Hopf monoidal comonads

Alain Bruguieres, in his talk [1], announced his work [2] with Alexis Virelizier and the second author which dealt with lifting closed structure on a monoidal category to the category of Eilenberg-Moore algebras for an opmonoidal monad. Our purpose here is to generalize that work to the context internal to an autonomous monoidal bicategory. The result then applies to quantum categories and bialgebroids.

💡 Research Summary

The paper “Hopf monoidal comonads” extends the previously known result on lifting closed (i.e., internal‑hom) structure from a monoidal category to the Eilenberg–Moore category of an op‑monoidal monad, by moving the entire discussion into the setting of an autonomous monoidal bicategory (a 2‑category equipped with a monoidal structure in which every object has left and right duals).
The authors begin by recalling the work of Bruguieres and Virelizier, who showed that if a monad (T) on a monoidal category (\mathcal{C}) is equipped with a strong op‑monoidal structure—meaning that the multiplication and unit of (T) are compatible with the tensor product in a way that respects the internal‑hom functors—then the closed structure on (\mathcal{C}) lifts to the category (\mathcal{C}^T) of (T)‑algebras. This result relies on the existence of natural transformations that intertwine the monad’s structure with the internal homs.
The present work generalizes this picture to a bicategorical context. The ambient setting is an autonomous monoidal bicategory (\mathcal{B}). In such a bicategory one has objects, 1‑cells (the morphisms equipped with a monoidal product), and 2‑cells (natural transformations). The authors define what it means for a 1‑cell (T\colon X\to X) to be an op‑monoidal monad in this higher‑dimensional environment and introduce the notion of a “Hopf monoidal comonad”. The Hopf condition is a set of coherence equations at the level of 2‑cells that guarantee the comultiplication (\delta) and counit (\varepsilon) of the comonad interact with the left and right internal‑hom functors (