Yes, generally. They will scan the cat for a microchip, match the cat against any 'lost cat' information they have, and then after the stray hold is up (generally 3-5 days) they will either offer him for adoption, or euthanize, depending on how full they are, what is wrong, etc. A shelter that is full can refuse to take a cat, but they can advise you on where to take him. The Animal Control in your area cannot refuse an animal that is brought in, as they are under contract with your county to take in strays, owner surrenders, etc. but they are also the ones that are quickest to euthanize due to lack of space.