Using titles in such comparisons is always a little bit risky – you have to deal with encodings and so on.
Much safer way is using of slugs in code, because they are url-safe.
So your code could look like this:
function posend_text_shortcode() {
$mycategory = get_the_category();
$slug = '';
if ( ! empty($mycategory) ) { // you have to check, if any category is assigned
$slug = $mycategory[0]->slug;
}
switch($slug){
case 'this-that': // change to real slug
get_template_part('/inc/style/check.php'); // you should use get_template_part instead of including template parts
break;
default:
get_template_part('/inc/style/default.php');
break;
}
}