Cartesian closed varieties II: links to algebra and self-similarity

This article is the second in a series investigating cartesian closed varieties. In first of these, we showed that every non-degenerate finitary cartesian variety is a variety of sets equipped with an action by a Boolean algebra B and a monoid M which interact to form what we call a matched pair ${\left[\smash{{B} \mathbin{\mid}{M} }\right]}$ . In this article, we show that such pairs ${\left[\smash{{B} \mathbin{\mid}{M} }\right]}$ are equivalent to Boolean restriction monoids and also to ample source-étale topological categories ; these are generalizations of the Boolean inverse monoids and ample étale topological groupoids used to encode self-similar structures such as Cuntz and Cuntz–Krieger $C^\ast$ -algebras, Leavitt path algebras, and the $C^\ast$ -algebras associated with self-similar group actions. We explain and illustrate these links and begin the programme of understanding how topological and algebraic properties of such groupoids can be understood from the logical perspective of the associated varieties.