  We relate shuffle algebras, as defined by Nichols, Feigin-Odesskii and Rosso,
to perverse sheaves on symmetric products of the complex line (i.e., on the
spaces of monic polynomials stratified by multiplicities of roots). More
precisely, we construct an equivalence between:
  (i) Braided Hopf algebras of a certain type.
  (ii) Factorizable collections of perverse sheaves on all the symmetric
products.
  Under this eqiuvalence, the Nichols algebra associated to an object V
corresponds to the collection of the intersection cohomology extensions of the
local systems on the open configuration spaces associated to the tensor powers
of V. Our approach is based on using real skeleta of complex configuration
spaces.
