I'm not sure how you can get all these "at a time". I believe you mean with a single request? The category value is stored as a URI in the work items, and you need to access this URI to get the category name and other information. So I believe some kind of iterations have to be done.

You can use the reportable REST API.
https://jazz.net/wiki/bin/view/Main/ReportsRESTAPI#category_type_com_ibm_team_worki

Or you can use the OSLC API to step through Project Area, Resource Shape till you get to the "allowedValues" of the category (it is a list of URIs mentioned above). The URI looks like this:
https://clm602:9443/ccm/oslc/context/_ftZ1gX57Eea3_ddS3_sNyg/shapes/workitems/defect/property/category/allowedValues

And each category URI looks like this:
https://clm602:9443/ccm/resource/itemOid/com.ibm.team.workitem.Category/__VI4Sn57Eea3_ddS3_sNyg