{
  "abstract": "Let $G$ be a group and let $n\\in\\mathbb N$. We consider one-dimensional cellular automata of length $n$ over the alphabet $G$, with periodic boundary conditions and with local rules $ϕ$ given by words. We construct a functor $F$ from $\\mathbf{Grp}$, the category of groups (or any similar algebraic structure), to the category of pointed digraphs, and show its faithfulness. We introduce the image category $\\mathcal C_{n,ϕ}$ of state transition graphs of all such group-alphabet cellular automata, and show $F$ is an equivalence onto this category. Consequently we investigate the translation of categorical structures in $\\mathbf{Grp}$ in terms of automata, and give examples of how standard methods of homological algebra may now be applied to automata. Additionally, a graph-theoretical construction of the product in $\\mathcal C_{n,ϕ}$ is given, together with a formula for cycle lengths and multiplicities in this product. When $G$ is abelian, further structural properties of state transition graphs follow, and we contrast these results with non-abelian counterexamples.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41013",
  "author_total": 2,
  "authors": [
    {
      "name": "François Huard"
    },
    {
      "name": "Nathan Dion Larivière"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dga3sfd6nv6fn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41013",
  "fulltext_url": null,
  "id": "dga3sfd6nv6fn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41013",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.FL"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.CT"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.GR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41013",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dga3sfd6nv6fn.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41013",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41013"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Functorial Analysis of Group-Alphabet Cellular Automata",
  "title_text": "Functorial Analysis of Group-Alphabet Cellular Automata",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/dga3sfd6nv6fn",
  "venue": "arXiv",
  "versions": []
}