2023/02/09 by Garner, Richard
#Category Theory (math.CT) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2302.04402
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.