  It is well known that central extensions of a group G correspond to
2-cocycles on G. Cocycles can be used to construct extensions of G-graded
algebras via a version of the Drinfeld twist introduced by Majid. We show how
2-cocycles can be defined for an abstract monoidal category C, following
Panaite, Staic and Van Oystaeyen. A braiding on C leads to analogues of Nichols
algebras in C, and we explain how the recent work on twists of Nichols algebras
by Andruskiewitsch, Fantino, Garcia and Vendramin fits in this context.
  Furthermore, we propose an approach to twisting the multiplication in braided
doubles, which are a class of algebras with triangular decomposition over G.
Braided doubles are not G-graded, but may be embedded in a double of a Nichols
algebra, where a twist may be carried out if careful choices are made. This is
a source of new algebras with triangular decomposition. As an example, we show
how to twist the rational Cherednik algebra of the symmetric group by the
cocycle arising from the Schur covering group, obtaining the spin Cherednik
algebra introduced by Wang.
