The tangent vector space of a space of monoidal structures on a
functor will be presented as a second cohomology of a (deformation)
complex. In a similar fashion  the tangent space of a space of
monoidal structures on a category can be identified with a third
cohomology. We will explain how the first obstruction to extending a
tangent monoidal structure to an honest one corresponds to a
Gerstenhaber (e_1-) type bracket in the first case and by an
e_2-bracket in the second. Some examples and applications will also be
discussed.