Let $k$ be a field of characteristic $0$, and let $\lambda\in k$. A $\lambda$-Drinfeld associator $\Phi = \Phi(a,b)$ is an element of the free completed associative $k$-algebra $k\langle\langle a,b\rangle\rangle$ on two variables $a,b$ (i.e. a formal power series in the noncommuting variables $a,b$ with coefficients in $k$) satisfying four properties:
- It is grouplike ($\Delta(\Phi) = \Phi\otimes\Phi$) for the coproduct $\Delta$ on $k\langle\langle a,b\rangle\rangle$ for which $a$ and $b$ are primitive elements;
- It is antisymmetric i.e. $\Phi(a,b) = \Phi(b,a)^{-1}$;
- It satisfies the pentagon axiom $$\Phi(t_{12}, t_{23}+t_{24})\Phi(t_{13}+t_{23},t_{34}) = \Phi(t_{23}, t_{34})\Phi(t_{12}+t_{13},t_{24}+t_{34})\Phi(t_{12}, t_{23});$$
- It satisfies the hexagon axioms $$\sigma(t_{13} + t_{23}) = \Phi(t_{13},t_{12})\sigma(t_{13})\Phi(t_{23}, t_{13})\sigma(t_{23}) \Phi(t_{12}, t_{23})$$ $$\sigma(t_{12} + t_{13}) = \Phi(t_{13},t_{23})\sigma(t_{13})\Phi(t_{12}, t_{13})\sigma(t_{12}) \Phi(t_{23}, t_{12}),$$ where $\sigma(\alpha) = \exp(\lambda \alpha/2)$.
The nLab page says that these equations are modelled on the defining conditions for a braided monoidal category, and provides Theorem 2.1 as a way of getting such a category from a lie algebra and a $\lambda$-Drinfeld associator.
I'm wondering about the converse; how did the pentagonal (and hexagonal) equations come about? I assume the pentagon equation arises somehow from MacLane's pentagon diagram describing the associator $$\alpha_{x y z}: (x\otimes y)\otimes z\xrightarrow{\sim}x\otimes (y\otimes z)$$ in a monoidal category, but I don't see how to explicitly obtain such an equation. Is it just in analogy with these diagrams or is there an actual monoidal category whose polygonal diagrams imply those for the Drinfeld associator? Moreover, what is the role of the mysterious $\sigma$ function occuring in the hexagonal equations?