  We introduce the symmetricity notions of symmetric h-monoidality,
symmetroidality, and symmetric flatness. As shown in our paper arXiv:1410.5675,
these properties lie at the heart of the homotopy theory of colored symmetric
operads and their algebras. In particular, the former property can be seen as
the analog of Schwede and Shipley's monoid axiom for algebras over symmetric
operads and allows one to equip categories of such algebras with model
structures, whereas the latter ensures that weak equivalences of operads induce
Quillen equivalences of categories of algebras. We discuss these properties for
elementary model categories such as simplicial sets, simplicial presheaves, and
chain complexes. Moreover, we provide powerful tools to promote these
properties from such basic model categories to more involved ones, such as the
stable model structure on symmetric spectra.
