Note on factorization categories

Abstract: We fill in what we consider as a gap in the construction of the factorizable Satake functor in the constructible setting in arXiv:1608.00284 and arXiv:1708.07205. To do so, we summarize four different constructions of factorization sheaves of categories on the Ran space and establish some relations between them. We also generalize the Drinfeld-Plucker formalism from arXiv:1708.07205, arXiv:2310.0638. In addition, we systematically study the commutative factorization categories Fact(C) assuming given an additional grading of C. We apply our results to relate the versions of the Satake functors for the Ran space with that of the configuration space of colored divisors on a curve.