Final Fantasy XIII upgrades

Most weapons and accessories in Final Fantasy XIII are capable of having their levels increased with components. Once the maximum level (indicated with a star as opposed to the item level appearing in the menu next to the item) has been reached many are upgradable to another weapon or accessory altogether with catalyst minerals. The ability to upgrade is given early in the game and accomplished at save stations.

Catalysts
Catalysts, which can not be improperly applied, are components used to upgrade weapons and accessories to their better counterparts. Catalysts can only be applied once the item has reached its maximum level, indicated by a star replacing the item's level in the menu. Trapezohedrons will always be used to transform a two tier weapon into its third tier form.