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  "exportedAt": "2026-11-04T08:44:11.739Z",
  "scope": "Reader-visible project fields and native live notes only; referenced notes in other projects, deleted notes and private AI context are excluded.",
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    "id": "2610.00003",
    "url": "https://sumire.aiformath.jp/p/2610.00003",
    "title": "Games as recursive coalgebras",
    "tagline": "Recursive coalgebras, game values, and the categorical structure of combinatorial games.",
    "visibility": "public",
    "owner": {
      "name": "hora",
      "handle": "ryuya"
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    "goal": "Explain how recursive game positions, game values, and operations such as Conway addition fit together, and identify what this explains about the Nim-sum.",
    "narrative": "The starting point is [Games as recursive coalgebras: A categorical view on the Nim-sum](https://arxiv.org/abs/2510.22886v3), available in [Main](https://sumire.aiformath.jp/p/2610.00003?view=main). The paper develops impartial games as recursive finite-powerset coalgebras, game values as recursive evaluations, and Bouton monoids as universal factorizations of game values. Conway addition is symmetric monoidal closed.\n\nThe next question is how much more explicit this framework can become. [The internal hom](https://sumire.aiformath.jp/p/2610.00003/n/4) has a proposed concrete cofree-subgame construction and proof in [note 29](https://sumire.aiformath.jp/p/2610.00003/n/29); its relationship to playing strategies remains separate. [Differential structures](https://sumire.aiformath.jp/p/2610.00003/n/6) asks for a common explanation of the Leibniz-type move formula and the Rota–Baxter-type law, with an explanation of Nim-sum as the target.\n\nTwo extensions have different starting points. [Partisan games](https://sumire.aiformath.jp/p/2610.00003/n/3) already recover short game forms, but values and strategies need additional structure. [A double category of games](https://sumire.aiformath.jp/p/2610.00003/n/1) still requires a precise strategy class and compatible squares; counterexamples for arbitrary relations do not settle that question.\n\nA settled boundary is that [Game itself is not a topos](https://sumire.aiformath.jp/p/2610.00003/n/16), by Remark B.27 of the paper. Questions about a topos completion are therefore distinct from that answered question.\n\nTo contribute, choose one of these linked questions and work against its stated remaining claim. For the internal hom, compute a small explicit example or clarify its game-theoretic meaning. For the double category, specify legal strategies and test compatibility with coalgebra maps before claiming a general construction or obstruction.",
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      "registeredInSumire": "2026-10-07T16:29:39.051Z",
      "updatedInSumire": "2026-10-30T14:35:21.579Z",
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        "title": "To understand why Nim-sum works",
        "body": "",
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        "title": "Categorical unification of recursive, geometric and logical aspects of combinatorial game theory",
        "body": "",
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    {
      "id": "n_H9eUCiXj1oc2",
      "number": 20,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/20",
      "title": "Partisan forms: what the coalgebra construction establishes",
      "author": {
        "name": "hora",
        "handle": "ryuya"
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      "body": "A coalgebra for \\(\\mathcal P_{\\mathrm{fin}}\\times\\mathcal P_{\\mathrm{fin}}\\) assigns Left and Right option sets to each position. In the recursive case, the terminal object recovers short partisan game forms. The relevant published statement is Proposition 5.4 in Section 5.2 of [Games as recursive coalgebras](https://arxiv.org/abs/2510.22886v3).\n\nThe four forms\n\\[\n0=(\\varnothing,\\varnothing),\\quad\n1=(\\{0\\},\\varnothing),\\quad\n-1=(\\varnothing,\\{0\\}),\\quad\n*=(\\{0\\},\\{0\\})\n\\]\nare distinct. This distinction is about game forms, not an identification of coalgebra maps with legal playing strategies.\n\nSections 3.5–3.6 of that paper concern game values and the terminal impartial game; they are not a source for a partisan Joyal duality or tensor construction.\n\nSum and negation can be defined recursively on partisan forms, as recorded in [note 3](https://sumire.aiformath.jp/p/2610.00003/n/3). Establishing a strategy category, closed structure, or double category requires the additional definitions and compatibility proofs stated there and in [note 1](https://sumire.aiformath.jp/p/2610.00003/n/1).",
      "dates": {
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      "projectLanguage": "ja",
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      "resolution": "[[2610.00003/3]] に統合する：論文 v3 Proposition 5.4 により終 recursive \\(\\mathcal P_{\\mathrm{fin}}\\times\\mathcal P_{\\mathrm{fin}}\\)-coalgebra は short partisan game forms の集合で，\\(0,1,-1,*\\) は形として相異なり，和と符号反転は左右の option の再帰で定まる．strategy の圏・閉構造・double category にはそれぞれ別の定義と証明が要る，という内容は同ノートの本文と同じである．",
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      "references": [
        {
          "url": "https://arxiv.org/abs/2510.22886v3",
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          "service": "arxiv",
          "identifier": "arXiv:2510.22886v3",
          "version": "v3",
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