  This paper 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
[B|M]. In this paper, we show that such pairs [B|M] are equivalent to Boolean
restriction monoids and also to ample source-\'etale topological categories;
these are generalisations of the Boolean inverse monoids and ample \'etale
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 to 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.
