The monoidal Eilenberg–Moore construction and bialgebroids

By Korn'el Szlach'anyi. 2002

46 citations recorded by the source corpus.

Abstract Strong monoidal functors U : C → M with left adjoints determine, in a universal way, monoids T in the category of opmonoidal endofunctors on M . Treating such opmonoidal monads as abstract “quantum groupoids” we derive Tannaka duality between right adjoint strong monoidal functors and opmonoidal monads. Bialgebroids, i.e., Takeuchi's ×R-bialgebras, appear as the special case when T has also a right adjoint. Street's 2-category of monads then leads to a natural definition of the 2-category of bialgebroids.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/06aaee243a7266a80b05d492ed02ee95b27b9c03

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Citations in the corpus

Quantum categories, star autonomy, and quantum groupoids (2003, methodology): “[Tak] (and developed by [Lu], [Xu], [Sch2], [BM] and [ Szl ]) we shall be more explicit about”
Hopf algebroids with bijective antipodes: Axioms, integrals, and duals (2003, background): “In [22] left bialgebroids over the base L have been characterized as opmono monadsT on the categoryLML such that their underlying functors T have right adjoints The endofunctor T is given by tensoring over L = L ⊗ Lop with theL-L-bimoduleA.”
Skew monoidales, skew warpings and quantum categories (2012, unknown): “It was shown in [27] that the bialgebroids (as considered by [29, 20, 30, 15]) with base ring R amount to cocontinuous opmonoidal monads on the category of two-sided R-modules.”
Skew-monoidal categories and bialgebroids (2012, background): “In order to relate this situation to that of earlier sections one may think E as the category of left E-objects in AbR, i.e., E is the bimodule category RAbR with ⊗ the tensor product ⊗ R .”
Hopf Algebroids (2008, background): “a Hopf monad in the terminology of [49] (called a bimonad in [68]).”
Hopf monads on monoidal categories (2010, methodology): “In this section, we study bialgebroids which, according to Szlachányi [21], are linear bimonads on categories of bimodules admitting a right adjoint.”
Skew-closed categories (2012, background): “He showed that bialgebroids H with base ring R (a concept from [15] and studied under this name in [13]) could be characterized in terms of skew-monoidal structures on the category of one-sided R-modules for which the lax unit was R itself.”