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Truncated factorized perverse sheaves on Sym(C)

Giovanna Carnovale, Francesco Esposito, Lleonard Rubio y Degrassi

TL;DR

The paper develops a comprehensive framework for truncated factorized perverse sheaves on Sym(C) with values in a braided category V, introducing d-truncated categories FP_{≤d} and FP^{≤d} and proving that these assemble into inverse-limit systems whose limit recovers the original FP. It establishes faithfulness of restriction maps and, in the pivotal d=1 case, an equivalence FP_{≤1} ≅ FP^{≤1} ≅ V, linking geometry on configuration/open strata to the ambient braided category. Building on Deligne’s operadic perspective and the little 2-cubes operad, the authors construct outer tensor products, monodromy, and factorization data, and show how restriction/extension functors interplay with these structures. The work lays the groundwork for a later equivalence with truncated bialgebra categories CERyD-A and CERyD-S, enabling a coherent algebraic-geometric bridge for d-th approximations of Nichols algebras and related Hopf-algebraic structures. Overall, the paper provides a robust categorical and geometric apparatus for studying truncated factorization on configuration/open strata and their algebraic manifestations.

Abstract

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.

Truncated factorized perverse sheaves on Sym(C)

TL;DR

The paper develops a comprehensive framework for truncated factorized perverse sheaves on Sym(C) with values in a braided category V, introducing d-truncated categories FP_{≤d} and FP^{≤d} and proving that these assemble into inverse-limit systems whose limit recovers the original FP. It establishes faithfulness of restriction maps and, in the pivotal d=1 case, an equivalence FP_{≤1} ≅ FP^{≤1} ≅ V, linking geometry on configuration/open strata to the ambient braided category. Building on Deligne’s operadic perspective and the little 2-cubes operad, the authors construct outer tensor products, monodromy, and factorization data, and show how restriction/extension functors interplay with these structures. The work lays the groundwork for a later equivalence with truncated bialgebra categories CERyD-A and CERyD-S, enabling a coherent algebraic-geometric bridge for d-th approximations of Nichols algebras and related Hopf-algebraic structures. Overall, the paper provides a robust categorical and geometric apparatus for studying truncated factorization on configuration/open strata and their algebraic manifestations.

Abstract

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.

Paper Structure

This paper contains 23 sections, 26 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Braided monoidal categories
  4. Operads
  5. Little $2$-cubes operad
  6. Deligne's interpretation of braided monoidal categories
  7. Inverse limits of categories
  8. Tensor products and braidings for perverse sheaves
  9. The outer tensor product $\boxtimes$
  10. Some equivalences
  11. Perverse sheaves on Sym(C)
  12. Basic geometric definitions
  13. Relevant categories and functors
  14. Restriction and extension
  15. The tensor product $\boxtimes_{\varphi}$
  16. ...and 8 more sections

Key Result

Theorem 2.11

(Deligne) Let $\mathcal{C}$ be a category. A (strict) braided category structure on $\mathcal{C}$ is equivalent to a structure of $\Pi_1 E_2$-algebra on $\mathcal{C}$ in the category of categories.

Theorems & Definitions (80)

  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 70 more