order for a given model category to be enriched over (∞,1)-categories, one should not ask for some property, but rather for some additional structure on it, along the lines of a 2-model structure.