Let k be a field, let k* = k \ {0} and let C2 be a cyclic group of order 2. We compute all of the braided monoidal structures on the category of k-vector spaces graded by the Klein group C2 × C2. For the monoidal structures we compute the explicit form of the 3-cocycles on C2 × C2 with coefficients in k*, while, for the braided monoidal structures, we compute the explicit form of the abelian 3-cocycles on C2 × C2 with coefficients in k*. In particular, this will allow us to produce examples of quasi-Hopf algebras and weak braided Hopf algebras with underlying vector space k[C2 × C2].
