  We define the notion of whiskered categories and groupoids, showing that
whiskered groupoids have a commutator theory. So also do whiskered
$R$-categories, thus answering questions of what might be `commutative
versions' of these theories. We relate these ideas to the theory of Leibniz
algebras, but the commutator theory here does not satisfy the Leibniz identity.
We also discuss potential applications and extensions, for example to
resolutions of monoids.
