Scalar extension of bicoalgebroids

TL;DR

This paper extends the theory of bicoalgebroids by defining modules, comodules, and Yetter--Drinfel'd modules, establishing their monoidal and pre-braided structures, and exploring scalar extensions and braided cocommutative coalgebras.

Contribution

It introduces a scalar extension construction for bicoalgebroids and analyzes their module and comodule categories using (co-)monadic frameworks.

Findings

Yetter--Drinfel'd category is monoidal and pre-braided

Scalar extension construction is generalized to bicoalgebroids

A weakened version of Schauenburg's theorem is obtained

Abstract

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.