A (not so) short note: the equivalence of various notions of symmetric monoidal category

Abstract

In this work, intended to be a companion note to a future preprint, we give a proof of the fact that the classical (biased) notion of symmetric monoidal category, the notion of unbiased symmetric monoidal category, and the notion of homotopy symmetric monoidal category are equivalent in a precise sense (in that suitably defined groupoid-enriched categories having, respectively, biased, unbiased, and homotopy symmetric monoidal categories as objects are equivalent as enriched categories).