  In this article, we characterise the operadic variety of commutative
associative algebras over a field via a (categorical) condition: the
associativity of the so-called cosmash product. This condition, which is
closely related to commutator theory, is quite strong: for example, groups do
not satisfy it. However, in the case of commutative associative algebras, the
cosmash product is nothing more than the tensor product; which explains why in
this case it is associative. We prove that in the setting of operadic varieties
of algebras over a field, it is the only example. Further examples in the
non-operadic case are also discussed.
