Cartesian closed varieties I: the classification theorem

TL;DR

This paper provides a classification theorem for cartesian closed varieties, showing they are precisely those with operations decomposable into hyperaffine-unary parts, characterized by actions of monoids and Boolean algebras.

Contribution

It improves Johnstone's 1990 characterization by establishing that equational theories are cartesian closed iff their operations have a unique hyperaffine-unary decomposition.

Findings

Non-degenerate cartesian closed varieties are varieties of sets with compatible monoid and Boolean algebra actions.

The classification theorem characterizes such varieties as those with operations decomposable into hyperaffine-unary forms.

The result unifies theories with monoid actions and Boolean algebra actions under a common framework.

Abstract

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.