Is it Haram to have a cat in my house?

All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.

Having cats in the house is permissible if there is a legal purpose acceptable in Islamic Sharee’ah, such as to keep insects and animals like rats, etc. away, and there is evidence that they are not a means of bringing contagious diseases.

It is obligatory for the one who keeps a cat to feed it properly or let it feed itself.

The Prophet said: “A woman was punished and entered Hellfire because she kept a cat in captivity until it died; she neither fed it nor gave it water to drink; nor did she set it free so that it could eat from the vermin of the earth.” [Al-Bukhari and Muslim]
Since a cat is pure, touching it does not nullify ritual purity. In the Hadeeth reported from Kabshah bint Ka'b who, when she was under the care of Abu Qataadah, entered the room to pour some water for him. A cat came, drank some of the water, and Qatadah proceeded to tilt the container so the cat could drink more. Kabshah said, “He noticed that I was watching him.” He asked, “Are you surprised, O Niece?” I answered, “Yes.” He said, “The Messenger of Allah said, 'It (the cat) is not impure; they intermingle with you.”

Allah knows best.