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I know of three homotopy theories of colored operads.

  • The (derived) localization category of Berger-Moerdijk's model structure on the category of strict simplicial (or topological) operads, with weak equivalences given by strict maps $O\to O'$ which induce weak equivalences on spaces of operations.
  • The "dendroidal" notion of infinity-operad, introduced by Moerdijk and Weiss and further studied in this paper by Cisinski and Moerdijk.
  • Lurie's infinity-operads, which are infinity categories fibered over the nerve of the category $\mathrm{Set}^*$ of pointed sets, satisfying certain Segal-style properties.

If all is well in the world, these three homotopy theories (viewed as e.g. $\infty$-categories) should be equivalent (or there should be a good reason for them not to be). But I can't find references for any equivalences between them. This paper seems to compare (1) and (2), via a Quillent adjunction, which it does not show is an equivalence. The obvious functor (1) $\implies$ (3) is written down in Lurie's Higher Algebra, but it is not (as far as I can tell) shown to be an equivalence.

Is more known about comparisons between these homotopy theories? I'm specifically interested in the comparison between (1) and (3). Are the corresponding $\infty$-categories equivalent, perhaps under additional restrictions? Is the functor (1) $\implies$ (3) fully faithful? Are there known examples where this functor is not an equivalence?

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The answer is yes: see the paper of Chu-Haugseng-Heuts, "Two models for the homotopy theory of ∞-operads", arXiv:1606.03826.

In brief, already Cisinski and Moerdijk ("Dendroidal sets and simplicial operads", arXiv:1109.1004) proved a Quillen equivalence between simplicial operads and dendroidal sets. In the paper of Cisinski and Moerdijk that you link to, they prove an equivalence between dendroidal sets and complete dendroidal Segal spaces. What Chu-Haugseng-Heuts prove is an equivalence between complete dendroidal Segal spaces and Barwick's complete Segal operads, and already before this Barwick had proved that complete Segal operads were equivalent to Lurie's ∞-operads.

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Dan's answer is comprehensive, as ever, but also see this paper of Moerdijk and Hinch.

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