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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? …

It is a result of May's work on operads that the homotopy category (or $\infty$-category, if you prefer) of connective spectra is equivalent to a full subcategory of the category of representations of …

Now the combinatorially defined infinity-category of operads is equivalent to the category of topological operads when the two sides are viewed as model categories or $\infty$-categories obtained from …

I know of three homotopy theories of colored operads. … 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. …

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? …