Braided categories of bimodules from stated skein TQFTs

For each braided category \mathcal{C} , we show that, under mild hypotheses, there is an associated category of “half braided algebras” and their bimodules internal to \mathcal{C} which is not only monoidal but even braided and balanced. We use this in the case where \mathcal{C} is the category of modules over a ribbon Hopf algebra to interpret stated skeins as a TQFT, namely, a braided balanced functor from a category of cobordisms to this category of algebras and their bimodules. Although our construction works in full generality, we relate in the special case of finite-dimensional ribbon factorizable Hopf algebras the stated skein functor to the Kerler–Lyubashenko TQFT by interpreting the former as the “endomorphisms” of the latter.