Cats spend more than 15 hours out of 24 hours a day sleeping or dozing. We know of five different categories of sleep: a dozy state, a light sleep and three levels of deep sleep.
A cat still moves if she is in a dozy state or light sleep. Part of her senses are still awake. If you disturb her, she will startle and be grumpy and defensive.
The real deep sleep lasts no longer than four hours a day. Your cat will look for an undisturbed, draft-free place to do it. Generally, she will place her paws under her body and curl herself up. The colder the sleeping place, the more tightly she will curl up, as her body temperature will fall while sleeping.
Cats dream a lot when they are in a deep sleep. They also make various sounds. Sometimes they even move and seem to hunt an invisible prey.
If it were up to cats, most would choose to sleep in the bed of their human. It's up to you whether you will allow your cat to do this. However, what you should consider is that cats like their private area to be respected. That means they should respect your private area, as well. One compromise could be to put a pillow on the windowsill of your bedroom. Your cat's sleeping place doesn't necessarily have to be the lap of luxury; however, it should be draft-free and warm. Naturally, many cats prefer a higher place for sleeping because of the better view, which makes them feel safer. If your cat sleeps in a basket, it should be large enough to allow her to have a good stretch. Use a washable blanket for cushioning.