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It is my understanding that the $\infty$-category of non-unital connected topological monoids is equivalent to the $\infty$-category of connected topological groups.

It follows that the functor from unital topological monoids to the category of non-unital topological monoids with unital monoid structure on $\pi_0$ is fully faithful. (Indeed, I seem to remember that it is an equivalence.)

My question is whether an analogous statement is true for operads: namely, is the functor from unital $\infty$-operads to non-unital operads with unital structure on $\pi_0$ fully faithful? If this is true, is there a reference? (Either in the $\infty$-categorical or the model theoretic framework.)

The reason I am asking this is that this would imply that formality for a unital operad of complexes follows from formality of the corresponding non-unital operad. A reference for this would be greatly appreciated as well.

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    $\begingroup$ You can find a proof of your last statement in my paper arxiv.org/abs/1811.10460 . It follows a more classical approach: first, I prove the existence of minimal models for unitary operads. Then the fact that formality for unitary operads $P_+$ follows indeed from formality of its truncation---the non-unitary operad $P$ you obtain by deleting $P_+(0)$. $\endgroup$
    – Agustí Roig
    Commented Sep 14, 2020 at 17:35
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    $\begingroup$ Thank you @AgustíRoig! This looks like exactly the kind of statement I need. $\endgroup$
    – Dmitry Vaintrob
    Commented Sep 14, 2020 at 17:52

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