Cartesian closed varieties I: the classification theorem

Richard Garner

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arXiv · 2302.04402

Cartesian closed varieties I: the classification theorem

Richard Garner

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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.

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Richard Garner. 2023-02-09. Cartesian closed varieties I: the classification theorem. https://arxiv.org/abs/2302.04402

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