Categories of representations of coalgebroids (Q1588380)

The paper generalizes the Tannaka-Krein duality for coalgebroids. A coalgebroid over a monoidal category \(\mathcal V\) is an object of the 2-category \({\mathcal V}^{\text{op}}\)-Cat of categories enriched in \({\mathcal V}^{\text{op}}\). A coalgebroid can also be described as a family of objects from a monoidal category \(\mathcal V\), double-indexed by a set \(I\), subject to certain coalgebra-like axioms. The 2-category of Comod\((C)\) of \(C\)-comodules is defined to be the 2-category of profunctors from \(I\) -- the unit \({\mathcal V}^{\text{op}}\)-category -- to \(C\). Its objects can be described as a family of objects from \(\mathcal V\), indexed by the set \(I\), subject to certain comodule-like axioms. It is shown that a coalgebroid \(C\) can be reconstructed from \(\text{Comod}(C)\) and the forgetful bi-functor to the 2-category \(\mathcal V\text{-Act}//\mathcal V\), (which plays the analogous role of the category of vector spaces in the classical Tannaka-Krein duality). Further, a characterization for an object of \(\mathcal V\text{-Act}//\mathcal V\) to be equivalent to \(\text{Comod}(C)\) for a certain coalgebroid \(C\), is given. Illustrating examples are given, which where mainly taken from a work of \textit{B. Pareigis} [J. Algebra 70, 356-374 (1981; Zbl 0463.18003)].