Clustered Hyperbolic Categories

We introduce a class of categories, called \emph{clustered hyperbolic categories}, which are generated by equivalent categories of representations of some Weyl cluster algebras. Every preseed $p$ gives rise to a \emph{categorical preseed} $\mathcal{P}$ which generates a clustered hyperbolic category through \emph{categorical mutations}. A "categorification" of Weyl cluster algebra is introduced in the sense of defining a map $\mathbf{F}_{p}$ from the clustered hyperbolic category $\mathfrak{C}(\mathcal{P})$ to the Weyl cluster algebra $\mathcal{H}(p)$ where image of $\mathbf{F}_{p}$ generates $\mathcal{H}(p)$.