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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.",
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    "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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        "id": "goal_71391ac0-c899-41e4-b6ed-e9d825d41001",
        "title": "To understand why Nim-sum works",
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        "title": "Categorical unification of recursive, geometric and logical aspects of combinatorial game theory",
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    {
      "id": "n_vnUUNhZqC02H",
      "number": 6,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/6",
      "title": "Differential structures of games",
      "author": {
        "name": "hora",
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      "body": "## Question\n\nIs there one categorical structure explaining both the Leibniz-type rule for moves under Conway addition and the Rota–Baxter-type rule for recursive game values? This is Question 5.1 of [Games as recursive coalgebras](https://arxiv.org/abs/2510.22886v3).\n\nThe target is an explanation of the Nim-sum calculation, not just a new name for an identity already obtained using the Sprague–Grundy theorem.\n\n## Established equations\n\nThe move map satisfies\n\\[\n\\theta_{X+Y}(x,y)\n=\\theta_X(x)\\times\\{y\\}\\;\\cup\\;\\{x\\}\\times\\theta_Y(y).\n\\]\nWhen a Bouton monoid is induced by a finite-powerset algebra \\(\\beta\\), Proposition 5.2 proves\n\\[\n\\beta(S)\\circledast\\beta(T)\n=\\beta\\bigl(S\\circledast\\beta(T)\\cup\\beta(S)\\circledast T\\bigr).\n\\]\n\nThese are the starting data. General differential-category terminology alone does not establish a deriving transformation or show that it recovers these formulas.\n\n## Next step\n\nState the candidate structure and derive both displayed equations from its axioms. Then give the Nim example and identify precisely which facts about Nim-sum are inputs and which are consequences. If the desired identification is assumed through Sprague–Grundy theory, the argument has not yet supplied an independent explanation.\n\nThis question is distinct from the [internal hom for Conway addition](https://sumire.aiformath.jp/p/2610.00003/n/4), which concerns an adjunction rather than a differential structure.\n\nComparison source: [Differential Categories Revisited](https://arxiv.org/abs/1806.04804).",
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