Scalar Extension of Bicoalgebroids

Abstract

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 [2] and [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. IntroductionBicoalgebroids were introduced by Brzeziński and Militaru in [2] as the structure that dualizes bialgebroids (in fact, Takeuchi's × R -bialgebras) in the sense of reversing arrows. This notion is not to be confused with the different kinds of bialgebroid-duals that were later introduced in [8]. It would seem that the study of bicoalgebroids hasn't been taken up vigorously since their inception; in our view, they merit attention for at least two reasons. First, it is well established that a bialgebroid may be thought of as a non-commutative analogue of the algebra of functions on a groupoid. It follows that a bicoalgebroid, in turn, should be regarded as a non-commutative analogue of the groupoid itself. This raises the hope that classical constructions on groupoids may find their non-commutative generalizations more easily in the context of bicoalgebroids. Secondly, just as bialgebroids play a fundamental role in depth-two extensions of algebras, it is expected that bicoalgebroids feature prominently in extensions of coalgebras (from a different approach, in [9] Kadison constructs bialgebroids from depth 2 extensions of coalgebras). To complete the picture, the dual Hopf-Galois theory of [16] for extensions of coalgebras should generalize (from bialgebras) to bicoalgebroids, giving a dual version of bialgebroid-Galois theory. Further work in this latter direction is deferred to a subsequent publication.Central to this paper is the introduction of scalar extension for bicoalgebroids. Incidentally, the construction that was shown in [1] to be a non-commutative version of scalar extension was defined (for Hopf-algebras) in [2] -alongside with bicoalgebroids. Bicoalgebroids; comodules and modulesThroughout, k will be a field and the category M = M k of k-modules will serve as our underlying category. The unadorned ⊗ always means ⊗ k .We use the ubiquitous Sweedler notation for coproducts and coactions. For a coalgebra C, ∆, ε , the coproduct ∆ : C → C ⊗ C on elements is denoted ∆(c) = c (1) ⊗ c (2) , with an implicit finite summation understood, ...

Skew-monoidal categories and bialgebroids

“…While the last two express only naturality of ψ the first two contain the monad data T q , µ q , η q . The difference disappears, however, if we introduce the lax monad T as a cosimplicial object ∆ → End M (2) by…”

“…It is easy to verify, using the bialgebroid axioms, that Ab R , * , R R , γ, η, ε is a right-monoidal category. One can notice that the skew-associator γ, which is uniquely determined by γ R,R,R , is, up to isomorphisms R * (R * R) ∼ = H ⊗ (2) of H as a left H-comodule algebra. Therefore the bialgebroid is a Hopf algebroid (or × R -Hopf algebra) in the sense of [23] precisely when the skew-associator γ is invertible.…”

“…Among the various multiple E-objects there are distinguished ones that behave nicely under the * -product. For each n > 0 let M (n) denote the category of (n, n − 1)-type of E-objects in M. (2) . The coproduct M (•) = n>0 M (n) is then closed under * but has no unit object.…”

Weak Hopf monoids in braided monoidal categories

“…A weak bialgebra [Böhm and Szlachónyi 1996;Nill 1998;Szlachányi 1997;Böhm et al 1999] is a generalization of a bialgebra with weakened axioms. These weakened axioms replace the three that follow by requiring that the unit be a coalgebra morphism and the counit be an algebra morphism.…”

“…This section is fairly standard in the ᐂ = Vect case -see for example [Böhm and Szlachányi 2000;Nill 1998;Nikshych and Vainerman 2002] -and carries over rather straightforwardly to the general braided ᐂ case; see [Day et al 2003]. …”