We refer to [JS97] for the notion of subfactor.  Yasuo Watatani proved the following result [W96, Theorem 2.2]:  

*Theorem*: Let $M$ be a type $\mathrm{II}_1$ factor and $N$ an irreducible subfactor of $M$ (i.e. $N' \cap M = \mathbb{C}$). If the Jones index $[M:N]$ is finite, then the intermediate
subfactor lattice $\mathcal{Lat}(N \subset M)$ is a finite set.  

We refer to [EGNO15] for the notion of tensor category. We are interested in extending Watatani's theorem to tensor categories. According to Dave Penneys (private communication), it reformulates in this framework as follows:  

> A connected unitary Frobenius algebra in a unitary tensor category has
> finitely many unitary Frobenius subalgebras.

**Question**: How to prove it directly in the tensor category framework? Is the unitary assumption required?  

A stronger version of Watatani's theorem was proved in [BDLR19], providing a bound for the cardinal of the lattice. Its proof involves a new notion of angle between two intermediate subfactors and Pimsner-Popa basis. Note that the arXiv version of [BDLR19] also contains a purely planar algebraic (short) proof only involving the notion of angle between two biprojections. Thus, a generalization of the notion of angle to between two Frobenius subalgebras can help answer the above question.
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*References*  


[BDLR19] Bakshi, Keshab Chandra; Das, Sayan; Liu, Zhengwei; Ren, Yunxiang. An angle between intermediate subfactors and its rigidity. Trans. Amer. Math. Soc. 371 (2019), no. 8, 5973--5991; arXiv:1710.00285.   
[EGNO15] Etingof, Pavel; Gelaki, Shlomo; Nikshych, Dmitri; Ostrik, Victor. Tensor categories. Mathematical Surveys and Monographs, 205. American Mathematical Society, Providence, RI, 2015. xvi+343 pp.    
[JS97] Jones, V.; Sunder, V. S. Introduction to subfactors. London Mathematical Society Lecture Note Series, 234. Cambridge University Press, Cambridge, 1997. xii+162 pp.  
[W96] Watatani, Yasuo. Lattices of intermediate subfactors. J. Funct. Anal. 140 (1996), no. 2, 312--334.