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

Cameron Kemp

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 $Y_4$.

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

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 , every modification endomorphic on the zero transformation vanishes if it is made up of the four-term relationators and whiskerings by . 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, . In particular, we make use of Bordemann, Rivezzi and Weigel's pentagon in .
Paper Structure (3 sections, 5 theorems, 119 equations)

This paper contains 3 sections, 5 theorems, 119 equations.

Table of Contents

  1. The fundamental conjecture
  2. Algebraic construction of the hexagonator series
  3. Construction of the pentagonator series

Key Result

Lemma 1.3

Given an infinitesimal 2-braiding $t$ on a symmetric strict monoidal $\mathsf{Ch}^{[-1,0]}$-category $(\mathsf{C},\otimes,I,\gamma)$, we have the following five relations:

Theorems & Definitions (26)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Remark 5
  • Remark 6
  • Definition 7
  • Definition 1.1
  • Example 1.2
  • Lemma 1.3
  • ...and 16 more