I would keep a small barn cat, one of those little 6-7 lb ones, that would not likely bother with a full-grown chicken. But not inside the coop. I would keep any chicks safe from all sides, including top and bottom, as predators come from all directions. Cats are predators. Cats can be trained, but it is pretty impossible to spend enough time with an outdoor cat to reliably train it to do anything.
Cats will reach through wire and grab a baby animal. Years ago, our spoiled and well-fed house cat was caught pulling a baby rabbit through an un-noticed broken wire in the back of the cage. Until then, we couldn't figure out what was happening to the babies. The cat viewed the hutch as his own private snack box.
If you are in a colder climate, please provide a snug place for the cat to sleep, and get two cats that get along. Provide feed and water to the cats. A healthy barn cat will prefer fresh mice over dry kibble any day, but needs strength to hunt when pickin's are lean....which they will be, if the cat does a good job. HAVE THEM NEUTERED/SPAYED!! PLEASE!!!
The two monster cats I have in the house could probably take down a full-grown chicken. If they were so inclined.
I bring my poodle into the coop when I clean it, and he cleans up the mice nests that are in there. He is trained to leave the hens alone, but I would not leave him in their pen unattended. The rooster starts it, and the dog must defend himself. Almost killed the roo one day....in self-defense. George's beak is still crooked from having his head in the dog's jaws.
Where there are chickens, there are rodents! Go get 'em! >^^<