Finitary Cartesian closed varieties and semigroup actions
Mark V Lawson
TL;DR
This work develops a categorical and algebraic bridge between Garner's matched-pair constructions $[B|M]$ and Boolean left restriction monoids $S$. By showing that $[B|M]$-Set is equivalent to a category of $S$-actions on supported sets, the authors unify the treatment of factorizable left restriction monoids with the $[E|M]$-set formalism, and they derive a Cartesian closed framework for these action categories. The key contributions are the structure theorem for factorizable LR-monoids via matched pairs, the equivalence between supported actions and presheaf-like data, and the Boolean/étale refinements that recover Garner's results and link to inverse- or étale-groupoid settings. Collectively, these results imply that every non-degenerate finitary Cartesian closed variety can be realized as a category of $S$-actions for some Boolean left restriction monoid $S$, providing a new perspective on Cartesian closedness and its algebraic semantics, with connections to topological full groups in $C^{*}$-algebra contexts.
Abstract
We build on some ideas of Richard Garner. Let $M$ be a monoid and $B$ a Boolean algebra. A `matched pair' $[B|M]$ consists of $B$ and $M$ and some mutual interactions. Garner showed that every such matched pair determines (what we shall call) a Boolean left restriction monoid $S = S[B|M]$. In this paper, we show that the data of a $[B|M]$-set (defined later) may be encoded by means of a certain kind of action by $S$. This means that the category $[B|M]$-{\bf sets} is equivalent to a category of {\bf $S$-actions}. We deduce, as a result of Garner's work, that every non-degenerate finitary Cartesian closed variety is equivalent to a special category of $S$-actions where $S$ is a Boolean left restriction monoid.