I am interested in state sum models and their relations with some other of TQFTs, especially the fully extended TQFTs and the Dijkgraaf-Witten TQFTs (generalized, in the sense that finite-group-bundles are replaced by higher bundles over higher algebraic structures). Forgive about my naiveness, but I immaturely suspect maybe three of them are the same. I hope I will get an answer one day, and this post is my first step.
I don't have much access to the experts of this field, and therefore am not sure how much this has been developed. Perhaps the answers are in written papers. In any case, I think a complete answer is too much to hope. If you have any relevant paper in mind, please point them out with short comments. Thank you so much in advance!
1. Crane-Yetter and Dijkgraaf-Witten
Crane-Yetter theory is a well-known $4$-D state sum model. According to Manuel Bärenz's edit on nLab, it can be realized as a generalized DW theory, based on quantum groups instead of finite groups.
Q1.1 Can you give a formal reference for that statement?
Q1.2 Can (m)any other state sum models be interpreted as a generalized DW theory? We have to be flexible here: fields can be higher bundles.
Q1.3 In contrast, can generalized Dijkgraaf-Witten theories be realized as state sum models?
2. Dijkgraaf-Witten and Fully Extended TQFTs
By Domenico Fiorenza's edit on nLab (sec. 2), Dijkgraaf-Witten models are fully extended.
Q2.1 Is there a formal reference for this statement?
Q2.2 Can fully extended TQFTs be realized as generalized Dijkgraaf-Witten models?
3. Fully Extended TQFTs and State Sum Models
Q3.1 I have been trying to find evidence why CY is fully extended and what the point is associated to, but in vain. The best answer I have heard is that physicists believe state sum models should automatically be fully extended. If this is true, I really want to know how and why it should work.
Q3.2 On the other hand, by cobordism hypothesis (proved by Lurie 2009), any fully extended TQFT is determined by the assignment at the point. I have a feeling that if the target category is "finite" enough, then this might be interpreted as a state sum model. Would you share your understanding? (EDIT: an answer by Kevin Walker suggested that some standard techniques bring you a state sum model from fully extended ones. Another possibly related post can be found here).