In this talk, we talk about the categorical crystal structure on the Hernandez-Leclerc category $\mathscr{C}_\mathfrak{g}^0$. We define extended crystals for quantum groups and show that there is a braid group action on extended crystals. We then explain how the set of the isomorphism classes of simple modules in $\mathscr{C}_\mathfrak{g}^0$ has an extended crystal structure, and discuss the braid group action from the viewpoint of the Hernandez-Leclerc category $\mathscr{C}_\mathfrak{g}^0$. This talk is based on a joint work with M. Kashiwara (arXiv: 2111.07255 and 2207.11644).
