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.
Best Answer
The answer to Question (ii) is positive. That is to say, there is a weak equivalences between the following functors from Segal topological categories to quasicategories: the composition of the singular complex functor with the homotopy coherent nerve functor, and the realization of the singular complex.
Indeed, the first step in both functors is the same: we take the singular complex of a Segal topological category, which yields a Segal category. Therefore, the problem reduces to establishing a weak equivalence between the homotopy coherent nerve functor N and the realization functor R, both considered as functors from Segal categories to quasicategories.
Both functors are homotopy cocontinuous: the homotopy coherent nerve is a Quillen equivalence, and the realization functor by construction.
Since the quasicategory of quasicategories is a reflective localization of ∞-presheaves on Δ, it suffices to construct a weak equivalence between the restructions of N and R along the Yoneda embedding of Δ into Segal categories.
Indeed, both restrictions are weakly equivalent to the Yoneda embedding of Δ into quasicategories, by construction.
Question (i) is not formulated rigorously, but there are rigorously defined functors from quasicategories to Segal categories. For example, one can use the left adjoint of the homotopy coherent nerve functor, which tautologically provides a positive answer.
Other constructions can be obtained by passing from quasicategories to Segal spaces, and then to Segal categories using the constructions of Joyal–Tierney and Bergner. Bergner's book has details of these construction. To see that the resulting functor is indeed the inverse of the functors considered above, consider the two compositions, and show they are weakly equivalent to the corresponding identity functor by restricting along the Yoneda embedding and observing that the resulting restrictions are weakly equivalent by construction.