While you may not be able to integrate your categories list directly into the slider (this is dependent on how the slider is set up and I haven’t used this particular theme), it’s possible to display a list of categories and their links that functions in the way you’ve asked and then apply the same (or similar) styles the slider’s tabs use to your categories links. Again, this wouldn’t interact with the slider at all, merely mimic its look.
To get a list of categories, you would need to call wp_list_categories (see Codex for more info). Where you would place this call will depend on where you want the categories links to show up (e.g., only on the front page, on all pages, on all posts, etc.).
Hope this gets you headed in the right direction.