Authors: M. D. Sheppeard
Important polytope sequences, like the associahedra and permutohedra, contain one object in each dimension. The more particles labelling the leaves of a tree, the higher the dimension required to compute physical amplitudes, where we speak of an abstract categorical dimension. Yet for most purposes, we care only about low dimensional arrows, particularly associators and braiding arrows. In the emerging theory of motivic quantum gravity, the structure of three dimensions explains why we perceive three dimensions. The Leech lattice is a simple consequence of quantum mechanics. Higher dimensional data, like the e8 lattice, is encoded in three dimensions. Here we give a very elementary overview of key data from an axiomatic perspective, focusing on the permutoassociahedra of Kapranov.
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