It is standard from work of Joyal and Lurie that there is a Quillen equivalence between the model category of simplicially enriched categories $Cat_\Delta$ and $\mathcal{S}\text{et}_\Delta$ with the Joyal model structure. This implies that for any fibrant simplicially enriched category, equivalently a topological category (in the sense of Segal), one gets a corresponding $\infty$-category by the coherent nerve functor $N$.

My question (coming out of some practical purposes) is whether one can establish an equivalence between $N$ and the standard coend construction using the usual nerve of a topological category. This sounds somewhat problematic, since the former preserves limits while the latter preserves colimits, but in the following we view $N$ as an equivalence $(Cat_\Delta)^{cf}\overset{\sim}{\to} (\mathcal{S}\text{et}_\Delta)^{J,cf}$ as $\infty$-categories, where $cf$=cofibrant and fibrant objects and $J$ stands for the Joyal model structure. So this seems not unreasonable.

More explicitly,

(1) for a topological category $\mathsf{T}$, the usual nerve is a simplicial space $\mathcal{P}_\bullet(\mathsf{T})$, with $\mathcal{P}_0(\mathsf{T})$ equal to the discrete set of objects in $\mathsf{T}$, and $$\mathcal{P}_n(\mathsf{T})=\coprod_{t_0,\cdots,t_n\in \mathcal{P}_0(\mathsf{T})} Maps_{\mathsf{T}}(t_0,t_1)\times\cdots\times Maps_{\mathsf{T}}(t_{n-1},t_n), n\geq 1$$ Here $$\mathcal{P}_n(\mathsf{T})\to \mathcal{P}_1(\mathsf{T})\underset{\mathcal{P}_0(\mathsf{T})}{\times}\cdots\underset{\mathcal{P}_0(\mathsf{T})}{\times} \mathcal{P}_1(\mathsf{T}), n\geq 1$$ is a strict isomorphism of topological spaces. But one can add the flexibility demanding it to be a weak homotopy equivalence, for which we call a "weak" topological category. From here one can take the coend $\int^{[n]\in \Delta}(\text{Sing}_\bullet\mathcal{P}_n(\mathsf{T}))\times N(\Delta^n)$ in $\text{Cat}_\infty$;

(2) conversely, for any $\infty$-category $\mathcal{C}$, by the work of Rezk, we can view it as a complete Segal space $\widetilde{\mathcal{P}}_{\bullet}$. One can cook up a simplicial space $\mathcal{P}_\bullet$ with a discrete 0-space (up to homotopy) as follows. Take a base point $x_i$ in each connected component of $\widetilde{\mathcal{P}}_0$, and let $\mathcal{P}_0\to \widetilde{\mathcal{P}}_0$ be the standard fibration from the (disjoint union of) path spaces based at $x_i$. Set $$\mathcal{P}_n=\widetilde{\mathcal{P}}_n\underset{\widetilde{\mathcal{P}}_0^{\times (n+1)}}{\times}\mathcal{P}_0^{\times (n+1)}.$$ Then $\mathcal{P}_\bullet$ gives a "weak" topological category.

My question is:

(i) Do the functors (1) and (2) give inverse equivalences between the $\infty$-category of "weak" topological categories and $Cat_\infty$, both modeled as full subcategories of the $\infty$-category of simplicial spaces? If I'm not mistaken, they give an adjunction pair.

(ii) If so, does the above coincide with the standard equivalence between $(Cat_\Delta)^{cf}$ and $(\mathcal{S}\text{et}_\Delta)^{J,cf}$? Intuitively, one would expect this to be true.

It is standard from work of Joyal and Lurie that there is a Quillen equivalence between the model category of simplicially enriched categories $Cat_\Delta$ and $\mathcal{S}\text{et}_\Delta$ with the Joyal model structure. This implies that for any fibrant simplicially enriched category, equivalently a topological category (in the sense of Segal), one gets a corresponding $\infty$-category by the coherent nerve functor $N$.

My question (coming out of some practical purposes) is whether one can establish an equivalence between $N$ and the standard coend construction using the usual nerve of a topological category. This sounds somewhat problematic, since the former preserves limits while the latter preserves colimits, but in the following we view $N$ as an equivalence $(Cat_\Delta)^{cf}\overset{\sim}{\to} (\mathcal{S}\text{et}_\Delta)^{J,cf}$ as $\infty$-categories, where $cf$=cofibrant and fibrant objects and $J$ stands for the Joyal model structure. So this seems not unreasonable.

More explicitly,

(1) for a topological category $\mathsf{T}$, the usual nerve is a simplicial space $\mathcal{P}_\bullet(\mathsf{T})$, with $\mathcal{P}_0(\mathsf{T})$ equal to the discrete set of objects in $\mathsf{T}$, and $$\mathcal{P}_n(\mathsf{T})=\coprod_{t_0,\cdots,t_n\in \mathcal{P}_0(\mathsf{T})} Maps_{\mathsf{T}}(t_0,t_1)\times\cdots\times Maps_{\mathsf{T}}(t_{n-1},t_n), n\geq 1$$ Here $$\mathcal{P}_n(\mathsf{T})\to \mathcal{P}_1(\mathsf{T})\underset{\mathcal{P}_0(\mathsf{T})}{\times}\cdots\underset{\mathcal{P}_0(\mathsf{T})}{\times} \mathcal{P}_1(\mathsf{T}), n\geq 1$$ is a strict isomorphism of topological spaces. But one can add the flexibility demanding it to be a weak homotopy equivalence, for which we call a "weak" topological category. From here one can take the coend $\int^{[n]\in \Delta}(\text{Sing}_\bullet\mathcal{P}_n(\mathsf{T}))\times N(\Delta^n)$ in $\text{Cat}_\infty$ (the $\infty$-category of $\infty$-categories);

(2) conversely, for any $\infty$-category $\mathcal{C}$, by the work of Rezk, we can view it as a complete Segal space $\widetilde{\mathcal{P}}_{\bullet}$. One can cook up a simplicial space $\mathcal{P}_\bullet$ with a discrete 0-space (up to homotopy) as follows. Take a base point $x_i$ in each connected component of $\widetilde{\mathcal{P}}_0$, and let $\mathcal{P}_0\to \widetilde{\mathcal{P}}_0$ be the standard fibration from the (disjoint union of) path spaces based at $x_i$. Set $$\mathcal{P}_n=\widetilde{\mathcal{P}}_n\underset{\widetilde{\mathcal{P}}_0^{\times (n+1)}}{\times}\mathcal{P}_0^{\times (n+1)}.$$ Then $\mathcal{P}_\bullet$ gives a "weak" topological category.

My question is:

(i) Do the functors (1) and (2) give inverse equivalences between the $\infty$-category of "weak" topological categories and $Cat_\infty$, both modeled as full subcategories of the $\infty$-category of simplicial spaces? If I'm not mistaken, they give an adjunction pair.

(ii) If so, does the above coincide with the standard equivalence between $(Cat_\Delta)^{cf}$ and $(\mathcal{S}\text{et}_\Delta)^{J,cf}$? Intuitively, one would expect this to be true.