On the global homotopy theory of symmetric monoidal categories

Abstract

Parsummable categories were introduced by Schwede as input for his global algebraic $K$-theory construction. We prove that their whole homotopy theory with respect to the so-called global equivalences can already be modelled by the more mundane symmetric monoidal categories. In another direction, we show that the resulting homotopy theory is also equivalent to the homotopy theory of a certain simplicial analogue of parsummable categories, that we call parsummable simplicial sets. These form a bridge to several concepts of 'globally coherently commutative monoids' like ultra-commutative monoids and global $\Gamma$-spaces, that we explore in arXiv:2012.12676.