Reuse the "category" slug for a custom post type

Question

Is it possible to reuse the "category" slug for a custom post type and still have the categories editable within the CPT menu?

With a CPT of "work" I'd like to be able to have URLS like this domain.com/work/category

I've tried this code but the taxonomy just doesn't show up in the CPT menu:

$args = array(
    'labels' => array(
        'name'                  => __( 'Work Projects' ),
        'singular_name' => __( 'Work Project' ),
        'add_new_item'  => __( 'Add New Work Project' ),
        'edit_item'         => __( 'Edit Work Project' )
    ),
    'public'                => true,
  'has_archive'         => true,
  'publicly queryable'  => true,
  'menu_icon'                       => 'dashicons-lightbulb',
  'show_in_nav_menus'       => true
);
register_post_type( 'work', $args );

$args = array(  
  'hierarchical' => true,  
  'labels' => array('name' => 'Work Categories', 'singular_name' => 'Work Category'), 
  'query_var' => true,
  'show_ui' => true,
  'show_in_menu' => true,
  'rewrite' => array(
      'slug' => 'type', 
      'with_front' => true 
  )
);  
register_taxonomy( 'category', 'work', $args );

Answer 1

To share a taxonomy between multiple post types when you register her, you have to pass an array with the slug if all posts types that she will belong to.

Example:

register_taxonomy( 'category', array( 'work', 'another_cpt' ), $args );

But in your case the category taxonomy is already registered, so you have to do the following:

function add_category_to_work_cpt(){
    register_taxonomy_for_object_type( 'category', 'work' );
}
add_action( 'init', 'add_category_to_work_cpt' );

See the Codex and this answer for reference.