{
  "format": "sumire-research-record",
  "version": 1,
  "exportedAt": "2026-11-04T08:43:27.192Z",
  "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",
  "mathematicalCorrespondence": "not_automatically_checked",
  "project": {
    "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": "",
    "dates": {
      "inspiration": null,
      "sourceRecorded": null,
      "evidence": null,
      "registeredInSumire": "2026-10-07T16:29:39.051Z",
      "updatedInSumire": "2026-10-30T14:35:21.579Z",
      "legacyRecordedAt": "2026-10-07T16:29:39.050Z",
      "legacyUpdatedAt": "2026-10-30T14:35:21.579Z"
    },
    "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_Jr8xtd9AJPkw",
      "number": 37,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/37",
      "title": "Reflected Day convolution",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "",
      "body": "## Result\n\n**定理（reflected Day convolution）．** \\((\\mathcal C,\\otimes,I)\\) を cocomplete monoidal category とし，\\(\\otimes\\) が各変数で小余極限を保つとする．\n\n\\[\nj:\\mathcal A\\hookrightarrow\\mathcal C\n\\]\nを小さい full dense monoidal subcategory で，unit を含み tensor に閉じるものとする．restricted Yoneda とその左随伴を\n\\[\nN:\\mathcal C\\to[\\mathcal A^{\\mathrm{op}},\\mathbf{Set}],\n\\qquad\nL(P)=\\int^{a\\in\\mathcal A}P(a)\\cdot a\n\\]\nとする．\\([\\mathcal A^{\\mathrm{op}},\\mathbf{Set}]\\) に \\(\\mathcal A\\) の tensor から誘導される Day convolution \\(\\star\\) を入れると，自然同型\n\\[\n\\boxed{L(P\\star Q)\\cong L(P)\\otimes L(Q)}\n\\]\nがある．特に\n\\[\n\\boxed{X\\otimes Y\\cong L(NX\\star NY)}\n\\]\nである．\n\n## Proof\n\nDay convolution の coend formula と co-Yoneda lemma により\n\\[\nL(P\\star Q)\n\\cong\n\\int^{a,b\\in\\mathcal A}P(a)\\times Q(b)\\cdot(a\\otimes b).\n\\]\n一方，\\(\\otimes\\) の各変数での cocontinuity により\n\\[\n\\begin{aligned}\nL(P)\\otimes L(Q)\n&=\\left(\\int^aP(a)\\cdot a\\right)\\otimes\n  \\left(\\int^bQ(b)\\cdot b\\right)\\\\\n&\\cong\n\\int^{a,b}P(a)\\times Q(b)\\cdot(a\\otimes b).\n\\end{aligned}\n\\]\n従って \\(L\\) は strong monoidal である．\\(\\mathcal A\\) の density により \\(LNX\\cong X\\) なので，第二の公式が従う．\n\n## Application to games\n\n\\(\\mathcal C=\\mathbf{Game}\\)，\\(\\mathcal A=\\mathbf{Game}_{\\mathrm{fp}}\\) を finitely presentable games の小 skeleton とし，Conway addition が各変数で cocontinuous かつ \\(\\mathcal A\\) を保つ場合，\n\\[\nX+Y\\cong L(NX\\star_{\\mathrm{Day}}NY).\n\\]\n従って Conway addition は Game 自身を前層圏と同一視した literal Day convolution ではないが，finite games 上の Day convolution を Gabriel--Ulmer 型 reflection で戻した **reflected Day convolution** である．\n\n## Interpretation\n\n\\[\n\\text{大きな圏の tensor}\n=\n\\text{小さな生成対象上の組合せ論}\n+\n\\text{free cocompletion/reflection}\n\\]\nである．Games と Species の接点は対象の同一視ではなく，tensor の普遍表示にある．\n\n## Status\n\n一般定理は標準的な coend argument で証明済み．Game への適用の仮定は成り立つ（人間による検証は未了）：finitely presentable なゲームは台集合が有限なゲームで（論文 v3 Lemma B.8），その充満部分圏は稠密で（Lemma B.7），1点ゲーム（unit）を含み Conway 和で閉じ（台集合は \\(UX\\times UY\\)），Conway 和は各変数で余極限を保つ（Corollary B.11 の証明）．",
      "dates": {
        "inspiration": null,
        "sourceRecorded": null,
        "evidence": null,
        "registeredInSumire": "2026-10-07T16:37:58.103Z",
        "updatedInSumire": "2026-10-30T14:35:21.579Z",
        "legacyRecordedAt": "2026-10-07T16:37:58.103Z",
        "legacyUpdatedAt": "2026-10-30T14:35:21.579Z"
      },
      "language": null,
      "projectLanguage": "ja",
      "languageSource": "not_recorded",
      "legacyStage": "proof",
      "classification": "unclassified",
      "classificationStatus": "unclassified",
      "importance": "unspecified",
      "section": "unassigned",
      "problemState": "unknown",
      "resolution": "",
      "verificationEvidence": {
        "ai": "一般定理（coend 計算）と，Game への適用の仮定が成り立つことの確認（論文 v3 Lemma B.7・B.8・Cor. B.11 による．2026-10-30 に追加）は AI による。人間未検証。",
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        "human": ""
      },
      "aiClaimModel": "",
      "formalization": "unknown",
      "humanReview": "unreviewed",
      "verification": "unverified",
      "ai": {
        "use": "assisted",
        "tools": "Codex, Claude",
        "additions": ""
      },
      "originalQuestion": "",
      "references": []
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    "mathlib": "https://leanprover-community.github.io/contribute/index.html"
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