Create “archives by year and week” by category template

Your English isn’t clear to me. What I understood is that you want to list archives of posts with specifications. Like, posts from the first week January, year 2013.

Well, there is a dedicated function for this purpose and it’s: wp_get_archives();

For example, if you want posts of the first week of January, year 2013:

$args = array(
'type' => 'daily',
'echo' => 0,
'year' => '2013',
'month' => '1'
);
echo '<ul>'.wp_get_archives($args).'</ul>';

Hope it helps.