Characteristic Functions of Smooth Fibrations and Induced Smooth Fibrations

The equivalent description of smooth fibration and smooth cofibration was studied in the category of di-ffeological spaces. By applying the smooth lifting function and the smooth retracting function, it was respectively shown that a smooth map is a smooth fibration if and only if it has a corresponding smooth lifting function and a smooth cofibration if and only if it has a corresponding smooth retracting function. Meanwhile, it was also shown that the smooth map between smooth mapping spaces induced by a smooth fibration or a smooth cofibration is again a smooth fibration.