Is the triple point of water exactly 0.01 $\deg$C? Is the triple point of water exactly $0.01 \deg$C, or is this an empirical value/is there some uncertainty to it? If so, how do we know it is exact, and why?
 A: As of today, February 11, 2019, the triple point of (pure) water is exactly 0.01 $^\circ$C because the Kelvin scale is defined by this value, and the Celsius scale is defined by its relation to the Kelvin scale.
This is similar to how the speed of light is exactly 299,792,458 m/s because the meter is defined to be the distance traveled by light in vacuum in 1/299,892,458 s.
On May 20, the definition of the Kelvin scale is to be changed so that the Boltzmann constant will be defined to be $1.380649×10^{−23}\ {\rm J⋅K}^{−1}$. After this date it is possible that more precise measurements than have previously been done will find that the triple point of water is not at exactly 0.01 $^\circ$C

isn’t the definition of Celsius based on melting and boiling point?

The problem is that the melting point of water changes slightly depending on the surrounding pressure of water vapor. It was found easier to reproduce the conditions of the triple point (and to know they were achieved by demonstrating the coexistence of three phases of water) than to accurately produce a 1 atm pressure as required for the old definition (1743-1954) which set the melting point of water at 0 $^\circ$C. 
But the triple point is actually the melting point with 61.657 Pa partial pressure of water vapor. It just happens also to be the boiling point at that pressure. So in a sense the Celsius scale is (for a few more months) defined by the melting point of water, just at a different pressure than previously.
