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  "exportedAt": "2026-11-04T08:46:42.910Z",
  "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"
    },
    "origin": "",
    "source": "",
    "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": "",
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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_isd0nWfTcj_2",
      "number": 26,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/26",
      "title": "AI evaluation report (2026-08-21): A double category of games",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "",
      "body": "# Independent review: 'finite-path strategy pushforward preserves interaction/hiding/identity' (a66892)\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対象論文 arXiv:2510.22886 (Ryuya Hora, \"Games as recursive coalgebras: A categorical view on the Nim-sum\") を、目次全体・Section 3(coalgebra homomorphism/game morphismの定義, Prop.3.10, Thm.3.14, Def.3.12, Cor.3.25, Prop.3.26)・Section 5(結び・未解決問題)を実際に読んで検証した。\n\n結果: 主張にある『interaction』『hiding』『identity strategy』という用語も、それらの保存に関する定理も、本文中に一切存在しない。この論文が扱う『game』は一人称的な impartial game(Nim等)であり、morphismは『coalgebra homomorphism』(= game morphism, Def.3.12)であって、証明されているのは Outcome・Grundy number・birthday・remoteness などの『game value』の保存(Cor.3.25, Prop.3.26)のみである。\n\n『interaction』『hiding』という語が唯一登場するのは Section 5.4.1, Question 5.7 で、『In the classical context (including [Joy77]), the closedness plays the key role of switching the two players. It makes their category of games compact closed...』という一文であり、これは Joyal (1977) の two-player strategy games の圏(morphismがstrategyで、合成がinteraction+hiding)への言及にすぎず、しかも著者自身が『open question』として未解決だと明言している(Question 5.7: 'What is the internal hom with respect to the Conway addition?')。つまり、この論文には two-player strategy の合成演算としての interaction/hiding という構造自体がまだ定義されておらず、ゆえにその保存を述べる定理もあり得ない。\n\nレビュー対象の主張は、(1) この論文の『finite rooted path 上で全単射を誘導する coalgebra 準同型』(= game morphismの特殊ケース)という枠組みと、(2) Joyal流のtwo-player game semantics における strategy の合成(interaction, hiding)・恒等 strategy、という無関係な2つの形式体系を混在させて構成された命題であり、どちらの evidence_ref にも証明はおろか定義さえ存在しない。evidence_type が 'proof_draft' とされているが、該当する証明はarXiv本文のどこにも見つからなかった。\n\n第二の evidence_ref(Joyalのゲーム講義ノートPDF, people.cs.uchicago.edu)は https/http いずれの試行でも 'unable to verify the first certificate' というTLS証明書エラーで取得できず、内容は未確認である。ただし主張の主節はarXiv論文側の枠組みに対する主張であるため、この未確認は結論を左右しない。",
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        "registeredInSumire": "2026-10-07T16:36:19.153Z",
        "updatedInSumire": "2026-10-10T08:51:05.371Z",
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      "language": null,
      "projectLanguage": "ja",
      "languageSource": "not_recorded",
      "legacyStage": "note",
      "classification": "unclassified",
      "classificationStatus": "unclassified",
      "importance": "unspecified",
      "section": "unassigned",
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        "use": "assisted",
        "tools": "Codex",
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