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There is a Quillen equivalence between the model category presenting Lurie's $\infty$-operads (which are inner fibrations $\mathcal{C}\to\mathrm{N}(\mathbf{F})$ satisfying certain conditions) and the model category of dendroidal sets. On the other hand, in his paper on operator categories and topological operads (http://arxiv.org/abs/1302.5756), Barwick proves that there is an equivalence of $\infty$-categories between complete Segal $\Phi$-operads and $\infty$-operads over $\Phi$ for a perfect operator category $\Phi$.

This provides a zig-zag of Quillen equivalences between Barwick's complete Segal $\Phi$-operads and dendroidal sets when $\Phi$ is the perfect operator category $\mathbf{F}$. Is there a direct construction of a Quillen equivalence between these two models (namely, complete Segal operads and dendroidal sets) for $\infty$-operads?

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    $\begingroup$ The comparison between dendroidal sets and Lurie's infinity-operads has actually not been established in complete generality yet: Heuts-Hinich-Moerdijk only prove it for non-unital operads. $\endgroup$ – Rune Haugseng Apr 18 '16 at 2:35
  • $\begingroup$ @RuneHaugseng Would it be possible to ask for a direct construction for a Quillen equivalence between those Barwick complete Segal $\Phi$-operads whose underlying $\infty$-operad is nonunital and dendroidal sets? $\endgroup$ – user62675 Apr 18 '16 at 22:59

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