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

We consider the space Z ( C , L ) Z(\mathbb {C},L) of 0 0 -cycles on the complex line C \mathbb {C} with coefficients in a commutative monoid L L subject to certain conditions. Such spaces include the symmetric products (for L = Z + L=\mathbb {Z}_{+} ) and the Ran space (for L = T = { T r u e , F a l s e } L=\mathbb {T}=\{ {\mathtt {True}}, {\mathtt {False}} \} being the Boolean algebra of truth values). We describe the appropriately defined category of perverse sheaves on Z ( C , L ) Z(\mathbb {C},L) in terms of the braided category (PROB) generated by the components of the universal L 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 L . The matrices in question are analogs of contingency tables familiar in statistics.