Classifying space via homotopy coherent nerve (Q6071800)

Just as the topological classifying space can be constructed by regarding a group as a category with a single object, taking its nerve and then applying the realization functor, taking the homotopy coherent nerve \(NG\)\ of a given simplicial group \(G\)\ gives the classifying space \(BG\)\ as its homotopy type. This paper aims to establish this result and a further generalization for simplicial groupoids. The main result (Theorem 3.6) of this paper has already appeared in [\textit{V. Hinich}, ``Homotopy coherent nerve in deformation theory'', Preprint, \url{arXiv:0704.2503}], where Hinich constructed a comparison map \[ BG\rightarrow NG \] without explaining why the map induces isomorphisms in the homotopy groups. The gap was remedied in [\textit{E. Minichiello} et al., ``Categorical models for path spaces'', Preprint, \url{arXiv:2201.03046}], but this paper takes an alternative path by comparing the corresponding left adjoints in place of comparison between \(BG\)\ and \(NG\).