Categorical crystals for quantum affine algebras

A new categorical crystal structure for the quantum affine algebras is presented. We introduce the extended crystal ${\widehat{B}}_{\mathfrak{g}}(\mathrm{\infty })$ for an arbitrary quantum group, which is the product of infinite copies of the crystal $B(\mathrm{\infty })$. For a complete duality datum in the Hernandez-Leclerc category ${\mathcal{C}}_{\mathfrak{g}}^0$ of a quantum affine algebra $U_q^{\mathrm{\prime }}(\mathfrak{g})$, we prove that the set of the isomorphism classes of simple modules in ${\mathcal{C}}_{\mathfrak{g}}^0$ has an extended crystal structure isomorphic to the extended crystal ${\widehat{B}}_{\mathfrak{g}}(\mathrm{\infty })$. An explicit combinatorial description of the extended crystal ${\widehat{B}}_{\mathfrak{g}}(\mathrm{\infty })$ for affine type $A_n^{(1)}$ is given in terms of affine highest weights.