Weapon upgrade screen.
Most weapons and accessories in Final Fantasy XIII can be leveled up with components. Once the maximum level (indicated with a ★ 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 is accomplished at save stations.
Weapons and accessories can also be dismantled, returning various rewards in the process. Dismantling something of a higher level gives a higher quantity of components, and dismantling something at ★ level will give additional rewards.
Catalysts[]
Catalysts, which cannot 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 second-tier weapon into its third-tier form.