  Racks and quandles are rich algebraic structures that are strong enough to
classify knots. Here we develop several fundamental categorical aspects of the
theories of racks and quandles and their relation to the theory of
permutations. In particular, we compute the centers of the categories and
describe power operations on them, thereby revealing free extra structure that
is not apparent from the definitions. This also leads to precise
characterizations of these theories in the form of universal properties.
