PROBs and perverse sheaves II. Ran spaces and 0-cycles with coefficients

TL;DR

This paper explores the structure of perverse sheaves on spaces of 0-cycles with coefficients in a monoid, generalizing symmetric products and Ran space, and introduces new categorical descriptions involving braided categories and Janus sheaves.

Contribution

It provides a novel categorical framework for perverse sheaves on 0-cycle spaces with monoid coefficients, including descriptions via braided categories and Janus sheaves.

Findings

Describes perverse sheaves on Z(C,L) in terms of braided categories generated by universal L-graded bialgebras.

Introduces Janus sheaves as objects with mixed functoriality on matrix categories with entries in L.

Connects the structure of these sheaves to contingency table analogs in statistics.

Abstract

We consider the space Z(C,L) of 0-cycles on the complex line C with coefficients in a commutative monoid L subject to certain conditions. Such spaces include the symmetric products (for L=Z_+) and the Ran space (for L=T={ True, False} being the Boolean algebra of truth values). We describe the appropriately defined category of perverse sheaves on Z(C,L) in terms of the braided category (PROB) generated by the components of the universal $L$-graded bialgebra. We give another description in terms of so-called Janus sheaves which are objects of mixed functoriality (data covariant in one direction and contravariant in the other) on a category formed by certain matrices with entries in L. The matrices in question are analogs of contingency tables familiar in statistics.