There is quite a bit of documentation in the WordPress site about categories, specifically how to retrieve data about them. Unfortunately all the available methods seem to focus on getting category data from a post, but it’s nearly impossible to figure out the category you are looking at when viewing the category page itself (i.e. the special “archive page” that lists all posts in a category).

The function is_category() falls short, since it only indicates whether or not the current page is a category page, but not which one. And get_cat_id(), get_category(), get_the_category(), and the half-dozen other “obviously”-named functions either require that you already know something about the category or provide information about a post.

So, the solution?

if (is_category()) {
   $categoryId = $GLOBALS["cat"];
}

Yes, it’s a global variable…