Cartier integration of infinitesimal 2-braidings via 2-holonomy of the CMKZ 2-connection, II: The pentagonator

TL;DR

This paper explores the structure of braided monoidal 2-categories, proposing that a certain Lie 2-algebra has trivial cohomology, and constructs the pentagonator using a 2-connection over the configuration space of four particles.

Contribution

It demonstrates that the Drinfeld-Kohno Lie 2-algebra has trivial cohomology, simplifying the axioms for braided monoidal 2-categories, and constructs the pentagonator via a 2-connection in this context.

Findings

The Drinfeld-Kohno Lie 2-algebra has trivial cohomology.

Constructed the pentagonator using the CMKZ 2-connection over Y4.

Modified the understanding of coherence in braided monoidal 2-categories.

Abstract

This is a continuation of the previous paper (arXiv:2508.01944) in this series. We recontextualise Cirio and Martins' work to motivate our fundamental conjecture that the Drinfeld-Kohno (Lie) 2-algebra has trivial cohomology. It is then shown that this conjecture implies the following: given a coherent totally symmetric infinitesimal 2-braiding $t$, every modification endomorphic on the zero transformation vanishes if it is made up of the four-term relationators and whiskerings by $t$. The power of such an implication is that, in our context, one need only construct the data of a braided monoidal 2-category and it will automatically satisfy the axioms. We thus conclude by constructing the pentagonator via Cirio and Martins' Knizhnik-Zamolodchikov 2-connection over the configuration space of 4 distinguishable particles on the complex line, $Y_4$. In particular, we make use of Bordemann, Rivezzi and Weigel's pentagon in $Y_4$.