stevecoy on "Category permalinks with custom post types"

I have a custom post type, Videos. It uses standard WP taxonomies (categories, tags).

$videoargs = array(
		'labels' => $labels,
		'public' => true,
		'publicly_queryable' => true,
		'show_ui' => true,
		'show_in_menu' => true,
		'query_var' => true,
		'rewrite' => array('slug' => 'video', 'with_front' => 'false'),
		'has_archive' => 'videos',
		'hierarchical' => false,
		'capability_type' => 'post',
		'menu_position' => 20,
		'menu_icon' => $icon,
		'taxonomies' => array('category'),
		'supports' => array('title','thumbnail','excerpt','comments')
	  ); 

	register_post_type( 'li_video', $videoargs);

The archive page for viewing all videos is at mysite.com/videos, using theme file archive-li_videos.php. I would like to be able to view different categories, showing videos only, at a URL like this:

mysite.com/videos/my-category-name
OR
mysite.com/videos/category/my-category-name

I know this is possible, because I've seen a plugin do it, but said plugin has really huge architecture and I can't figure it out.

I know I could easily pass a query string parameter like mysite.com/category/my-category-name?view=videos and alter the query parameters accordingly, but I'd like to avoid this. I also do not want to register a separate taxonomy for my custom post type, because then users will have an extra thing to keep track of. And it seems like this is doable.

Thanks for your help!