Cat sleep with baby

So I have a 6 month old. She sleeps in the side sleeper right next to our bed. We are going to start transitioning to her crib.soon. My cat has always tried to sleep with her but I haven’t let her just because my fear of my daughter suffocating. My cat has seemed Interested in her. We can’t keep my cat out of my daughters room as we keep the door open due to no insulation in there at the moment and the tall gate we have in the doorway my cat just learned to jump over. Has anyone’s cat slept with their baby? Has anything bad happened?