Two-cocycles and cleft extensions in left braided categories

We define two-cocycles and cleft extensions in categories that are not necessarily braided, but where specific objects braid from one direction, like for a Hopf algebra H a Yetter-Drinfeld module braids from the left with H-modules. We will generalize classical results to this context and give some application for the categories of Yetter-Drinfeld modules and H-modules. In particular, we will describe liftings of coradically graded Hopf algebras in the category of Yetter-Drinfeld modules with these techniques.