Skew braces have recently attracted attention as a method to study set-theoretical solutions of the Yang-Baxter equation. Here, we present a new approach to these solutions by studying Hopf algebras in the category, SupLat, of complete lattices and join-preserving morphisms. We connect the two methods by showing that any Hopf algebra, H in SupLat, has a corresponding group, R(H), which we call its remnant and a co-quasitriangular structure on H induces a YBE solution on R(H), which is compatible with its group structure. Conversely, any group with a compatible YBE solution can be realised in this way. Additionally, it is well-known that any such group has an induced secondary group structure, making it a skew left brace. By realising the group as the remnant of a co-quasitriangular Hopf algebra, H, this secondary group structure appears as the projection of the transmutation of H. Finally, for any YBE solution, we obtain a SupLat-FRT Hopf algebra in SupLat, whose remnant recovers the universal skew brace of the solution.