What is the purpose of the capacitor across the hysteresis resistor?

 

Sounds like homework, especially using a 741 opamp. Is it?

Regardless, what you your thoughts on what it's purpose might be? First consider what impact it has on the circuit operation as the signal frequency increases.

 

It helps speed the output transition at the trigger point.

 

The capacitor between the output and the non-inverting input provides a path for the feedback to make the transition happen. Without that lower coupling impedance provided by the capacitor the circuit may not even respond quickly to a change in the input. It might even oscillate.