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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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        "title": "To understand why Nim-sum works",
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      {
        "id": "goal_4d2683f0-360b-4edd-b553-ca70400f6943",
        "title": "Categorical unification of recursive, geometric and logical aspects of combinatorial game theory",
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    {
      "id": "n_FfIj28O6bieP",
      "number": 9,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/9",
      "title": "double categoriesと組合せゲーム",
      "author": {
        "name": "hora",
        "handle": "ryuya"
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      "body": "gameをdouble categoryとして扱う既存研究には少なくとも次の二つがある。\n\n- Jules Hedges, *Morphisms of Open Games*  \n  lensをvertical directionに用い、equilibriumを扱えるdouble categoryを与える。\n\n- Neil Ghani, Clemens Kupke, Alasdair Lambert, Fredrik Nordvall Forsberg, *A compositional treatment of iterated open games*  \n  coalgebraを使ってiterated open gamesを扱い、repeated prisoner’s dilemmaを合成的に記述する。\n\n一方で、open games in economicsとcombinatorial gamesは、現在の研究コミュニティではかなり別の対象として扱われている。この分断は、洞の *Games as recursive coalgebras* とJoyal strategiesをdouble-categoricalに結ぶ研究にとって明確な空白地帯である。\n\n## Idea\n\n組合せゲームに特化したdouble category / virtual equipmentを作る。\n\n- objects: recursive coalgebraとしてのgame frames\n- vertical arrows: option-preservingかつoption-liftingなcoalgebraic game morphisms\n- horizontal arrows: Joyal型のstrategies\n- 2-cells: coalgebraic mapsに沿ったstrategy simulations\n- companions/conjoints: copycat-through-$f$\n- horizontal composition: swivel-chair / interaction-and-hiding\n\n既存のopen gamesのdouble categoryをそのまま特殊化するのではなく、\n\n\\[\n\\text{coalgebraic combinatorial game structure}\n\\quad+\\quad\n\\text{Joyal strategy semantics}\n\\]\n\nを直接統合する。\n\nこれはeconomic game theoryのopen gamesと、Conway–Joyal型combinatorial game theoryの間を埋める構成になる可能性がある。\n\n## Relation to P-positions and game semantics\n\n*The game semantics of game theory*（arXiv:1904.11287）では、combinatorial gameのP-positionを、複数のplayerが協力して外界と戦うゲームにおけるNash equilibriumとして理解する議論があるらしい。\n\nこの見方が正しければ、\n\n\\[\n\\text{P-position}\n\\]\n\nは単なるrecursive outcome classではなく、\n\n\\[\n\\text{internal coalition vs. environment}\n\\]\n\nというopen-game的意味論を持つ。\n\n調べるべき問い：\n\n- P-positionをcoalgebraic invariantとして記述できるか。\n- そのP-position conditionは、strategy equipment内のequilibrium-like universal propertyになるか。\n- Bouton’s theoremは、そのequilibrium predicateをNim-sumによって計算するintegration theoremと解釈できるか。\n- Joyalのwinning strategyとopen gamesのbest-response/equilibriumを同一のdouble-categorical構造で比較できるか。\n\n## Evolutionary games and compositionality\n\n単にすべてをlambda termとして符号化しても、それぞれの相互作用のinterfaceやcomposition lawは説明されない。\n\nしかしDawkins的な観点では、economic agents、organisms、genes、memesは同じ世界の中で相互作用している。したがって必要なのは単一のuntyped syntaxではなく、異なるgame semanticsを接続できるheterogeneous compositional frameworkである可能性が高い。\n\n候補：\n\n- virtual double categories / equipments\n- decorated or indexed open games\n- coalgebraic game frames with varying strategy doctrines\n- fibrations of game semantics\n- double categories whose objects encode interfaces and whose horizontal arrows encode heterogeneous games\n- cybernetic systems combining biological fitness, economic utility, and informational replication\n\n## Goal\n\n第一段階では、combinatorial gamesに対するvirtual equipmentを厳密に構成し、既存のopen gamesのdouble categoriesとの比較定理を与える。\n\n第二段階では、P-position、Nash equilibrium、coalgebraic recursion、Bouton’s theoremを同じcompositional languageで関係づける。\n\n最終的には、economic gamesとbiological gamesを同じsyntaxへ潰すのではなく、異なる意味論を保ったまま合成できる「進化論の構造的モデル」へ接続する。\n\n## David Jaz Myersのcategorical systems theoryとの関係\n\nDavid Jaz Myersのbook draft *Categorical Systems Theory* を参照する。Julesのdouble-categorical approachは、Myersのcategorical systems theoryにおけるdouble-categorical approachと関係しているらしい。\n\nしたがって、この草稿は単なる一般的なsystems theoryの背景ではなく、Julesのopen gamesにおけるdouble-categorical構成が属する系譜と、組合せゲーム側で構成したいdouble categoryあるいはvirtual double categoryを比較するための直接的な参照点として読むべきである。ただし、両者の正確な技術的対応関係は未確認であり、今後、用いるlens、interface、horizontal/vertical morphism、compositionの対応を文献上で検証する必要がある。",
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