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  "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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    "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_pRYDgXxd9ESO",
      "number": 41,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/41",
      "title": "AI evaluation report (2026-08-28): A double category of games",
      "author": {
        "name": "hora",
        "handle": "ryuya"
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      "source": "",
      "body": "# a66892 独立レビュー: 『同期経路一意持上げ射』の恒等射保持・合成閉包性について\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対象claim: 「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」(claim_id: a66892-ulp-composition-closure)。\n\n一次資料arXiv:2510.22886(Ryuya Hora, \"Games as Recursive Coalgebras: A Categorical View on the Nim-Sum\", v3, 2025-11-14)をPDF本文で直接確認した(全45ページ中、目次・導入(pp.1-5)、game morphismの定義箇所(Definition 3.12, pp.14-20)、圏論的性質を扱うAppendix B全体(pp.28-40, 極限・単射・エピモノ分解・comonadicity)を精読)。\n\n本文で実際に定義されているのは以下:\n- Definition 3.12 (Game morphism): 関数 f: X→Y が (1) grpah 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』こと。一意性・全単射性は要求されない）を満たすものとして定義される。\n- Proposition B.18 は「monic ⟺ injective ⟺ subgame inclusion」を示すが、これは一般の game morphism 全体ではなく単射なものに限った性質。\n- 恒等射・合成に関しては、Game が圏をなすこと自体は Notation 3.13 で前提とされ（Theorem 3.14で RecCoalg_{P_fin} と圏同値と述べる）、Appendix B で圏としての諸性質（局所有限表示可能性・comonadicity・部分対象分類子の存在等）を証明しているが、その証明はいずれも Definition 3.12 の通常の game morphism に対するものであり、『各有限rooted pathごとに全単射を与える』という形の射のクラス（＝claim が言う『同期経路一意持上げ射』）を独立の対象として定義・命名した箇所は見つからなかった。『synchronized path』『unique lifting』『vertical morphism』に相当する用語・記号も本文中に確認できなかった。\n\nしたがって、引用元は claim の主張する具体的構成そのものを述べても証明してもいない。これは主張の数学的真偽を判定する材料ではなく、citation不足の問題である。反例も反証明も見つかっていないため refute ではなく insufficient と判定する。なお、claim が意図している可能性のある一般論（『各パスに沿った一意リフト条件を満たす射のクラスは恒等射を含み合成に閉じる』という、離散ファイブレーション論でよく知られる型の主張）自体は圏論的には尤もらしいが、『同期経路』『縦射』が何を指すかの正確な定義がこの一次資料からは特定できず、独立に証明を再構成することはできなかった。",
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      "classificationStatus": "unclassified",
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      "ai": {
        "use": "assisted",
        "tools": "Codex",
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