Description
is a fixed Hopf algebra over a fixed ground ring .
Objects
Pairs of a right -comodule algebra and a convolution invertible map of -comodules .
1-cells
A 1-cell from to is a --bimodule in the category of -comodules.
2-cells
A 2-cell is a morphism in the category .
Vertical composition
Composition in the category .
Horizontal composition
Tensor product over the middle -comodule algebra.
Base category (for coinvariants)
Objects
An algebra with -measuring and (normalized) cocycle .
Cocycle means:
Morphisms
Bimodules with (left) bimodule measuring compatible with the cocycle.
2-cells
Morphisms of bimodules commuting with bimodule measuring.
of --bimodules.
Equivalence of bicategories
, .
2-cell to restriction.
.
Other direction:
Tensoring with for objects.
On 1-cells: has --bimodule structure
for the case of trivial cocycle for
Discuss the cocycle for the bimodule measuring. Then instead of expression like we need
Need a compatibility condition here (to have a bimodule!).
Left action only:
Right action only: