Comparison of homotopy coherent nerves

Remark 24.3.3. The fact that we refer to \(\Ntop (D)\) as the ‘homotopy coherent nerve’ is somewhat abusive: this terminology is often used instead for the simplicial set \(N^{\Delta }(\bC )\) associated to a simplicially enriched category, cf. Example 1.6.14. The two constructions are closely related: given a topologically enriched category \(D\), we may obtain a simplicially enriched category \(\bC \) by taking \(\Hom ^{\Delta }_{\bC }(X,Y) := \Sing (\Homtop _{D}(X,Y))\), i.e. we apply the singular simplicial complex construction from Example 1.6.8 to each of the hom spaces of \(D\). Then the underlying \(\infty \)-category of the quasicategory \(N^{\Delta }(\bC )\) is equivalent to \(\Ntop (D)\). The proof of this fact is quite technical: it relies on the fact that there is a model structure (due to Bergner (2007)) on the category of (small) simplicially enriched categories whose localization at the weak equivalences is \(\Cat _{\infty }\). A reference for this claim in the literature may be found in Gepner and Meier (2023), Corollary B.4.