Truncated factorized perverse sheaves on Sym(C)

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 U_{n\leq d}Sym^{n}(C) 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 prove that the natural restriction functor from FP_{\leq d} to \FP^{\leq d} is an equivalence for any d in N, and that FP^{\leq 1} and \FP_{\leq 1} are equivalent to V. We show that the full direct image *, the extension by zero ! and the intermediate extension !* induce functors from FP_{\leq d} to \FP. In addition, we show that the families (FP^{\leq d})_{d in N} and (\FP_{\leq d})_{d in N} fit into systems of categories, compatible with restrictions and extensions, whose inverse limit is FP.