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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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        "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_czu70ozJYZsi",
      "number": 16,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/16",
      "title": "Game is not a topos",
      "author": {
        "name": "hora",
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      "body": "## Answer\n\nThe category \\(\\mathbf{Game}\\) is not cartesian closed and therefore is neither an elementary nor a Grothendieck topos. This is Remark B.27 of [Games as recursive coalgebras](https://arxiv.org/abs/2510.22886v3).\n\nThe obstruction is failure of a product functor to preserve colimits. For the game \\(\\mathbb S_2\\) used in the paper, the quotient of its canonical \\(\\mathbb Z/2\\mathbb Z\\)-action is \\(\\mathbb S_1\\), whereas the corresponding quotient after multiplying by \\(\\mathbb S_2\\) has six elements and is not \\(\\mathbb S_2\\times\\mathbb S_1\\). Hence \\(\\mathbb S_2\\times(-)\\) has no right adjoint.\n\nThis does not contradict the existence of a subobject classifier or local finite presentability. It also does not contradict closedness for Conway addition: the categorical product and Conway addition are different operations.\n\n## Separate question\n\nA suitable completion might be a topos even though \\(\\mathbf{Game}\\) itself is not. [Note 42](https://sumire.aiformath.jp/p/2610.00003/n/42) concerns such a completion. Any claim about it must specify the completion and embedding and provide its own proof; it does not reopen the answered question above.",
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