Describing ends and tangles (and their edge variants) through Boolean algebras and functors

Abstract: The end space of an infinite graph arises naturally in many contexts as an important invariant and an interesting construction. It compactifies a locally finite graph and Diestel shows how to extend the end space to a larger space, called the tangle space, which is able to compactify any infinite graph. In both ends and tangles, it is the vertex-connectivity structure of the graph that is being studied. If we switch our attention to edge-connectivity, we can analogously define edge-ends. There is a space known as the edge-direction space which turns out to play an analogous role as the tangle space in its relationship with the end space: the edge-direction space provides a larger compact space in which the not necessarily compact space of edge-ends lives in. In this paper, we make this analogy precise, providing a natural edge analogue definition of tangles and proving they result in exactly the edge-directions. We also describe a combinatorial construction of certain Boolean algebras which give rise, via Stone duality, to the tangle and the edge-direction spaces. Finally, we pursue functorial definitions of the combinatorial constructions used in the paper, inspired by the famously functorial nature of Stone duality and by previous work by one of the authors and colleagues on trying to functorialize the end space construction. We hope our work will provide foundation and inspiration for further work on infinite graph theory that makes ample use of category theory and powerful algebraic constructions such as Boolean algebras and Stone duality.