The Operadic Nerve, Relative Nerve, and the Grothendieck Construction

Abstract

We relate the relative nerve ${\mathrm{N}}_f(\mathcal{D})$ of a diagram of simplicial sets $f:\mathcal{D}\to \mathsf{sSet}$ with the Grothendieck construction $\mathsf{Gr}F$ of a simplicial functor $F:\mathcal{D}\to \mathsf{sCat}$ in the case where $f=\mathrm{N}F$. We further show that any strict monoidal simplicial category $\mathcal{C}$ gives rise to a functor ${\mathcal{C}}^{\bullet }:{\Delta }^{\mathrm{op}}\to \mathsf{sCat}$, and that the relative nerve of $\mathrm{N}{\mathcal{C}}^{\bullet }$ is the operadic nerve ${\mathrm{N}}^{\otimes }(\mathcal{C})$. Finally, we show that all the above constructions commute with appropriately defined opposite functors.

arXiv commentImprovements to exposition, citations and introduction thanks to referee suggestions

References in corpus (5)

• Higher Topos Theory
• Derived Algebraic Geometry III: Commutative Algebra
• On the Unicity of the Homotopy Theory of Higher Categories
• Quillen adjunctions induce adjunctions of quasicategories
• The Enriched Grothendieck Construction

Cited by in corpus (1)

• The Enriched Grothendieck Construction