The genuine operadic nerve

Peter Bonventre

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.

Keywords: infinity operads, equivariant operads,symmetric monoidal categories

2010 MSC: 55P91,55P48,19D23,18D30

Theory and Applications of Categories, Vol. 34, 2019, No. 26, pp 736--780.

Published 2019-09-11.

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