Categories over quantum affine algebras and monoidal categorification

Let $U_q'(\mathfrak{g})$ be a quantum affine algebra of untwisted affine $ADE$ type, and $\mathcal{C}_{\mathfrak{g}}^0$ the Hernandez-Leclerc category of finite-dimensional $U_q'(\mathfrak{g})$-modules. For a suitable infinite sequence $\widehat{w}_0= \cdots s_{i_{-1}}s_{i_0}s_{i_1} \cdots$ of simple reflections, we introduce subcategories $\mathcal{C}_{\mathfrak{g}}^{[a,b]}$ of $\mathcal{C}_{\mathfrak{g}}^0$ for all $a \le b \in \mathbb{Z}\sqcup\{ \pm \infty \}$. Associated with a certain chain $\mathfrak{C}$ of intervals in $[a,b]$, we construct a real simple commuting family $M(\mathfrak{C})$ in $\mathcal{C}_{\mathfrak{g}}^{[a,b]}$, which consists of Kirillov-Reshetikhin modules. The category $\mathcal{C}_{\mathfrak{g}}^{[a,b]}$ provides a monoidal categorification of the cluster algebra $K(\mathcal{C}_{\mathfrak{g}}^{[a,b]})$, whose set of initial cluster variables is $[M(\mathfrak{C})]$. In particular, this result gives an affirmative answer to the monoidal categorification conjecture on $\mathcal{C}_{\mathfrak{g}}^-$ by Hernandez-Leclerc since it is $\mathcal{C}_{\mathfrak{g}}^{[-\infty,0]}$, and is also applicable to $\mathcal{C}_{\mathfrak{g}}^0$ since it is $\mathcal{C}_{\mathfrak{g}}^{[-\infty,\infty]}$.