# Balanced Coalgebroids

A balanced coalgebroid is a ${\cal V}^{op}$-category with extra structure ensuring that its category of representations is a balanced monoidal category. We show, in a sense to be made precise, that a balanced structure on a coalgebroid may be reconstructed from the corresponding structure on its category of representations. This includes the reconstruction of dual quasi-bialgebras, quasi-triangular dual quasi-bialgebras, and balanced quasi-triangular dual quasi-bialgebras; the latter of which is a quantum group when equipped with a compatible antipode. As an application we construct a balanced coalgebroid whose category of representations is equivalent to the symmetric monoidal category of chain complexes. The appendix provides the definitions of a braided monoidal bicategory and sylleptic monoidal bicategory.

Keywords: Symmetric monoidal bicategories, balanced Vop-categories, coalgebras, quantum groups.

2000 MSC: 18D20, 18D05, 16W50, 81R50.

Theory and Applications of Categories, Vol. 7, 2000, No. 6, pp 71-147.

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