Instead of building a separate page, build a custom TEMPLATE page. Then it will be inside your theme and already have access to all the functions. You do this by copying the page.php file from your theme into a new file and putting the template name up at the top in comments like this:
/* Template Name: Category Page */
Now you can create a fresh page and set the template of that page to "Category Page," which will now appear in the drop down for template selection.
You can strip out and add whatever you want as far as code goes, to get rid of menus and other things you may not want.