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Balanced coalgebroids. (English) Zbl 0960.18005

This paper sets up the higher-categorical context for the transfer of structure back and forth between coalgebras and their categories of representations. A coalgebra is a monoid in the opposite \({\mathcal V}^{\text{op}}\) of a symmetric monoidal category \({\mathcal V}\) such as vector spaces. Noting that a monoid is a one-object category, the author looks at coalgebroids, which are categories enriched in \({\mathcal V}^{\text{op}}\) rather than mere coalgebras. The category \({\mathcal C}omod(C)\) of representations of such a coalgebroid \(C\) is considered along with its action by \({\mathcal V}\) (making it a so-called \({\mathcal V}\)-actegory) and with its evaluation \({\mathcal V}\)-actegory morphisms \({\mathcal C}omod(C) \to{\mathcal V}\), one for each object of \(C\). This makes \({\mathcal C}omod(C)\) an object of the 2-category \({\mathcal V}\text{-Act}//{\mathcal V}\).
A major achievement of the paper is a symmetric monoidal structure on the 2-category \({\mathcal V}\text{-Act}\); tensor product over \({\mathcal V}\) requires a descent construction (rather than a mere coequalizer); the unit for the tensor product is \({\mathcal V}\).
The structure transfer alluded to above is achieved by studying properties of the 2-functor \[ {\mathcal C}omod: {\mathcal V}^{\text{op}}-{\mathcal C}at^{\text{op}} \to{\mathcal V}\text{-Act}//{\mathcal V}. \] It is symmetric monoidal with a left biadjoint. The author defines the notion of monoidal bifull faithfulness and shows that \({\mathcal C}omod\) has the property. It is this property that determines the bijection between structures such as pseudomultiplications, braidings, symmetries, and balancings on \(C\) with the corresponding structures on \({\mathcal C}omod(C)\).
There are appendices giving full definitions of braided and sylleptic monoidal bicategories.

MSC:

18D20 Enriched categories (over closed or monoidal categories)
16W50 Graded rings and modules (associative rings and algebras)
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
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