{ "format": "sumire-research-record", "version": 1, "exportedAt": "2026-11-04T08:42:23.576Z", "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": [] } ], "goalDependencies": [] }, "notes": [ { "id": "n_55acKb5mKfVi", "number": 1, "url": "https://sumire.aiformath.jp/p/2610.00003/n/1", "title": "A double category of games", "author": { "name": "hora", "handle": "ryuya" }, "source": "", "body": "## Question\n\nCan recursive-coalgebra maps and playing strategies be combined in a double category of games? The intended vertical maps are exact coalgebra morphisms; the horizontal maps should come from an explicitly chosen category of games and strategies in the tradition of Joyal.\n\n## Current boundary\n\nA coalgebra map preserves the complete set of options. A playing strategy chooses responses along legal plays. These are different structures, and their interaction must be defined rather than identified.\n\nThe work in [note 13](https://sumire.aiformath.jp/p/2610.00003/n/13) shows why a counterexample with arbitrary binary relations cannot automatically be treated as a counterexample with legal total deterministic strategies. The elementary relational obstruction is recorded in [note 23](https://sumire.aiformath.jp/p/2610.00003/n/23); it does not settle this double-category question.\n\n## Next step\n\nFix the objects (including any Left/Right or turn data), the legal strategy class, and the proposed squares. On a small example, verify that vertical maps transport legal strategies and that horizontal composition is compatible with the squares. State any restrictions needed on the vertical maps.\n\nA construction must verify units, both compositions, and interchange. A no-go claim must exhibit a counterexample inside the chosen legal strategy class.\n\nSources: [Games as recursive coalgebras](https://arxiv.org/abs/2510.22886v3); [Joyal's games and strategies](https://people.cs.uchicago.edu/~brady/CSPP50102/notes/Joyal-games.pdf).", "dates": { "inspiration": null, "sourceRecorded": null, "evidence": null, "registeredInSumire": "2026-10-07T16:29:54.551Z", "updatedInSumire": "2026-10-30T04:49:21.632Z", "legacyRecordedAt": "2026-10-07T16:29:54.548Z", "legacyUpdatedAt": "2026-10-30T04:49:21.632Z" }, "language": null, "projectLanguage": "ja", "languageSource": "not_recorded", "legacyStage": "conjecture", "classification": "unclassified", "classificationStatus": "unclassified", "importance": "unspecified", "section": "open-questions", "problemState": "open", "resolution": "", "verificationEvidence": { "ai": "", "lean": "", "human": "" }, "aiClaimModel": "", "formalization": "unknown", "humanReview": "unreviewed", "verification": "unverified", "ai": { "use": "assisted", "tools": "Codex", "additions": "" }, "originalQuestion": "", "references": [ { "url": "https://arxiv.org/abs/2510.22886v3", "verification": "not_checked", "service": "arxiv", "identifier": "arXiv:2510.22886v3", "version": "v3", "pinned": true, "correspondence": { "status": "reader_memo_unverified", "assumptions": "", "conclusions": "", "evidence": "", "differences": "", "unchecked": "", "declaration": "" } } ] } ], "noteDependencies": [], "guides": { "arxiv": "https://info.arxiv.org/help/submit/index.html", "formal_conjectures": "https://github.com/google-deepmind/formal-conjectures/blob/main/CONTRIBUTING.md", "mathlib": "https://leanprover-community.github.io/contribute/index.html" }, "search": { "arxiv": "https://arxiv.org/search/", "formal_conjectures": "https://google-deepmind.github.io/formal-conjectures/browse/", "mathlib": "https://leanprover-community.github.io/mathlib4_docs/" } }