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It is now known that the $\infty$-category of $\infty$-operads as defined by Lurie is equivalent to the underlying $\infty$-category of the model category of simplicial operads, see http://arxiv.org/pdf/1606.03826.pdf. However, the proof in loc. cit. is obtained via a zig-zag of Quillen equivalences (going through Moerdijk and Weiss's theory of dendroidal sets and their variants, and Barwick's theory of perfect operator categories), making the direct comparison somewhat non-explicit. On the other hand, given a fibrant simplicial operad, we may associate to it very explicitly an $\infty$-operad, namely, its operadic nerve, as described in section 2.1.1 of Higher Algebra. The operadic nerve functor is not just explicit, it is also the one used in various works relating algebras over simplicial operads and algebras over $\infty$-operads, as in here. Being quite a natural construction, it seems very likely that the operadic nerve functor is actually the one that induces the equivalence above between simplicial operads and $\infty$-operads. Since the theory of $\infty$-operads has no self automorphisms (see here), this is the same as just saying that the operadic nerve induces an equivalence. However, since it is neither a left nor a right Quillen functor, it might be more accessible to tackle the question by comparing it to some zig zag of Quillen equivalences. Either way, my question is:

Is the operadic nerve functor an equivalence of $\infty$-categories?

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  • $\begingroup$ Cisinski and Moerdijk's complete dendroidal Segal spaces model makes it clear that for an adjunction of ∞-categories both of which are equivalent to the ∞-category of ∞-operads to be an equivalence of ∞-categories it suffices that the generators are mapped to generators by the left adjoint. In our case, the generators of the ∞-category of ∞-operads can be chosen to be trees (or rather their images in the two chosen models), and the problem boils down to showing that the left adjoint of the operadic nerve preserves them, which can be done by direct inspection. $\endgroup$
    – Dmitri Pavlov
    Commented Sep 17, 2016 at 20:53
  • $\begingroup$ That's a good point, but since the operadic nerve is not a right Quillen functor, how do you show that it induces a right adjoint functor on the level of $\infty$-categories? $\endgroup$
    – Yonatan Harpaz
    Commented Sep 18, 2016 at 9:20
  • $\begingroup$ Why shouldn't the operadic nerve be a right Quillen functor? It's the composition of the forgetful functor from simplicial multicategories to simplicial categories and the homotopy coherent nerve functor from simplicial categories to simplicial sets. Both are right Quillen functors. $\endgroup$
    – Dmitri Pavlov
    Commented Sep 18, 2016 at 13:45
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    $\begingroup$ The only problem is that in the end you need to mark the coCartesian edges lying above inert edges of $Fin_{\ast}$, and this operation doesn't preserve limits (only homotopy limits). $\endgroup$
    – Yonatan Harpaz
    Commented Sep 18, 2016 at 20:23

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Yes, precisely because it is a chain of Quillen equivalences of model categories.

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    $\begingroup$ I think that statement should include a reference. Clearly the OP was aware that a chain of Quillen equivalences gives rise to an equivalence of $\infty$-categories. He said as much in his post. But, he didn't know that there was a chain of Quillen equivalences and now you say there is one. Why? $\endgroup$
    – David White
    Commented Apr 3, 2023 at 11:42

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