  We classify braided tensor categories over C of exponential growth which are
quasisymmetric, i.e., the squared braiding is the identity on the product of
any two simple objects. This generalizes the classification results of Deligne
on symmetric categories of exponential growth, and of Drinfeld on
quasitriangular quasi-Hopf algebras. In particular, we classify braided
categories of exponential growth which are unipotent, i.e., those whose only
simple object is the unit object. We also classify fiber functors on such
categories. Finally, using the Etingof-Kazhdan quantization theory of Poisson
algebraic groups, we give a classification of coconnected Hopf algebras, i.e.
of unipotent categories of exponential growth with a fiber functor.
