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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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      "name": "hora",
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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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      "number": 7,
      "url": "https://sumire.aiformath.jp/p/2610.00003/n/7",
      "title": "Generalized games framework",
      "author": {
        "name": "hora",
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      "body": "## Question\n\nHow far does the recursive-coalgebra framework extend from sets and the finite-powerset functor to a locally presentable category \\(\\mathcal C\\) and an accessible endofunctor \\(T:\\mathcal C\\to\\mathcal C\\)? This is Question 5.9 of [Games as recursive coalgebras](https://arxiv.org/abs/2510.22886v3).\n\n## Separate the claims\n\nThe framework contains several ingredients: recursive evaluation into algebras; a terminal recursive coalgebra; game-combining operations; and universal factorizations of values. A construction of one ingredient is not by itself a construction of all the others for arbitrary \\((\\mathcal C,T)\\).\n\nOne concrete extension is already in the paper: replacing \\(\\mathcal P_{\\mathrm{fin}}\\) with \\(\\mathcal P_{\\mathrm{fin}}\\times\\mathcal P_{\\mathrm{fin}}\\) recovers short partisan forms (Proposition 5.4). [Note 3](https://sumire.aiformath.jp/p/2610.00003/n/3) states the remaining boundary for values and strategies.\n\nThe factorization argument recorded in [note 2](https://sumire.aiformath.jp/p/2610.00003/n/2) is another partial direction; it does not, on its own, establish the entire framework for every locally presentable base and accessible endofunctor.\n\n## Next step\n\nFormulate one generalization theorem at a time. State the hypotheses needed for recursive evaluation and the terminal object, then separately give the additional monoidal or value data used by a Bouton-type factorization. Test those hypotheses against the impartial and partisan examples before making the universal claim.",
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          "url": "https://arxiv.org/abs/2510.22886v3",
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