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We construct a generalization of the operadic nerve, providing a translation between the equivariant simplicially enriched operadic world to the parametrized $\infty$-categorical perspective. This naturally factors through genuine equivariant operads, a model for ``equivariant operads with norms up to homotopy''. We introduce the notion of an op-fibration of genuine equivariant operads, extending Grothendieck op-fibrations, and characterize fibrant operads as the image of genuine equivariant symmetric monoidal categories. Moreover, we show that under the operadic nerve, this image is sent to G-symmetric monoidal G-$\infty$-categories. Finally, we produce a functor comparing the notion of algebra over an operad in each of these two contexts.
@article{TAC_2019_34_a25,
author = {Peter Bonventre},
title = {The genuine operadic nerve},
journal = {Theory and applications of categories},
pages = {736--780},
publisher = {mathdoc},
volume = {34},
year = {2019},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2019_34_a25/}
}
Peter Bonventre. The genuine operadic nerve. Theory and applications of categories, Tome 34 (2019), pp. 736-780. http://geodesic.mathdoc.fr/item/TAC_2019_34_a25/