Zoran Skoda bicategory of cleft extensions

Description

$H$ is a fixed Hopf algebra over a fixed ground ring $k$.

Objects

Pairs $(E,\gamma)$ of a right $H$-comodule algebra and a convolution invertible map of $H$-comodules $\gamma:H\to E$.

1-cells

A 1-cell from $(E,\gamma)$ to $(E',\gamma')$ is a $E$-$E'$-bimodule in the category of $H$-comodules.

2-cells

A 2-cell is a morphism $D$ in the category ${}_E\mathcal{M}^H_{E'}$.

Vertical composition

Composition in the category ${}_E\mathcal{M}^H_{E'}$.

Horizontal composition

Tensor product over the middle $H$-comodule algebra.

Base category (for coinvariants)

Objects

An algebra $U$ with $H$-measuring $\triangleright$ and (normalized) cocycle $\sigma:H\otimes H\to U$.

Cocycle means:

Morphisms

Bimodules with (left) bimodule measuring compatible with the cocycle.

2-cells

Morphisms of bimodules commuting with bimodule measuring.

$\phi: C\to C'$ of $U$-$U'$-bimodules.

Equivalence of bicategories

$(E,\gamma)\mapsto (E^{co H},\triangleright_\gamma)$, $h\triangleright u = \gamma(h_{(1)})u\gamma^{-1}(h_{(2)})$.

${}_U D_V\mapsto {}_U D^{co H}_V$

$h\triangleright c = \gamma_U(h_{(1)})u\gamma^{-1}_V (h_{(2)})$

2-cell to restriction.

$\sigma(h,k) = \gamma(h_{(1)})\gamma(k_{(1)})\gamma^{-1}(h_{(2)}k_{(2)})$.

Other direction:

Tensoring with $H$ for objects.

On 1-cells: $C\otimes H$ has $U\sharp H$-$U'\sharp H$-bimodule structure

for the case of trivial cocycle for $U'$

Discuss the cocycle for the bimodule measuring. Then instead of expression like $(h_{(1)}\triangleright c)\otimes h_{(2)}g$ we need $(h_{(1)}\triangleright c)\sigma'(h_{(2)},g_{(1)})\otimes h_{(3)}g_{(2)}$

Need a compatibility condition here (to have a bimodule!).

Left action only:

Right action only: