coCartesian fibration of (∞,1)-operads in nLab

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Definition

Let $\mathcal{O}^\otimes$ be an (∞,1)-operad. A coCartesian fibration of (∞,1)-operads is

a coCartesian fibration $p : \mathcal{C}^\otimes \to \mathcal{O}^\otimes$ of the underlying quasi-categories;
such that the composite

$\mathcal{C}^\otimes \stackrel{p}{\to} \mathcal{O}^\otimes \to FinSet^{*/} = Comm^\otimes$

exhibits $\mathcal{C}^\otimes$ as an (∞,1)-operad.

In this case we say that the underlying (∞,1)-category

\mathcal{C} = \mathcal{C}^\otimes \times_{\mathcal{O}^\otimes} \mathcal{O}

is equipped by $p$ with the structure of an $\mathcal{O}$ -monoidal (∞,1)-category (see remark below).

For $\mathcal{C}^\otimes \to \mathcal{O}^\otimes$ a coCartesian fibration of $(\infty.1)$ -operads by def. , the underlying map

\mathcal{C} \to \mathcal{O}

\chi \colon \mathcal{O} \to (\infty,1)Cat \,.

In fact, this is restricted from the (∞,1)-Grothendieck construction $\mathcal{O}^{\otimes} \rightarrow (\infty,1)Cat$ ; this functor is an $\mathcal{O}$ -monoid, and hence an $\mathcal{O}$ -algebra with respect to the cartesian symmetric monoidal (∞,1)-category structure on (∞,1)Cat. This way coCartesian fibrations of $(\infty,1)$ -operads over some $\mathcal{O}^\otimes$ are equivalently $\mathcal{O}$ -algebras in (∞,1)Cat, i.e. $\mathcal{O}$ -monoidal (∞,1)-categories.