Equivariant Eilenberg-Watts theorem for module coalgebras

Abstract

For coalgebras $C$ and $D$, Takeuchi proved that the category of linear functors from ${\mathfrak{M}}^C$ to ${\mathfrak{M}}^D$ preserving small coproducts is equivalent to the category of $C$-$D$-bicomodules, where ${\mathfrak{M}}^C$ for a coalgebra $C$ means the category of right $C$-comodules. We formulate and prove an equivariant version of this result for module coalgebras over a bialgebra. As an application, for a bialgebra $H$, we establish an equivalence of the 2-category of a particular class of module categories over the monoidal category ${\mathfrak{M}}^H$ and the 2-category of a particular class of module categories over the monoidal category ${}_H\mathfrak{M}$ of left $H$-modules.

22 pages