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

TL;DR

This paper explores the connections between cartesian closed varieties, algebraic structures like Boolean restriction monoids, and topological categories, linking logical, algebraic, and topological frameworks for self-similar structures.

Contribution

It establishes the equivalence between matched pairs [B|M], Boolean restriction monoids, and ample source-étale topological categories, advancing the understanding of their interrelations.

Findings

Matched pairs [B|M] are equivalent to Boolean restriction monoids.

Boolean restriction monoids correspond to ample source-étale topological categories.

These structures encode self-similar algebraic and topological objects like Cuntz algebras.

Abstract

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.