Geodesic
The operadic nerve, relative nerve and the Grothendieck construction
Theory and applications of categories, Tome 34 (2019), pp. 349-374 Cet article a éte moissonné depuis la source Theory and Applications of Categories website

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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.

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Classification : 55U40, 55U10, 18D20, 18D30
Keywords: simplicial categories, Grothendieck construction, higher category theory, operads 
@article{TAC_2019_34_a12,
     author = {Jonathan Beardsley and Liang Ze Wong},
     title = {The operadic nerve, relative nerve and the {Grothendieck} construction},
     journal = {Theory and applications of categories},
     pages = {349--374},
     year = {2019},
     volume = {34},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2019_34_a12/}
}
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Jonathan Beardsley; Liang Ze Wong. The operadic nerve, relative nerve and the Grothendieck construction. Theory and applications of categories, Tome 34 (2019), pp. 349-374. http://geodesic.mathdoc.fr/item/TAC_2019_34_a12/