[Paper Review] A diagrammatic approach to Hopf monads

Simon Willerton|arXiv (Cornell University)|Jul 3, 2008

Homotopy and Cohomology in Algebraic Topology7 references3 citations

TL;DR

This paper introduces a diagrammatic approach to Hopf monads using three-dimensional string diagrams to provide an object-free, visual formalism for monoidal categories with duals. It demonstrates that the composition of a strong monoidal functor and its left adjoint yields a Hopf monad, with the antipode structure verified via diagrammatic manipulation, offering a clearer, more intuitive framework for understanding Hopf monad constructions in higher category theory.

ABSTRACT

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.

Motivation & Objective

• To provide a diagrammatic, object-free formulation of Hopf monads in monoidal categories with duals.
• To extend string diagrammatics to include dinatural transformations, enabling three-dimensional reasoning.
• To clarify the construction of Hopf monads arising from strong monoidal functors with left adjoints.
• To offer a visual and conceptual framework that makes the three-dimensional nature of Hopf monad structures more apparent.
• To re-derive and simplify results from Bruguières and Virelizier using diagrammatic techniques.

Proposed method

• Using three-dimensional string diagrams to represent functors, natural transformations, and dinatural transformations in monoidal 2-categories.
• Formulating evaluation and coevaluation as dinatural transformations to eliminate reliance on objects.
• Applying diagrammatic reasoning to verify the antipode condition for bimonads.
• Utilizing the adjunction between a strong monoidal functor and its left adjoint to construct a Hopf monad.
• Verifying the Hopf monad axioms (HM1 and HM2) through diagrammatic isotopy and coherence.
• Leveraging the fact that strong monoidal functors commute with duality structures to define the antipode.

Experimental results

Research questions

• RQ1 How can dinatural transformations in monoidal 2-categories be effectively represented and manipulated using diagrammatic calculus?
• RQ2 In what way does the composition of a strong monoidal functor and its left adjoint give rise to a Hopf monad?
• RQ3 How can the antipode of a bimonad be constructed and verified diagrammatically?
• RQ4 What is the role of three-dimensionality in capturing the coherence of Hopf monad structures?
• RQ5 Can the abstract constructions of Bruguières and Virelizier be simplified and made more transparent through diagrammatic reasoning?

Key findings

• The natural transformation S defined as a composition of adjunctions and duals acts as a left antipode for the bimonad U∘F.
• The diagrammatic proof of the antipode axioms (HM1 and HM2) is verified through isotopy and coherence, showing that the Hopf monad condition holds.
• The construction of a Hopf monad from a strong monoidal functor with a left adjoint is made explicit and diagrammatically transparent.
• The use of three-dimensional string diagrams allows for a fully object-free treatment of duality and monoidal structures.
• The framework successfully captures the three-dimensional nature of dinatural transformations and their interactions in monoidal 2-categories.
• The method simplifies and reinterprets prior results from Bruguières and Virelizier using visual reasoning, enhancing clarity and intuition.