A Primer on Chainmails: Structures for Point-free Connectivity
J. F. Du Plessis, Zurab Janelidze, Bernardus A. Wessels
TL;DR
This work generalizes the notion of connectivity in a point-free setting by introducing chainmails, a poset-theoretic abstraction of connected subsets. It develops the core theory of mails, mail-connectedness, and locally connected lattices, and then establishes a categorical adjunction between chainmails and join-complete lattices, culminating in an equivalence between chainmails and locally connected lattices via the D and K constructions. The main contribution is a precise, structural correspondence: chainmails ↔ locally connected lattices, enabling a point-free treatment of connectivity across graphs, spaces, and image-processing contexts. The results pave the way for new applications in discrete-continuous modeling and inform potential connections to physical continuum notions and upcoming work on frames and connectivity in a point-free setting.
Abstract
In point-free topology, one abstracts the poset of open subsets of a topological space, by replacing it with a frame (a complete lattice, where meet distributes over arbitrary join). In this paper we propose a similar abstraction of the posets of connected subsets in various space-like structures. The analogue of a frame is called a chainmail, which is defined as a poset admitting joins of its mails, i.e., subsets having a lower bound. The main result of the paper is an equivalence between a subcategory of the category of complete join-semilattices and the category of chainmails.