It all depends on what do you mean under "manage a category". If under manage you anticipate the ability to manage posts/pages that belong to that category, then YES you can do that. To learn more about this subject, please refer to this Youtube video.
However, if you are looking to restrict the ability to add new categories and also edit all existing categories, then we would have to disappoint you. Unfortunately, the way the WordPress core is implemented, there is no way to restrict adding new categories and at the same time prevent editing existing.
In other words - you cannot prevent a user from adding new categories and allow editing existing. Plugins or custom solutions that offer this type of solution, do not do this correctly and compromising WordPress core integrity.