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Both relative categories and marked simplicial sets (over Δ^0) present the ∞-category of ∞-categories.

Naturally, one could ask whether there is a reasonably direct way to pass between these two models. There are some obvious candidates, e.g., the nerve of a relative category is a marked simplicial set.

Is the functor that sends a relative category (C,W) to the marked simplicial set (NC,NW) a weak equivalence from the relative category of relative categories to the relative category of marked simplicial sets?

The closest reference I could find is Remark 1.3.4.2 in Higher Algebra, which states it on the level of individual objects, not relative categories.

Note that although the right adjoint functor (C,W)↦(NC,NW) does compute the correct answer, its left adjoint is badly behaved because it only depends on the marked 2-skeleton. Thus there is no chance that this right adjoint functor is a right Quillen equivalence of model categories.

However, Barwick and Kan treat the case of nonmarked simplicial sets in §6.6 of their paper on relative categories, and they recover Thomason’s theorem (Theorem 6.7 in their paper), which can be seen as the analog of the above question in the nonmarked setting. Their result is stated as a Quillen equivalence, which is achieved by replacing the naive cosimplicial object (n↦(0→⋯→n)) in relative categories with its double subdivision as a relative poset. The resulting nerve functor N_ξ is weakly equivalent to the naive nerve functor N, but their left adjoints are not weakly equivalent, and it is the nonnaive left adjoint that computes the correct category that corresponds to a simplicial set.

Thus one can refine the above question from relative categories to model categories as follows.

Is the subdivided relative nerve (N_ξ C,N_ξ W) weakly equivalent to the naive relative nerve (NC,NW) as a marked simplicial set? Is (C,W)↦(N_ξ C,N_ξ W) a right Quillen equivalence?

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Concerning the first question: the simplicial localization functor $L^H$ induces an equivalence from the relative category of small relative categories to the relative category of small simplicial categories (see, e.g., Barwick and Kan http://arxiv.org/pdf/1012.1540.pdf). The right derived coherent nerve functor $\mathbb{R}N$ induces an equivalence from the relative category of small simplicial categories to the relative category of quasi-categories. The functor $X \mapsto X^{\natural}$ induces an equivalence from relative category of quasi-categories to the relative category of marked simplicial sets over $\Delta^0$ (and Cartesian equivalences). According to Hinich (http://arxiv.org/pdf/1311.4128v4.pdf, Proposition 1.2.1), there is a natural equivalence of marked simplicial sets $$ (N(C),N(W)) \stackrel{\simeq}{\to} \mathbb{R}N(L^H(C,W))^{\natural}. $$ It follows that the association $(C,W) \mapsto (N(C),N(W))$ induces an equivalence from the relative category of small relative categories to the relative category of marked simplicial sets.

Concerning the second question: I think it's highly likely that the answer is yes (on both counts), but as far as I know it has not been written up anywhere yet (though someone will probably write a paper about it soon).

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  • $\begingroup$ Indeed, these results of Hinich and Barwick-Kan do answer the first part affirmatively. $\endgroup$ – Dmitri Pavlov Apr 9 '16 at 15:12

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