Hello to all

I still don't understand the use of properties. My problem is quite
easy, I think, but I can't figure it out in the documentation.

I have my "areas of responsibility" on the first level, the second
level represents projects, which are tagged with :PROJECT:. The third level
represents either the list of next action items (tagged with contexts)
or -- if the project is lager -- some milestones are interpolated:

* Teaching
#+CATEGORY: Teaching
** Sanskrit I (WS08)                        :PROJECT:
#+CATEGORY: Sanskrit I
*** NEXT Look for Introductions             :LIBRARY:
*** NEXT Borrow the chosen books            :LIBRARY:
*** NEXT Ask students for convenient date   :EMAIL:
** History of Indian Religion (WS08)        :PROJECT:
#CATEGORY: Indian Religion
*** Plan part on Buddhism                   :MILESTONE:
**** NEXT Look for Books                    :LIBRARY:
**** NEXT Chose readings for students       :OFFICE:
*** Plan part on Hinduism                   :MILESTONE:
**** NEXT Look for books                    :LIBRARY:
**** NEXT Invite guest-speaker              :PHONE:
* Misc Office
#+CATEGORY: Misc Office
** Proofread Sociology-Article              :PROJECT:
#+CATEGORY: Sociology-Article
*** NEXT Make a printout                    :COMPUTER:
** Install Emacs on every colleague's PC    :PROJECT:
#+CATEGORY: Bring joy to the world

This works quite good. I can produce agenda-views that show, for
example, each next action with LIBRARY context, while the #+CATEGORY
property yields the associated project's name:

,----
| Sanskrit:        NEXT Look for books        :LIBRARY:
| Sanskrit:        NEXT Borrow chosen books   :LIBRARY:
| Indian Religion: NEXT Look for books        :LIBRARY:
`----

THE PROBLEM:

When viewing projects with C-c C-a P or stuck projects with C-c C-a #
the first column is wrong, because the #+CATEGORY property has been
changed by a preceding project. For example, the project "Install
Emacs on every colleague's PC" is stuck, because it doesn't contain
any TODOs (you see how urgent it is to help me). So, I would like it
to appear in the stuck-projects-view or in the general projects' view
as:

,----
| Misc Office:  Install Emacs on every colleague's PC  :PROJECT:
`----

Instead it reads:

,----
| Sociology-Article: Install Emacs on every colleague's PC :PROJECT:
`----

I've tried to mark my projects with :CATEGORY: since the documentation
says:

,----
| If you would like to have a special CATEGORY for a single entry or a
| (sub)tree, give the entry a :CATEGORY: property with the location as
| the value.
`----

But the problem remains. 

What have I misunderstood? In other words, how can I set the category
for my projects (2nd level with sub-trees) while keeping the category
of the "areas of responsibility" (1st level) intact over the whole
tree?

Thanks for help

Sven