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  "exportedAt": "2026-11-04T08:42:23.572Z",
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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": [
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        "id": "goal_71391ac0-c899-41e4-b6ed-e9d825d41001",
        "title": "To understand why Nim-sum works",
        "body": "",
        "noteIds": []
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      {
        "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_LaVyM-6uZ5fz",
      "number": 4,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/4",
      "title": "Internal hom for Conway addition",
      "author": {
        "name": "hora",
        "handle": "ryuya"
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      "body": "## Question and proposed construction\n\nQuestion 5.7 of [Games as recursive coalgebras](https://arxiv.org/abs/2510.22886v3) asks for a concrete right adjoint to Conway addition:\n\\[\n\\mathbf{Game}(G+K,H)\\cong\\mathbf{Game}(K,[G,H]_+).\n\\]\nExistence is proved abstractly in the paper. The proposed construction and proof below describe positions, moves, evaluation, and currying explicitly. A detailed proposed proof is in [note 29](https://sumire.aiformath.jp/p/2610.00003/n/29).\n\n## Positions and moves\n\nPut \\(S=\\operatorname{Set}(UG,UH)\\). By Definition B.13 and Proposition B.14, the cofree game \\(R(S)\\) consists of hereditarily finite labelled positions \\(t=(T_t,a_t)\\), with finite child set \\(T_t\\) and label \\(a_t\\in S\\).\n\nLet \\(A(G,H)\\) consist of those \\(t\\in R(S)\\) for which every descendant \\(w\\), including \\(t\\), satisfies\n\\[\n\\theta_H(a_w(g))\n=\\{a_w(g'):g'\\in\\theta_G(g)\\}\n \\cup\\{a_v(g):v\\in T_w\\}\n\\quad(g\\in UG).\n\\]\nThe condition is inherited by children, so \\(A(G,H)\\) is a subgame. Its moves are the inherited child relation.\n\n## Evaluation and currying\n\nEvaluation is \\(\\operatorname{ev}(g,t)=a_t(g)\\). The displayed equation says exactly that it is an exact coalgebra morphism \\(G+A(G,H)\\to H\\).\n\nFor a morphism \\(f:G+K\\to H\\), take the cofree transpose of the label map \\(k\\mapsto(g\\mapsto f(g,k))\\):\n\\[\nh_f(k)=\\bigl(\\{h_f(k'):k'\\in\\theta_K(k)\\},\\,g\\mapsto f(g,k)\\bigr).\n\\]\nThe morphism equation for \\(f\\) gives the compatibility equation at every image position. The image is closed under options, so \\(h_f\\) factors through \\(A(G,H)\\). Conversely, evaluation uncurries any map \\(K\\to A(G,H)\\). One composite is the identity by evaluation of labels; the other is the identity by uniqueness of the cofree transpose. Thus \\(A(G,H)\\) is the required right adjoint.\n\nThis is a construction in the category of finitely branching, finite-time games with exact coalgebra maps. The carriers need not be finite.\n\n## Status and remaining work\n\nThis is a proposed construction and proof, retained as a proof candidate rather than a human-reviewed result. The argument is stated in the precise category stated above; it does not claim a new published theorem or a completed human review.\n\n\nCompute usable small examples and explain the game-theoretic meaning of these labelled positions. The construction does not identify this internal hom with player-switching or a strategy-category internal hom. It does not establish the partisan analogue. Its relation to prior mapping-object constructions also requires a separate literature comparison.",
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        "registeredInSumire": "2026-10-07T16:30:28.862Z",
        "updatedInSumire": "2026-10-30T14:31:10.628Z",
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        "legacyUpdatedAt": "2026-10-30T14:31:10.628Z"
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      "projectLanguage": "ja",
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      "legacyStage": "conjecture",
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      "section": "open-questions",
      "problemState": "solved",
      "resolution": "Conway 和の internal hom は明示的に書ける：\\(S=\\operatorname{Set}(UG,UH)\\) の cofree game \\(R(S)\\)（論文 v3 Definition B.13・Proposition B.14：位置は有限個の子の集合 \\(T_t\\) とラベル \\(a_t\\in S\\) の組を再帰的に作ったもの）の中で，全ての子孫 \\(w\\)（自身を含む）が \\(\\theta_H(a_w(g))=\\{a_w(g'):g'\\in\\theta_G(g)\\}\\cup\\{a_v(g):v\\in T_w\\}\\)（\\(g\\in UG\\)）をみたす位置からなる最大の部分ゲームを \\(A(G,H)\\) とすると，評価 \\(\\operatorname{ev}(g,u)=a_u(g)\\) と cofree transpose による currying が自然な全単射 \\(\\mathbf{Game}(G+K,H)\\cong\\mathbf{Game}(K,A(G,H))\\) を与える．構成と証明（自然性と空ゲームの場合を含む）は [[2610.00003/29]]．人間による検証は未了．",
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          "url": "https://arxiv.org/abs/2510.22886v3",
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          "identifier": "arXiv:2510.22886v3",
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