Scalar extension of bicoalgebroids

After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter–Drinfel’d modules over a bicoalgebroid. It is proved that the Yetter–Drinfel’d category is monoidal and pre–braided just as in the case of bialgebroids, and is embedded into the one–sided center of the comodule category. We proceed to define Braided Cocommutative Coalgebras (BCC) over a bicoalgebroid, and dualize the scalar extension construction of Brzezinski and Militaru [2] and Balint and Slachanyi [1], originally applied to bialgebras and bialgebroids, to bicoalgebroids. A few classical examples of this construction are given. Identifying the comodule category over a bicoalgebroid with the category of coalgebras of the associated comonad, we obtain a comonadic (weakened) version of Schauenburg’s theorem. Finally, we take a look at the scalar extension and braided cocommutative coalgebras from a (co–)monadic point of view.

💡 Research Summary

The paper begins by revisiting the notion of a bicoalgebroid, which can be regarded as the coalgebraic dual of a bialgebroid. A bicoalgebroid consists of two base coalgebras (C) and (D) together with a total coalgebra (\mathcal{H}) equipped with a coproduct (\Delta:\mathcal{H}\to\mathcal{H}\otimes_{C}\mathcal{H}) and counit (\epsilon:\mathcal{H}\to C) that are compatible with the base coalgebra structures via appropriate left and right coactions. After fixing this definition, the authors introduce left‑ and right‑(\mathcal{H})-comodules as objects carrying coactions that respect the base coalgebra, and analogously define left‑ and right‑(\mathcal{H})-modules via actions.

A central technical achievement is the construction of a monoidal structure on the category (\mathcal{M}^{\mathcal{H}}) of (\mathcal{H})-comodules. By using the relative tensor product (\otimes_{C}) over the base coalgebra, the authors show that the coaction on a tensor product of two comodules is well defined and that the unit object is precisely the base coalgebra (C). The associativity constraints follow from the coassociativity of (\Delta).

Next, Yetter‑Drinfel’d modules over a bicoalgebroid are defined. An object that is simultaneously an (\mathcal{H})-comodule ((M,\rho)) and an (\mathcal{H})-module ((M,\triangleright)) is required to satisfy the compatibility condition
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