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.
Categories: math.AG math.CT