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).

65 pages. After receiving a remark we have amended Definition 2.10; Corollary 2.11 has been stricken and replaced by a counterexample which shows which the changes to Definition 2.10 (which do not affect the results of later sections) were indeed necessary. Further comments are welcome!