I forgot to ask whether you wanted to change the background of the
entry field or the listbox. For a ttk::combobox, the ttk::style options
generally work on the entry field. But it's a bit confused, as there are
multiple elements in the ttk::combobox (entry field, arrow, listbox).
There is a writeup here:
http://wiki.tcl.tk/37973
that I worked on, on how to change colors for various widgets.
The example below changes all the background colors to cyan:
-background changes the base background, and that will show as the background
of the arrow.
-fieldbackground changes the entry field's background (for state: normal).
The 'ttk::style map' command with -fieldbackground changes the entry field's
background when it is in the readonly state.
And the 'option add' command changes the listbox's background.
package require Tk
ttk::style theme use default
set category_list {student ex_student visitor collaborator}
set category_pt student
ttk::combobox .category \
-textvariable category_pt \
-state readonly \
-values $category_list
ttk::style configure TCombobox -background cyan
ttk::style configure TCombobox -fieldbackground cyan
ttk::style map TCombobox -fieldbackground [list readonly cyan]
option add *TCombobox*Listbox.background cyan
pack .category