  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.
