The crystal of the categorified quantum group

A categorification of the crystal $B({\tilde{U}}_q(\mathfrak{g}))$ of the modified quantized enveloping algebra associated to a symmetrizable Kac-Moody algebra $\mathfrak{g}$ is given. A crystal is constructed from a categorical $\mathfrak{g}$-action on abelian categories without finiteness conditions, and is used to construct the bicrystal structure on the set of self-dual indecomposable objects of the categorified quantum group. Crystals from Webster’s generalized tensor product categorification are identified with tensor products of lowest and highest weight crystals. As an application, a categorification of the extremal weight module is obtained.