Do monsters kill cats in minecraft?

I wanted to build a creeper mob farm in minecraft and put cats in the middle so that the creepers could escape. In order to save resources, I wanted to leave out the trap doors at the top so that other monsters can also spawn. Are these monsters going to kill the cat?

No they won't

The zombies won't kill the cat

Thank you!

The problem is if you don't use trap doors and anti-spinning carpets, the farm is less efficient as many mob spawn areas are used for other mobs