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).**

anyreasonable target. $\endgroup$ – Arun Debray Apr 20 at 17:42