You would have to enable auditting on that business object. Then you will see the change of category in the audit tab of the incident or in the audit table (not sure the exact table at the moment).
I have a collection object for Category in my attributes list on my default Incident window, but when I select it to add or select Category.Title it only adds it to the window and now in the area where I can select it to be added to the incident tree.
Hopefully that makes sense, so any further assistance would be appreciated.
You would need to do this at the object level. So go into Object Designer and find the module for which you are referencing. Find the "Category" object and open it then there should be an option for "Is Auditable". I believe this is where you need to set it to true. At this point it'll show in your Audit Trail for each ticket.