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This question is answered in the affirmative in a recent paper of mine with Liang Ze Wong. In fact, we prove it more generally for a (strictly) monoidal simplicially enriched category. As any simplici …

Given a symmetric monoidal category $C$, we can construct its endomorphism operad (or multicategory) $End(C)$ whose objects are the objects of $C$, and for which the multimorphisms from $\{c_1,\ldots, …

Is there any kind of "straightening" construction for $\infty$-operads? … My real goal here to ask if there is a way in which I can think of $\infty$-operads as (possibly some weakened version of) commutative algebra objects in some $\infty$-category. …

This question was answered in the affirmative by Rune Haugseng in Section 4 of https://arxiv.org/pdf/1708.09632.

Let $A$ be a connected $E_{n+1}$-ring spectrum and let $\alpha\in\pi_k(A)$. I am having trouble showing that attaching an $E_n$-cell along $\alpha$ will necessarily not kill an element $\beta\in\pi_k( …

an object in a symmetric monoidal quasicategory $\mathscr{C}$), things can be slightly more complicated, and we should probably have $H$ with monoidal structure and comonoidal structure given by some operads … like the little $n$-disk operads $\mathbb{E}_n.$ It's basically formal when working in quasicategories to say that $H$ is an $\mathbb{E}_n$-algebra with a compatible $\mathbb{E}_m$-coalgebra structure …

In other words, $H$ is determined by a functor of $\infty$-operads $H^\otimes\colon Assoc^\otimes\to CoAlg(C)^\otimes$. …

In Lurie's DAG II, a notion of monoidal $\infty$-category is given that differs from the notion given in his later book Higher Algebra. In the former, the relevant structure is a cocartesian fibratio …

So, this ends up being simpler than I realized, and is in some sense this question's existence is purely a result of me not reading the above cited DAG II closely enough. In the first section of DAG …