# When is a quasicategory over $N(\Delta)^{op}$ a planar $\infty$-operad?

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 fibration of simplicial sets $C^\otimes\to N(\Delta)^{op}$, whereas the latter replaces $N(\Delta)^{op}$ with $\mathcal{A}ss^\otimes$, Lurie's associative $\infty$-operad. The former is perhaps more easily used because of Proposition 1.6.3 of DAG II which allows us to produce a cocartesian fibration of simplicial sets $N(M^\otimes)\to N(\Delta^{op})$ whenever we started with a topologically enriched monoidal category $M$.

Unfortunately, this does not immediately provide us with a cocartesian fibration $N(M^\otimes)\to \mathcal{A}ss^\otimes$, where $\mathcal{A}ss^\otimes$ is the associative $\infty$-operad of Higher Algebra. There is certainly a morphism of simplicial sets $N(M^\otimes)\to \mathcal{A}ss^\otimes$, but it's not clear to me that it's necessarily easy to check whether or not this actually gives the structure I desire (that is, a monoidal structure on the $\infty$-category $N(M)$ corresponding to the monoidal structure on $M$). It happens to follow from Theorem 4.7.1.10 of Higher Algebra that if the map of marked simplicial sets $N(M^\otimes)^\natural\to (N(\Delta)^{op})^\natural$ (where I've used $\natural$ to indicate marking the inert morphisms in $N(\Delta)^{op}$ according to Definition 4.7.1.9 of HA and the inert morphisms of $N(M^\otimes)$ simply being the ones that project onto inert morphisms of $N(\Delta)^{op}$) is fibrant in the model structure on $(\mathcal{P}Op_\infty)_{/(N(\Delta)^{op})^\natural}$ induced from the model structure on $\mathcal{P}Op_\infty$, then the above composition produces a planar $\infty$-operad. However, it seems like the only way to check if something is fibrant is to know beforehand that it actually comes from a planar $\infty$-operad.

So my question is the following: given a topologically enriched monoidal category $M$, is there a planar $\infty$-operad $p:N(M^\otimes)\to\mathcal{A}ss^\otimes$ such that the fiber over $\langle 1\rangle$ is equivalent to $N(M)$ and such that a strict algebra in $M$ gives an associative algebra of $p:N(M^\otimes)\to\mathcal{A}ss^\otimes$?

Thanks! :-)

In the first section of DAG II it's proven that if we start with a simplicial monoidal category in which the monoidal functor $C\times C\to C$ is simplicially enriched and such that $C$ is fibrant in the Bergner model category of simplicial categories in which the fibrant objects are enriched in Kan complexes (e.g. we might take $C$ to be topologically enriched from the start) then we can obtain a cocartesian fibration of simplicial sets $C^\otimes\to N(\Delta)^{op}$ whose fiber over $[1]$ is equivalent to $N(C)$. Later, it's proven that there is an equivalence of $\infty$-categories between the cocartesian fibrations $C^\otimes\to N(\Delta)^{op}$ and the cocartesian fibrations $D^\otimes\to \mathcal{A}ss^\otimes$ (actually Lurie calls these things Segal monoids, but you can check that these correspond to the $\mathcal{A}ss^\otimes$-monoids from Higher Algebra). What's important however is that this equivalence is given by composing with an approximation map" $N(\Delta)^{op}\to \mathcal{A}ss^\otimes$. In other words we have complete control over what the fiber over $\langle 1\rangle$ is in the $\mathcal{A}ss^\otimes$-monoid associated to $C^\otimes\to N(\Delta)^{op}$, and indeed it remains $N(C)$.
What's even more important is that later, it is shown that there is an equivalence of $\infty$-categories between the algebras of $C$ with respect to the fibration $C^\otimes\to N(\Delta)^{op}$ and the associated composition $C^\otimes\to \mathcal{A}ss^\otimes$. Thus, we still have complete control over specific algebras for the latter (Segal) monoidal structure. And these are indeed the monoids of the original category $C$.