  We develop the theory of the higher commutator for Taylor varieties. A new
higher commutator operation called the hypercommutator is defined using a type
of invariant relation called a higher dimensional congruence. The
hypercommutator is shown to be symmetric and satisfy an inequality relating
nested terms. For a Taylor algebra the term condition higher commutator and the
hypercommutator are equal when evaluated at a constant tuple, and it follows
that every supernilpotent Taylor algebra is nilpotent. We end with a
characterization of congruence meet-semidistributive varieties in terms of the
neutrality of the higher commutator.
