  In 1990, Johnstone gave a syntactic characterisation of the equational
theories whose associated varieties are cartesian closed. Among such theories
are all unary theories -- whose models are sets equipped with an action by a
monoid M -- and all hyperaffine theories -- whose models are sets with an
action by a Boolean algebra B. We improve on Johnstone's result by showing that
an equational theory is cartesian closed just when its operations have a unique
hyperaffine-unary decomposition. It follows that any non-degenerate cartesian
closed variety is a variety of sets equipped with compatible actions by a
monoid M and a Boolean algebra B; this is the classification theorem of the
title.
