Truncated factorized perverse sheaves on Sym(C)

TL;DR

This paper studies categories of factorized perverse sheaves on symmetric powers of the complex line, introducing truncated versions and analyzing their relationships and limits, with a focus on the case d=1.

Contribution

It defines truncated categories of factorized perverse sheaves on Sym(C), establishes their inverse limit structure, and proves equivalences in the case d=1.

Findings

The categories form inverse systems with compatible limits.

The restriction functor is faithful and compatible with limits.

For d=1, the categories are equivalent to the braided monoidal category V.

Abstract

Kapranov and Schechtman defined the category FP of factorized perverse sheaves on Sym(C) smooth along the stratification given by multiplicities and with values in a braided monoidal category V. We define for each d\in N the category FP^{\leq d} of factorized perverse sheaves on the disjoint union of Sym^n(C) for n\leq d and the category FP_{\leq d} of factorized perverse sheaves on the open subset of Sym(C) consisting of multi-sets with multiplicities bounded by d. We show that the families (FP^{\leq d})_{d in N} and (FP_{\leq d})_{d in N} fit into systems of categories whose inverse limit is FP, and that for each d the natural restriction functor from FP_{\leq d} to FP^{\leq d} is faithful and compatible with taking the limit. For d=1 we prove that the natural restriction functor is an equivalence and that FP^{\leq 1} and FP_{\leq 1} are equivalent to V.