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  "exportedAt": "2026-11-04T08:28:34.444Z",
  "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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      "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": [
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        "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_LL8DLDfUFKqK",
      "number": 28,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/28",
      "title": "AI evaluation report (2026-08-22): A double category of games",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "",
      "body": "# claim a66892-ulp-finite-strategy-preservation の独立レビュー\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: 各長さの有限 rooted path 上で全単射を誘導する coalgebra 準同型について、有限経路 strategy の直像は interaction、hiding、および恒等 strategy を保存する。\n\n挙げられた2つの一次資料を会話履歴に頼らず独立に検証した。\n\n(1) arXiv:2510.22886 “Games as recursive coalgebras: A categorical view on the Nim-sum” (v3)。abstractページとHTML full-text版(https://arxiv.org/html/2510.22886v3)の両方をfetchした。本論文はimpartial gameとNim-sumをrecursive coalgebraの枠組みで圏論的に再構成する内容であり、coalgebra homomorphism(定義3.2)・game homomorphism(定義3.12)は登場するが、主張の核である 'strategy'・'interaction'・'hiding' という語・概念は本文中に一度も現れない。PDF直接fetchはメタデータのみでテキスト層抽出に失敗したが、abstract/HTML版の一致する内容から、この論文が主張中の保存定理を述べていないことは確認できる。\n\n(2) people.cs.uchicago.edu の Joyal-games.pdf。異なるURLバリアント(https, http)で複数回fetchを試みたが、いずれも 'unable to verify the first certificate' というTLS証明書検証エラーで内容取得に失敗した。archive.orgのキャッシュもこのharnessでは取得不可(ツール側の制限)だった。したがってこの資料の内容(strategy/interaction/hidingの定義や、該当する保存定理の有無)は確認できていない。\n\n結論: 挙げられた一次資料のうち一方は主張と無関係な内容(該当概念を含まない)であり、他方はアクセス不能で確認できなかった。反例の構成にも、否定の証明にも至っていないため、この主張を積極的に誤りと断定する根拠(refute)はない。一方で主張を裏付ける一次資料での確認もできていないため、supportとも言えない。よって insufficient と判定する。",
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        "registeredInSumire": "2026-10-07T16:36:37.634Z",
        "updatedInSumire": "2026-10-10T08:51:07.204Z",
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          "url": "https://arxiv.org/html/2510.22886v3",
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