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The operadic nerve, relative nerve and the Grothendieck construction

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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.

http://www.tac.mta.ca/tac/volumes/34/13/34-13.pdf

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