PROBs and perverse sheaves I. Symmetric products

Algebraic structures involving both multiplications and comultiplications (such as, e.g., bialgebras or Hopf algebras) can be encoded using PROPs (categories with PROducts and Permutations) of Adams and MacLane. To encode such structures on objects of a braided monoidal category, we need PROBs (braided analogs of PROPs). Colored PROBs correspond to multi-sorted structures. In particular, we have a colored PROB B governing non-negatively graded bialgebras in braided categories. As a category, B splits into blocks B_n according to the grading. We relate B_n with the category P_n of perverse sheaves on the n-th symmetric product of the complex line, smooth with respect to the natural stratification by multiplicities. More precisely, we show that P_n is equivalent to the category of functors from B_n to vector spaces. This gives a natural quiver description of P_n.