we will be covering some parts of chapter VIII on braided bialgebras, and tit -bits of chapter XIII on braidings for module categories.
in general:
recall that if B is a bialgebra, then the category Mod(B) is monoidal in such a way that the forgetful functor U: Mod(B) –→ Vect preserves just about everything. We shall show that the converse is also true. that is, if Mod(B) is monoidal in such a way that the forgetful functor preserves just about everything, then B is a bialgebra, and the monoidal structure is induced by the coalgebra structure on B. the proof will involve many lovely string diagrams.
we will then define the notions of quasicommutative bialgebras, and braided bialgebras. we shall show that if B is a braided bialgebra, then Mod(B) is a braided monoidal category in such a way that the forgetful functor preserves just about everything. We remark that the converse is also true. that is, if Mod(B) is braided monoidal in such a way that the forgetful functor preserves just about everything, then B is a braided bialgebra, and the monoidal structure is induced by the braiding structure on B. the proof may be found in ross streets lecture notes on quantum groups.
while the above description was a story about monoids, comonoids etc in the base category Vect, we shall prove the results are still true with Vect replaced by any symmetric monoidal category.