Show category to visitors in different name or different language without changing basic category

You need to make your category name translation ready. You can do so by using WordPress functions such as __('Your category','your-text-domain'), _x('Your Category', 'your context', 'text domain') and more click here. After that, you can use poedit or other tool to extract translatable strings form your code and create your_locale.po file which contain your custom translation strings.