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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.

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    $\begingroup$ Well, one sufficient condition is that $\mathcal{C}$ is a model category enriched over $\mathbf{sSet}_\mathrm{Joyal}$. This is the case for $\mathbf{sSet}_\mathrm{Joyal}$ itself. $\endgroup$ – Zhen Lin Feb 4 '15 at 13:35
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    $\begingroup$ Model categories enriched over SSet with the Quillen model structure are presentations of (infinity,1)-categories. Model categories enriched over SSet with the Joyal model structure are presentations of (infinity,2)-categories. See Remark 0.0.4 in [Jacob Lurie, (Infinity,2)-Categories and the Goodwillie Calculus I], arxiv.org/abs/0905.0462. $\endgroup$ – AAK Feb 4 '15 at 13:36
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    $\begingroup$ Anyway, then the natural requirement (in my opinion) is that C be a Joyal-model category in the sense of Hovey. $\endgroup$ – Fernando Muro Feb 4 '15 at 15:33
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    $\begingroup$ Hovey defines enriched model categories in his textbook. This is a special case (which he does not address – but then again, the Joyal model structure wasn't well known when it was written!). $\endgroup$ – Zhen Lin Feb 4 '15 at 16:05
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    $\begingroup$ The notion of 'model category' is not perfect. All of them should morally be simplicial, and to a certain extent they are (the homotopy enrichment, Quillen equivalences with simplicial models, etc.) So I wouldn't say that lifting the enrichment to models is a very strong condition. $\endgroup$ – Fernando Muro Feb 4 '15 at 16:30
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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.

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  • $\begingroup$ Hovey's book finishes with a list of Vistas, and introducing the correct notion of a 2-model category was a top priority. That was in 1998 and many people have tried to find a good definition. As far as I know, no one has succeeded. Everyone knows what the weak equivalences should be, but not the (co)fibrations. I would not advise this approach. $\endgroup$ – David White Feb 10 '15 at 22:16
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    $\begingroup$ @DavidWhite, the question of Hovey is about model structures on 2-categories, not about model structures on 2-relative categories. How is it relevant to this discussion? $\endgroup$ – AAK Feb 10 '15 at 22:25

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