The permutoassociahedron, Mac Lane's coherence theorem and asymptotic zones for the KZ equation

AbstractWe construct a combinatorial CW-complex KPn whose vertices correspond to all possible bracketings of all possible permutations of n letters A1,…, An. This structure is implicitly present in Mac Lane's coherence theorem for symmetric and braided monoidal categories. It also fits very naturally in the framework of the study of Knizhnik-Zamolodchikov (KZ) equations initiated by V. Drinfel'd. We show that KPn is a combinatorial (n − 1)-ball and establish its connection with the Grothendieck-Knudsen moduli space of stable n-pointed curves of genus 0