Are inflatable beds and cats compatible?

It depends on the bed. I have an old Nautilus air mattress that has a heavy cover and a thick foam top which helps protect against most sharp objects. That bed originally cost a couple thousand bucks. I believe the higher-end Sleep Number beds also have foam tops. A cheaper bed like an AeroBed has no cover at all and might be vulnerable to sharp claws, though a mattress pad or heavy comforter would provide some protection.