# The operadic nerve, relative nerve and the Grothendieck construction

## Jonathan Beardsley and Liang Ze Wong

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

Keywords: simplicial categories, Grothendieck construction, higher category theory, operads

2010 MSC: 55U40, 55U10, 18D20, 18D30

Theory and Applications of Categories, Vol. 34, 2019, No. 13, pp 349-374.

Published 2019-04-22.

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