Hypercubes, $n$-groupoids, and mixtures
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913503512625152 |
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| author | Epstein, Marcelo |
| author_facet | Epstein, Marcelo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10730 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hypercubes, $n$-groupoids, and mixtures Epstein, Marcelo Category Theory Mathematical Physics 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. |
| title | Hypercubes, $n$-groupoids, and mixtures |
| topic | Category Theory Mathematical Physics |
| url | https://arxiv.org/abs/2409.10730 |