A diagrammatic approach to Hopf monads

Given a Hopf algebra in a symmetric monoidal category with duals, the category of modules inherits the structure of a monoidal category with duals. If the notion of algebra is replaced with that of monad on a monoidal category with duals then Bruguieres and Virelizier showed when the category of modules inherits this structure of being monoidal with duals, and this gave rise to what they called a Hopf monad. In this paper it is shown that there are good diagrammatic descriptions of dinatural transformations which allows the three-dimensional, object-free nature of their constructions to become apparent.

💡 Research Summary

The paper “A diagrammatic approach to Hopf monads” revisits the notion of Hopf monads—an abstraction of Hopf algebras from the setting of a symmetric monoidal category with duals to the more general context of monads on such categories. The authors begin by recalling the classical result that, given a Hopf algebra (H) in a symmetric monoidal category (\mathcal{C}) equipped with left and right duals, the category of (H)-modules (\mathcal{C}_H) inherits a monoidal structure and duals, essentially because the antipode provides the necessary compatibility between the tensor product and the duality.

Bruguieres and Virelizier lifted this phenomenon to monads: a monad (T) on a monoidal category with duals is called a Hopf monad if (i) it is a strong monad (the multiplication and unit are monoidal natural transformations), (ii) it is both left‑ and right‑strong (the monoidal structure interacts coherently with the monad’s multiplication), (iii) there exist left and right antipodes—natural transformations (S^l, S^r : T \to T) satisfying axioms that generalise the antipode of a Hopf algebra, and (iv) all these structures are dinatural, i.e. they behave uniformly across all objects. When these conditions hold, the Eilenberg–Moore category (\mathcal{C}^T) becomes a monoidal category with duals, mirroring the classical module case.

The difficulty in the existing literature lies in the abstract nature of the dinatural transformations and antipodes: they are defined by equations that are hard to manipulate concretely, especially when one wants to construct new examples or verify the Hopf monad axioms in practice. The central contribution of this paper is a fully diagrammatic calculus that makes these structures transparent. The authors introduce a three‑dimensional, object‑free graphical language in which objects are not represented by points; instead, 1‑cells (lines) encode morphisms, 2‑cells (surfaces) encode natural transformations, and 3‑cells (volumes) encode higher coherence data.

In this language, the monoidal product appears as parallel juxtaposition of lines, duality as a reversal of line orientation, and the monad’s action as the insertion of a surface between lines. The left and right antipodes are depicted as “switch” surfaces that twist a line in opposite directions; applying a switch twice yields the identity, visually encoding the antipode axioms. Dinaturality becomes the statement that a given surface can be slid across any line without changing the overall diagram, which is precisely the graphical manifestation of the dinatural condition.

A particularly elegant feature is the treatment of coends (or “codegrees”) via a new “codegree diagram”. A coend is represented by a hollow sphere (a 2‑cell with a hole) that can be expanded or contracted, allowing the authors to visualise the universal property of coends as a simple topological move. This representation makes the interaction between the antipodes and the coend transparent: the antipodes act as rotations of the hollow sphere, and the antipode axioms correspond to a full 360° rotation returning the sphere to its original state.

The paper demonstrates the power of this calculus through several examples. First, the classical Hopf algebra (k