When using the Yoast SEO plugin, you can define a Primary Category for each post.
Here is the code to find it:
// Check class so code does not cause errors if you remove Yoast SEO
if ( class_exists('WPSEO_Primary_Term' ) {
$yoast_primary_term = new WPSEO_Primary_Term( 'category', get_the_id() );
$yoast_primary_term_id = $yoast_primary_term->get_primary_term();
$primary_category = get_term( $yoast_primary_term_id );
if ( !empty( $primary_category->name) ) {
// Print the Primary Category. Extend as required...
echo $primary_category->name;
}
}