A quantum double construction in Rel. (Q2909732)

The author's idea is to show that parts of quantum group theory can be carried out in a category used for the semantics of computation and logic, namely the compact closed category \(\mathbf{Rel}\) of sets and binary relations. This is a ribbon category with a symmetric braiding. The author studies bialgebras and Hopf algebras in this category, and shows that their (co)module categories are monoidal categories with extra structures, such as traces and autonomy. The principal example is to take in \(\mathbf{Rel}\) the group algebra of a group \(G\), and apply the Drinfeld quantum double construction. The resulting Hopf algebra has a universal R-matrix and a universal twist. The category of its modules is the category of crossed \(G\)-modules, i.e., sets \(X\) with a \(G\)-action, and a function \(|\;|\) from \(X\) to \(G\) such that \(|g.x|=g.|x|.g^{-1}\). This category, with suitable binary relations, has non-symmetric braiding and non-trivial twist.