Genuine versus naïve symmetric monoidal $G$-categories

Abstract

We prove that through the eyes of equivariant weak equivalences the genuine symmetric monoidal $G$-categories of Guillou and May [Algebr. Geom. Topol. 17 (2017), no. 6, 3259–3339] are equivalent to just ordinary symmetric monoidal categories with $G$-action. Along the way, we give an operadic model of global infinite loop spaces and provide an equivalence between the equivariant category theory of genuine symmetric monoidal $G$-categories and the $G$-parsummable categories studied by Schwede [J. Topol. 15 (2022), no. 3, 1325–1454] and the author [New York J. Math. 29 (2023), 635–686].