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  "exportedAt": "2026-11-04T08:43:27.194Z",
  "scope": "Reader-visible project fields and native live notes only; referenced notes in other projects, deleted notes and private AI context are excluded.",
  "submission": "not_submitted",
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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.",
    "history": "",
    "dates": {
      "inspiration": null,
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      "registeredInSumire": "2026-10-07T16:29:39.051Z",
      "updatedInSumire": "2026-10-30T14:35:21.579Z",
      "legacyRecordedAt": "2026-10-07T16:29:39.050Z",
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    "goals": [
      {
        "id": "goal_71391ac0-c899-41e4-b6ed-e9d825d41001",
        "title": "To understand why Nim-sum works",
        "body": "",
        "noteIds": []
      },
      {
        "id": "goal_4d2683f0-360b-4edd-b553-ca70400f6943",
        "title": "Categorical unification of recursive, geometric and logical aspects of combinatorial game theory",
        "body": "",
        "noteIds": []
      }
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  "notes": [
    {
      "id": "n_AOLnldYHQBQI",
      "number": 30,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/30",
      "title": "AI evaluation report (2026-08-23): A double category of games",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "",
      "body": "# a66892-ulp-composition-closure の独立レビュー（insufficient）\n\n> この報告はcommit時点では独立reviewを経ていない。書誌情報と主張の検証状態は`research/runs/`のRunEventを正とし、人間の確認前の内容として読むこと。\n\n主張: 「有限 rooted path ごとの全単射として定義した同期経路一意持上げ射は、恒等射を含み、縦射の合成について閉じる」。\n\n引用元 arXiv:2510.22886v3（Ryuya Hora, \"Games as recursive coalgebras: A categorical view on the Nim-sum\"）を、目次・Introduction・Section 2（games as graphs）・Section 3.3–3.6（recursive coalgebra、game morphism の定義 3.12、Bouton monoid 手前まで）・Section 4（monoidal structures）・Appendix B.3（limits: inverse image, monomorphisms, equalizer）・B.4（comonadicity）まで通読した。\n\n本文に実在する唯一の関連定義は Definition 3.12（game morphism）で、条件は (1) graph morphism 条件と (2) path-lifting 条件（f(x)→y ならば x→x' かつ f(x')=y となる x' が存在する）のみである。これは「存在」条件であって、各 finite rooted path に対する「全単射」や「一意な持上げ」を要求するものではない。また Proposition B.18 は monic な morphism（=injective=subgame inclusion）を特徴づけているが、これも claim の言う「finite rooted path ごとの全単射」とは別の概念である。\n\n本文中に「同期経路」「一意持上げ」（synchronized path / unique lifting as bijection）に相当する語や、「縦射」（vertical morphism）という語は一箇所も見当たらない。つまり claim が前提とする構成そのものが、この一次資料には（少なくとも探索した範囲では）存在しない。\n\nGame が圏をなすこと（Notation 3.13, U: Game→Set が comonadic 等）自体は本文で証明されているので、「恒等射を含み合成に閉じる」という性質一般は圏の公理として当然成り立つ。しかし claim が指す特定の射のクラス（bijection per finite rooted path として定義されるもの）が本文中で定義・命名されていない以上、この claim を本文の定理として支持することも、反証することもできない。\n\n結論: これは反例や矛盾ではなく、引用元に主張・定義そのものが見当たらないケース（citation/definition mismatch）である。ルールに従い refute ではなく insufficient とする。",
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        "registeredInSumire": "2026-10-07T16:37:05.830Z",
        "updatedInSumire": "2026-10-10T08:51:16.248Z",
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      "classificationStatus": "unclassified",
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