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

TL;DR

This paper explores the relationship between 2-shifted Poisson structures and braided monoidal 2-categories, demonstrating how certain shifted Poisson structures induce syllepses and satisfy coherence conditions.

Contribution

It establishes that 2-shifted Poisson structures induce totally symmetric infinitesimal 2-braidings and relate coherency to the Maurer-Cartan equation, extending the understanding of syllepses in higher categories.

Findings

Hexagonators satisfy braided monoidal cochain 2-category axioms under symmetry and coherence.

Infinitesimal 2-braidings from 2-shifted Poisson structures are totally symmetric.

3-shifted Poisson structures induce syllepses.

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.