Hypercubes, $n$-groupoids, and mixtures

TL;DR

This paper explores the mathematical structure of composite mixtures using $n$-groupoids, analyzing hypercubes and their skeletons to understand the conditions for material uniformity and conservativity.

Contribution

It introduces a novel framework linking $n$-groupoids with hypercubes to model composite mixtures and identifies conditions for conservativity and uniformity.

Findings

Double groupoids with commuting squares lead to conservative $n$-groupoids.

Material uniformity corresponds to transitivity of the $n$-groupoid core.

Hypercube analysis provides insights into the structure of composite mixtures.

Abstract

The theory of composite mixtures consisting of $n$ constituents is framed within the schema provided by the notion of $n$-groupoid. The point of departure is the analysis of $n$-dimensional hypercubes and their skeletons, to each of whose edges an element (an arrow) of one of $n$ given material groupoids is assigned according to the coordinate class to which it belongs. In this way a $GL(3,{\mathbb R})$-weighted digraph is obtained. It is shown that if the double groupoid associated with each pair of constituents consists of commuting squares, the resulting $n$-groupoid is conservative. The core of this $n$-groupoid is transitive if, and only if, the mixture is materially uniform.