The Short-Term and Long-Term Challenges
The most immediate challenge is to protect the building from further water saturation. The water used to extinguish the fire has soaked the vaults. Characteristically, there is a pocket of rubble/mortar at the point where the arches of the vault spring away from the walls. These areas will have absorbed an enormous amount of water and will need to dry out. I would anticipate the rapid installation of a light-weight metallic roof frame covered in waterproof sheets.
Simultaneously, the surviving masonry must be inspected for damage—it can be cracked and rendered fragile by the heat of the fire. This will apply especially to the slender sticks of stone that make up the tracery of the windows. A careful archaeological inspection should take place as the fire-damaged elements of the roof are removed.
Some will argue that all elements that were damaged or destroyed should be replaced with identical forms. Others will point to the concrete roof of Reims Cathedal, which burned in World War I, the iron roof of Chartres Cathedral or the iron-and-copper roof of Saint-Denis.
There will be a competition to allow architects to present their various visions.
I would argue that the “Gothic” movement, launched largely by Notre-Dame cathedral, brought revolutionary new ways to build, and we should apply 21st-century solutions to the problem of the future cathedral.