Update 3/5:
When I originally answered this question I didn’t realize that your code was almost an exact copy of the wp_get_archives function. There is nothing in your code or the original wp_get_archives code that supports categories. The function was written to get date archives and your defaults are missing the type, limit, before and after arguments.
If you want to build categories into this function you will need to write your own where and join filter clauses.
Source: wp-includes/general-template.php
/**
* Display archive links based on type and format.
*
* The 'type' argument offers a few choices and by default will display monthly
* archive links. The other options for values are 'daily', 'weekly', 'monthly',
* 'yearly', 'postbypost' or 'alpha'. Both 'postbypost' and 'alpha' display the
* same archive link list, the difference between the two is that 'alpha'
* will order by post title and 'postbypost' will order by post date.
*
* The date archives will logically display dates with links to the archive post
* page. The 'postbypost' and 'alpha' values for 'type' argument will display
* the post titles.
*
* The 'limit' argument will only display a limited amount of links, specified
* by the 'limit' integer value. By default, there is no limit. The
* 'show_post_count' argument will show how many posts are within the archive.
* By default, the 'show_post_count' argument is set to false.
*
* For the 'format', 'before', and 'after' arguments, see {@link
* get_archives_link()}. The values of these arguments have to do with that
* function.
*
* @since 1.2.0
*
* @param string|array $args Optional. Override defaults.
*/
function wp_get_archives($args="") {
global $wpdb, $wp_locale;
$defaults = array(
'type' => 'monthly', 'limit' => '',
'format' => 'html', 'before' => '',
'after' => '', 'show_post_count' => false,
'echo' => 1
);