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

Question

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! :-)

Answer 1

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 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 $C$.

It seems possible to me that this can be determined from some of the stuff in Higher Algebra, but for this particular question this seems like the quickest route to getting an answer (in particular there is a Quillen equivalence between categories of preoperads, as I discuss above, but this introduces questions of fibrancy and so forth).

I'd really like to know if there are other ways to answer this question!