Graphical Methods for Tannaka Duality of Weak Bialgebras and Weak Hopf Algebras in Arbitrary Braided Monoidal Categories

Tannaka Duality describes the relationship between algebraic objects in a given category and their representations; an important case is that of Hopf algebras and their categories of representations; these have strong monoidal forgetful “fibre functors” to the category of vector spaces. We simultaneously generalize the theory of Tannaka duality in two ways: first, we replace Hopf algebras with weak Hopf algebras and strong monoidal functors with separable Frobenius monoidal functors; second, we replace the category of vector spaces with an arbitrary braided monoidal category. To accomplish this goal, we introduce a new graphical notation for functors between monoidal categories, using string diagrams with coloured regions. Not only does this notation extend our capacity to give simple proofs of complicated calculations, it makes plain some of the connections between Frobenius monoidal or separable Frobenius monoidal functors and the topology of the axioms defining certain algebraic structures. Finally, having generalized Tannaka to an arbitrary base category, we briefly discuss the functoriality of the construction as this base is varied.

💡 Research Summary

The paper presents a comprehensive generalization of Tannaka duality by simultaneously extending two fundamental aspects of the classical theory. First, it replaces ordinary Hopf algebras with weak Hopf algebras, which relax the strict unit and counit conditions of a bialgebra. Second, it replaces strong monoidal fibre functors (the usual “forgetful” functors to Vectₖ) with separable Frobenius monoidal functors, and it allows the base category to be any braided monoidal category V rather than just the category of vector spaces.

To manage the added complexity, the authors introduce a novel graphical calculus that augments traditional string diagrams with coloured regions representing functorial domains. In this notation, the monoidal structure maps ϕ: F(x)⊗F(y)→F(x⊗y) and the comonoidal maps ψ: F(x⊗y)→F(x)⊗F(y) appear as internal “splittings” and “mergings” of the coloured region, making naturality, associativity, and unit axioms visually evident as continuous deformations of the diagram. This visual language clarifies the Frobenius equations (ϕ∘ψ = id, ψ∘ϕ = id) and the separability condition (ϕ₀ and ψ₀ are mutual inverses), and it provides an intuitive way to verify the weak unit and weak counit axioms that involve the braiding.

The paper begins with a careful recollection of the algebraic structures involved. In a braided monoidal category V, a weak bialgebra H carries an algebra (μ, η) and a coalgebra (Δ, ε) that satisfy the usual bialgebra compatibility together with weakened unit and counit axioms. Four canonical idempotents (s, t, r, z) arise from the weak structure; their Karoubi‑splittings give rise to an internal separable Frobenius algebra inside H. A weak Hopf algebra is then a weak bialgebra equipped with an antipode S satisfying equations that involve these idempotents, ensuring that S behaves as a weak convolution inverse of the identity.

With the graphical toolkit in place, the authors define a separable Frobenius monoidal functor F: A→V as a functor equipped with both monoidal and comonoidal structure maps satisfying the Frobenius and separability equations. They prove that every strong monoidal functor is automatically separable Frobenius, and that such functors preserve duals, a crucial property for the reconstruction process.

The central construction—the Tannaka reconstruction—takes a separable Frobenius functor F and forms the coend
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