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 \(\mathfrak {B}\) governing \(\mathbb {Z}_{\ge 0}\) -graded bialgebras in braided categories. As a category, \(\mathfrak {B}\) splits into blocks \(\mathfrak {B}_n\) according to the grading. We relate \(\mathfrak {B}_n\) with the category \(\mathfrak {P}_n\) of perverse sheaves on the symmetric product \({\operatorname {Sym}}^n(\mathbb {C})\) smooth with respect to the natural stratification by multiplicities. More precisely, we show that \(\mathfrak {P}_n\) is equivalent to the category of functors \(\mathfrak {B}_n\rightarrow \operatorname {Vect}\) . This gives a natural quiver description of \(\mathfrak {P}_n\) .
