*Categories are pushed along with content as a dependency. So if a new piece of content is labeled with a new category and pushed, then both the content and the category will be added to the receiving server. In addition, the parent heirarchy chain for the pushed category will also be pushed/preserved. However, sibling categories will not also be sent, unless specifically added to the content being pushed.
All Categories can be synchronized from one Dynamic Push Publishing Endpoint to another using the Content Types → Categories manager, in addition to being pushed as a dependency of content.
*Categories can NOT be pushed to a Static Push Publishing Endpoint - attempting this will result in an empty bundle.