Set the number of posts per page in all WordPress category archives using pre_get_posts.

wordpress-change-category-posts-per-page.php

<?php

add_action( 'pre_get_posts', 'prefix_category_query' );
/**
 * Customize category query using pre_get_posts.
 * 
 * @author     FAT Media <http://youneedfat.com>
 * @copyright  Copyright (c) 2013, FAT Media, LLC
 * @license    GPL-2.0+
 * @todo       Change prefix to theme or plugin prefix
 *
 */
function prefix_category_query( $query ) {
  
	if ( $query->is_main_query() && ! $query->is_feed() && ! is_admin() && is_category() ) {
		$query->set( 'posts_per_page', '10' ); //Change this number to anything you like.
	}

}