A categorical reconstruction of crystals and quantum groups at $q=0$

The quantum co-ordinate algebra $A_q(\mathfrak{g})$ associated to a Kac-Moody Lie algebra $\mathfrak{g}$ forms a Hopf algebra whose comodules are precisely the $U_q(\mathfrak{g})$ modules in the BGG category ${\mathcal{O}}_{\mathfrak{g}}$. In this paper we investigate whether an analogous result is true when $q=0$. We classify crystal bases as coalgebras over a comonadic functor on the category of pointed sets and encode the monoidal structure of crystals into a bicomonadic structure. In doing this we prove that there is no coalgebra in the category of pointed sets whose comodules are equivalent to crystal bases. We then construct a bialgebra over $\mathbb{Z}$ whose based comodules are equivalent to crystals, which we conjecture is linked to Lusztig’s quantum group at $v=\mathrm{\infty }$.