Note on factorization categories
arXiv:2404.11561
The paper develops a systematic theory of commutative factorization categories over the Ran space, clarifies constructions of factorizable Satake functors, extends the Drinfeld‑Plücker formalism, and studies graded versions of these categories.
Abstract
We systematically study the commutative factorization categories over the Ran space. We fill in what we consider as a gap in the construction of the factorizable Satake functor in the constructible setting in arXiv:1708.07205, arXiv:1608.00284. To do so, we summarize four different constructions of commutative factorization sheaves of categories on Ran and establish some relations between them. We also generalize the Drinfeld-Plücker formalism from arXiv:1708.07205, arXiv:2310.0638. In addition, we study the commutative factorization categories Fact(C) with 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. This paper is a companion of arXiv:2508.01527.
152 pages, v.14: a mistake in Corollary 2.7.7 is corrected, an appendix on relative rigidity is added