Extend the Hopf-ideal correspondence to ind-completions

Extend Theorem CSA-Hopf-ideal-bijection, after suitable modifications, to the case where the braided tensor category is an ind-completion of a braided tensor category, thereby recovering Takeuchi’s correspondence when the category is the category of vector spaces.

Background

The paper proves an order-preserving bijection between Hopf ideals of a Hopf algebra in a braided finite tensor category and normal left coideal subalgebras. The proof relies essentially on finiteness, including the finite-tensor-category Galois correspondence and related categorical dimension arguments.

The authors explicitly ask whether, after suitable modifications, this result can be extended beyond the finite setting to ind-completions of braided tensor categories. Such an extension would include Takeuchi’s classical correspondence for ordinary Hopf algebras as a special case.