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In the "ordinary" operad category, it is known that there is a colored operad $Op$ with set of colors $\mathbb{N}$ corresponding to "degrees" of vertices and with operations indexed by trees, such that algebras over $Op$ in $\mathrm{Set}$ (or more generally any symmetric monoidal category) correspond to monochromatic operads.

Is it known that $\infty$-algebras over $Op$ are equivalent to monochromatic $\infty$-operads?

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Yes, combine Corollary 9.4.1 and Theorem 7.11 of arXiv:1410.5675, for example.

This topic is also examined more explicitly in the work of Chu and Haugseng, arXiv:1707.08049.

Corollary 5.1.13 shows that enriched ∞-operads are equivalent to ∞-algebras over the operad of colored operads.

Theorem 5.2.10 proves (using arXiv:1410.5675) that the underlying ∞-category of the relative category of ordinary enriched colored operads is equivalent to the ∞-category of enriched colored ∞-operads.

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  • $\begingroup$ Thank you for the reference! Can you give a rough idea of how these imply the result? Am I correct in understanding that you observe that simplicial operads and sOper-algebras give two model category structures on the same category and the free object transferability result of Corollary 9.4.1 implies they are Quillen equivalent? $\endgroup$
    – Dmitry Vaintrob
    Commented Apr 24, 2021 at 18:25
  • $\begingroup$ @DmitryVaintrob: I added references to the work of Chu and Haugseng, which combines these results to get more-or-less exactly the statement that you asked for. $\endgroup$
    – Dmitri Pavlov
    Commented Apr 24, 2021 at 19:09
  • $\begingroup$ Thank you! That certainly seems to settle it. Interestingly enough, I had recently read the first couple of sections of the Chu-Haugseng paper, but didn't notice this theorem. $\endgroup$
    – Dmitry Vaintrob
    Commented Apr 24, 2021 at 23:59

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