  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.
