I am started to creating a shopping cart mobile application. I decided Opencart as backend. So I am checking default apis provided by Opencart. But I cannot find any information about category listing api. Is there any api for that?
No, there is no API for that. You can get the categories programmatically by querying the database and it would be relatively easy to write an API to provide query data to an outside source.
Take a look at catalog/controller/api/currency.php for a simple example of how that can be accomplished.