Add custom attribute for category

Follow bellow steps

Step : 1 app\etc\modules\AR_Dbchange.xml

<?xml version="1.0"?>
<config>
    <modules>
        <AR_Dbchange>
            <active>true</active>
            <codePool>local</codePool>
        </AR_Dbchange>
    </modules>
</config> 

Step : 2 app\code\local\AR\Dbchange\etc\config.xml

<?xml version="1.0"?>
<config>
    <modules>
        <AR_Dbchange>
            <version>0.1.0</version>
        </AR_Dbchange>
    </modules>
    <global>
        <resources>
            <ar_dbchange_setup>
              <setup>
                   <module>AR_Dbchange</module>
                   <class>Mage_Catalog_Model_Resource_Eav_Mysql4_Setup</class>
              </setup>
          </ar_dbchange_setup>
        </resources>  
    </global>
</config> 

Step : 3 app\code\local\AR\Dbchange\sql\ar_dbchange_setup\mysql4-install-0.1.0.php

<?php

$installer = $this;
$installer->startSetup();

$attribute  = array(
    'group'                     => 'General',
        'input'                     => 'select',
        'type'                      => 'int',
        'label'                     => 'Mobile (api)',
        'source'                    => 'eav/entity_attribute_source_boolean',
        'global'                    => Mage_Catalog_Model_Resource_Eav_Attribute::SCOPE_GLOBAL,
        'visible'                   => 1,
        'required'                  => 0,
        'visible_on_front'          => 0,
        'is_html_allowed_on_front'  => 0,
        'is_configurable'           => 0,
        'searchable'                => 0,
        'filterable'                => 0,
        'comparable'                => 0,
        'unique'                    => false,
        'user_defined'              => false,
        'default'           => '0',
        'is_user_defined'           => false,
        'used_in_product_listing'   => true
);
$installer->addAttribute('catalog_category', 'mobile_api', $attribute);
$installer->endSetup();

?>

Step : 4 After refresh all magento cache to admin side (System -> Cache Management) and logout ,login again