Truncated factorized perverse sheaves on Sym(C)

Kapranov and Schechtman defined the category FP of factorized perverse sheaves on Sym© 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© for n\leq d and the category FP_{\leq d} of factorized perverse sheaves on the open subset of Sym© 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.