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  "exportedAt": "2026-11-04T08:44:03.910Z",
  "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.",
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      "registeredInSumire": "2026-10-07T16:29:39.051Z",
      "updatedInSumire": "2026-10-30T14:35:21.579Z",
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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_sh7k6jLmnuKk",
      "number": 18,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/18",
      "title": "AI evaluation report (2026-08-16): A double category of games",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "",
      "body": "# claim a66892-state-objects の独立反対検討\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: 「double category の縦圏(vertical category)と横bicategory(horizontal bicategory)は共通の対象族を持つため、recursive-coalgebraic games と Joyal games の候補構成には対象同定または共通部分への制限が必要」。\n\n1) 一般命題の検証: nLab の double category の項目、Wikipedia の Double category 項目、および Verity double bicategory の定義を確認したところ、double category（および double bicategory）は定義上、vertical 側の圏と horizontal 側の(bi)category が同一の対象(0-cell)族を持つ構造であることが明記されている。この点は標準的でtrivialに近い定義的事実であり、真であると独立に確認できた。\n\n2) 一次資料としての引用の検証(ここが問題):\n  - evidence_refs の一つ http://www.tac.mta.ca/tac/volumes/12/8/12-08.pdf (証明書エラーのため https://tac.mta.ca/... で再取得) は、実際には Michael Barr『On Subgroups of the Lambek Pregroup』(TAC vol.12 no.8, 2004, pp.262-269) であり、pregroup(半順序モノイド)に関する論文で、double category にも games にも一切言及がない。**この引用は主張とは無関係の誤った一次資料である。**\n  - https://arxiv.org/abs/2510.22886 (『Games as recursive coalgebras』) は recursive coalgebra を用いてゲームを圏 Game として再定義する論文で、圏 Game が局所有限表示可能・対称モノイダル閉・Set上comonadic等の性質を持つことを示すが、要旨レベルでは double category・bicategory・Joyal games への言及は確認できなかった(全文までは検証していない)。\n  - https://people.cs.uchicago.edu/~brady/CSPP50102/notes/Joyal-games.pdf は証明書エラーで直接取得できなかったが、Web検索により該当ノートの内容として Joyal の games の圏では対象=games、射=strategies であることが裏付けられた。これは recursive-coalgebraic games 側の対象(coalgebra的なgame表現)と一致するとは限らず、対象クラスが異なりうるという主張の前提は支持的だが、直接この一次資料内で「対象同定または制限が必要」と明言されているかは未確認。\n\n結論: 一般的な圏論的事実(1)は真であり独立に確認できたが、主張を支える3件の evidence_refs のうち1件は完全に無関係な論文であり、残り2件も「候補構成に対象同定または制限が必要」という具体的帰結を直接述べているとは確認できなかった。従って証拠の連鎖として不十分。",
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      "classificationStatus": "unclassified",
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      "aiClaimModel": "",
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        "use": "assisted",
        "tools": "Codex",
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        {
          "url": "https://arxiv.org/abs/2510.22886",
          "verification": "not_checked",
          "service": "arxiv",
          "identifier": "arXiv:2510.22886",
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