PROBs and perverse sheaves I. Symmetric products

TL;DR

This paper develops a categorical framework using PROBs to encode algebraic structures like bialgebras within braided monoidal categories, and relates these structures to perverse sheaves on symmetric products of the complex line.

Contribution

It introduces a colored PROB B for non-negatively graded bialgebras in braided categories and establishes an equivalence between categories of perverse sheaves and functor categories from B_n.

Findings

P_n is equivalent to functor categories from B_n

Provides a quiver description of P_n

Connects algebraic structures with geometric sheaf categories

Abstract

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.