∞ -operads as symmetric monoidal ∞ -categories

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Rune Haugseng
Norwegian University of Science and Technology
https://orcid.org/0000-0001-6468-362X
Joachim Kock
Universitat Autònoma de Barcelona. Departament de Matemàtiques
https://orcid.org/0000-0003-3358-2812

We use Lurie’s symmetric monoidal envelope functor to give two new descriptions of ∞-operads: as certain symmetric monoidal ∞-categories whose underlying symmetric monoidal ∞-groupoids are free, and as certain symmetric monoidal ∞-categories equipped with a symmetric monoidal functor to finite sets (with disjoint union as tensor product). The latter leads to a third description of ∞-operads, as a localization of a presheaf ∞-category, and we use this to give a simple proof of the equivalence between Lurie’s and Barwick’s models for ∞-operads.

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∞-operads, symmetric monoidal ∞-categories

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Haugseng, Rune; Kock, Joachim. «∞ -operads as symmetric monoidal ∞ -categories». Publicacions Matemàtiques, 2024, vol.VOL 68, núm. 1, p. 111-37, http://raco.cat/index.php/PublicacionsMatematiques/article/view/422922.
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