$\newcommand{\abs}[1]{|#1|}$The classifying space $\abs{\mathcal{C}}$ of a category $\mathcal{C}$ is the geometric realisation $\abs{\mathrm{N}_{\bullet}(\mathcal{C})}$ of its nerve $\mathrm{N}_\bullet(\mathcal{C})$.
- When $\mathcal{C}$ is monoidal, we can however first deloop it into a bicategory $\mathbf{B}\mathcal{C}$, which also has a classifying space, this time defined as the geometric realisation of its Duskin nerve $|\mathrm{N}^{\mathrm{D}}_\bullet(\mathbf{B}\mathcal{{C}})|$.
(The simplicial set $\mathbf{B}_{\bullet}\mathcal{C}\overset{\mathrm{def}}{=}\mathrm{N}^{\mathrm{D}}_{\bullet}(\mathbf{B}\mathcal{C})$ is called the classifying simplicial set of the monoidal category $\mathcal{C}$; see Kerodon, Tag 00FJ.)
- Similarly, if $\mathcal{C}$ is braided monoidal, then we can deloop it twice, obtaining a tricategory $\mathbf{B}\mathbf{B}\mathcal{C}$, which again has a classifying space $\abs{\mathrm{N}^{\mathrm{S}}_{\bullet}(\mathbf{B}\mathbf{B}\mathcal{C})}$, the geometric realisation of its Street Nerve.
- (Finally, we could in principle do the same for $\mathcal{C}$ symmetric monoidal, delooping it into a tetracategory and then taking the geometric realisation $\abs{\mathrm{N}^{?}_{\bullet}(\mathbf{B}\mathbf{B}\mathbf{B}\mathcal{C})}$ of its nerve, but the latter hasn't been constructed yet.)
Question. What is the relation between the homotopy types of the three spaces $\abs{\mathrm{N}_{\bullet}(\mathcal{C})}$, $|\mathrm{N}^{\mathrm{D}}_\bullet(\mathbf{B}\mathcal{{C}})|$, and $\abs{\mathrm{N}^{\mathrm{S}}_{\bullet}(\mathbf{B}\mathbf{B}\mathcal{C})}$?
