<p>I personally call it UC Berkeley, just because many people I talk to don’t know the nickname “Cal.”</p>
<p>But “Cal” is fine, as long as the audience knows what you’re referring to. I don’t think “Cal” is any more “arrogant” or more “ambiguous” than “Xerox machine,” or “Kleenex,” or “dial” (someone on your smart phone), or (buying a) “PC.” It’s part of the language dynamics.</p>