  This article is devoted to the investigation of the deformation (twisting) of
monoidal structures, such as the associativity constraint of the monoidal
category and the monoidal structure of monoidal functor. The sets of twistings
have a (non-abelian) c ohomological nature. Using this fact the maps from the
sets of twistings to some cohomology groups (Hochschild cohomology of K-theory)
are constructed. The examples of monoidal categories of bimodules over some
algebra, modules and comodules over bialgeb ra are examined. We specially
concentrate on the case of free tensor category.
