TITLE: Glueing triangulated categories
QUESTION [13 upvotes]: Hello!
Given a triangulated category, one can look for semiorthogonal decompositions into (simpler?) triangulated subcategories. 
I'd like to know if there's a way to attack the opposite problem, i.e. to classify the ways two given triangulated categories can be composed to give a big triangulated category which decomposes semiorthogonally into the given ones. Does anybody know about this? 
I hope this question isn't too vague.
Thank you!

REPLY [3 votes]: There is an article by Kuznetsov and Lunts which lifts the glueing of triangulated categories to differential graded categories: http://arxiv.org/abs/1212.6170.