You can apply for a .cat domain if you belong to the Catalan cultural and linguistic community in the Internet.
The main prerequisite to registering a .cat domain is that there is an intention to use Catalan on the website. Catalan does not need to be the primary language of the website. To this effect there is a 6 month deadline in place by which the website should be using Catalan.
Do you intend to use Catalan on the site? If so check if your domain is available.
The Fundacio puntCAT may suspend or delete any domain name that does not provide valid contact data, that has no significant contents in Catalan, or that may be found in breach of the terms established by the registration agreement. You may find the text of the agreement here - fundacio.cat/en/domini/about-cat/domain-rules
A video in Catalan about who uses .cat