Cartesian closed varieties I: the classification theorem

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.