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Syllepses from 3-shifted Poisson structures and second-order integration of infinitesimal 2-braidings

Cameron James Deverall Kemp

TL;DR

The paper advances higher-categorical deformation quantisation by constructing and analyzing second-order structures in braided strictly-unital monoidal cochain 2-categories, guided by shifted Poisson geometry. It establishes that second-order hexagonators arise as obstructions unless a strict infinitesimal 2-braiding is totally symmetric and coherent, and it demonstrates that 2-shifted Poisson structures yield such coherent data, while 3-shifted and coboundary 2-shifted Poisson structures give rise to syllepses. The work provides a concrete framework for second-order deformation quantisation, including explicit 4-term relationators and infinitesimal hexagonators, and shows that the pentagonator remains trivial at this order, with coherence encoded by the Breen polytope. These results illuminate the link between shifted Poisson theory and higher categorical deformation quantisation, offering a path toward a KZ-type 2-connection and potential applications in derived algebraic geometry. The findings lay groundwork for extending to higher orders and exploring the holonomy of 2-connections in this 2-categorical setting.

Abstract

This paper follows on from ``Infinitesimal 2-braidings from 2-shifted Poisson structures". It is demonstrated that the hexagonators appearing at second order satisfy the requisite axioms of a braided monoidal cochain 2-category provided that the strict infinitesimal 2-braiding is totally symmetric and coherent (in Cirio and Faria Martins' sense). We show that those infinitesimal 2-braidings induced by 2-shifted Poisson structures are indeed totally symmetric and we relate coherency to the third-weight component of the Maurer-Cartan equation that a 2-shifted Poisson structure must satisfy. Furthermore, we show that 3-shifted Poisson structures and ``coboundary" 2-shifted Poisson structures induce syllepses.

Syllepses from 3-shifted Poisson structures and second-order integration of infinitesimal 2-braidings

TL;DR

The paper advances higher-categorical deformation quantisation by constructing and analyzing second-order structures in braided strictly-unital monoidal cochain 2-categories, guided by shifted Poisson geometry. It establishes that second-order hexagonators arise as obstructions unless a strict infinitesimal 2-braiding is totally symmetric and coherent, and it demonstrates that 2-shifted Poisson structures yield such coherent data, while 3-shifted and coboundary 2-shifted Poisson structures give rise to syllepses. The work provides a concrete framework for second-order deformation quantisation, including explicit 4-term relationators and infinitesimal hexagonators, and shows that the pentagonator remains trivial at this order, with coherence encoded by the Breen polytope. These results illuminate the link between shifted Poisson theory and higher categorical deformation quantisation, offering a path toward a KZ-type 2-connection and potential applications in derived algebraic geometry. The findings lay groundwork for extending to higher orders and exploring the holonomy of 2-connections in this 2-categorical setting.

Abstract

This paper follows on from ``Infinitesimal 2-braidings from 2-shifted Poisson structures". It is demonstrated that the hexagonators appearing at second order satisfy the requisite axioms of a braided monoidal cochain 2-category provided that the strict infinitesimal 2-braiding is totally symmetric and coherent (in Cirio and Faria Martins' sense). We show that those infinitesimal 2-braidings induced by 2-shifted Poisson structures are indeed totally symmetric and we relate coherency to the third-weight component of the Maurer-Cartan equation that a 2-shifted Poisson structure must satisfy. Furthermore, we show that 3-shifted Poisson structures and ``coboundary" 2-shifted Poisson structures induce syllepses.
Paper Structure (30 sections, 16 theorems, 188 equations)

This paper contains 30 sections, 16 theorems, 188 equations.

Table of Contents

  1. Conventions and notations
  2. Introduction
  3. Summary
  4. Setup
  5. The closed symmetric strict monoidal tricategory of cochain 2-categories
  6. Pseudonatural transformations, modifications and their compositions
  7. The closed symmetric strict monoidal structure
  8. Braided strictly-unital monoidal cochain 2-categories
  9. The associator and pentagonator
  10. The braiding and hexagonators
  11. Syllepses
  12. Symmetric syllepses on braided strictly-unital monoidal cochain 2-categories
  13. Infinitesimal syllepses on symmetric strict monoidal cochain 2-categories
  14. Syllepses from 3-shifted Poisson structures
  15. Infinitesimal 2-braidings and coboundary infinitesimal syllepses
  16. ...and 15 more sections

Key Result

Lemma 4.4

If an infinitesimal 2-braiding $\otimes\xRightarrow{t}\otimes$ is symmetric then t_U(VW)$\iff$t_(UV)W.

Theorems & Definitions (82)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.6
  • Remark 2.7
  • Remark 2.9
  • Remark 2.10
  • Remark 2.12
  • Definition 2.15
  • ...and 72 more