{ "format": "sumire-research-record", "version": 1, "exportedAt": "2026-11-04T08:44:35.346Z", "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_88KP72pSSYxI", "number": 17, "url": "https://sumire.aiformath.jp/p/2610.00003/n/17", "title": "box積はDay convolutionか", "author": { "name": "hora", "handle": "ryuya" }, "source": "", "body": "## Idea\n\n\\(\\mathbf{Game}\\) 上の box product（Conway addition を与える monoidal 構造）を、ある small monoidal category 上の Day convolution として実現できるかを問う。\n\nDay convolution は small monoidal category \\(\\mathcal A\\) に対し前層圏 \\([\\mathcal A^{\\mathrm{op}},\\mathbf{Set}]\\) へ monoidal 構造を誘導する。\\(\\mathbf{Game}\\) が前層圏あるいはその reflective 部分圏として書けるなら（この点は [Gameの圏はtoposか](https://sumire.aiformath.jp/p/2610.00003/n/16) の問いと繋がる）、box product が \\(\\mathcal A\\) の monoidal 構造から Day convolution として誘導されるかを問える。\n\n原メモが挙げる比較対象は次の三つ。\n\n- \\(\\omega\\) や \\(\\mathbb N\\) を添字に取ったときの chain のテンソル\n- topos of trees の Day convolution\n- species の圏。\\([\\mathbb B,\\mathbf{Set}]\\) 上の Day convolution が species の積を与えるのと同じ形で、2-rig 構造を作るならこれとパラレルに構成する\n\n## Goal\n\nbox product が Day convolution として書けるかを決定する。書けるなら、添字圏 \\(\\mathcal A\\) とその上の monoidal 構造を同定する。そのうえで、Conway addition の右随伴（[Internal hom for Conway addition](https://sumire.aiformath.jp/p/2610.00003/n/4) の internal hom）が Day convolution の閉構造として自動的に出るかを確かめる。\n\n[Closed structures on games](https://sumire.aiformath.jp/p/2610.00003/n/5) は「\\(\\mathbf{Game}\\) 上の monoidal closed structures を全分類せよ」という問いだが、こちらは「既知の box product を Day convolution という具体的な手段で構成できるか」という構成の問いであり、分類問題への攻め口になる。また [Internal hom for Conway addition](https://sumire.aiformath.jp/p/2610.00003/n/4) は internal hom の具体形を cofree game の部分として探しているが、Day convolution の一般論から閉構造が出るなら別ルートでの記述が得られる。\n\n## species と differential 2-rig の材料（2026-08-16 追記）\n\n原メモの「2rig にしたいなら、species とパラレルに！」に対応する既知の結果：\n\n- Loregian--Trimble, *Differential 2-rigs* ([arXiv:2103.00938](https://arxiv.org/abs/2103.00938)) は differential 2-rig を定義し（Def. 4.1）、1生成元上の free differential 2-rig を与える（Thm. 5.16）。そこでは「ある colored species の圏が1生成元上の free differential 2-rig である」ことが示されている\n- substitution / plethystic product に対する chain rule も同論文にある（Rem. 5.3、Thm. 5.18、証明は Appendix B）\n- Joyal 1981（Adv. Math. **42**(1):1--82）の Proposition 5 が species の Leibniz rule にあたる\n\nしたがって「species とパラレルに 2-rig を作る」という筋は、Loregian--Trimble の free differential 2-rig の構成と直接比較できる。box product が Day convolution として書けたとき、それが colored species 側の構成とどう対応するかが具体的な問いになる。\n\n## 2026-08-26 AI研究監査：reflected Day convolution\n\n$\\mathcal A=\\mathbf{Game}_{\\mathrm{fp}}$ を finitely presentable games の小 skeleton，$N:\\mathbf{Game}\\to[\\mathcal A^{op},\\mathbf{Set}]$ を restricted Yoneda，\n\\[\nL(P)=\\int^{A\\in\\mathcal A}P(A)\\cdot A\n\\]\nをその左随伴とする．$\\star$ が各変数で cocontinuous で finitely presentable games を保つ tensor なら，$\\mathcal A$ 上の制限から Day convolution $\\star_{\\mathrm{Day}}$ が定まり，\n\\[\nL(P\\star_{\\mathrm{Day}}Q)\\cong L(P)\\star L(Q),\n\\qquad\nX\\star Y\\cong L(NX\\star_{\\mathrm{Day}}NY)\n\\]\nが自然に成り立つ．証明は Day convolution の coend 展開と cocontinuity：\n\\[\n\\int^{A,B}P(A)\\times Q(B)\\cdot(A\\star B)\n\\cong\n\\left(\\int^AP(A)\\cdot A\\right)\\star\n\\left(\\int^BQ(B)\\cdot B\\right).\n\\]\n\nしたがって box product / Conway addition は Game 自身を前層圏と同一視した literal Day convolution ではない（Game は topos でない）が，Gabriel--Ulmer の reflective presentation を通した canonical reflected Day convolution である．", "dates": { "inspiration": null, "sourceRecorded": null, "evidence": null, "registeredInSumire": "2026-10-07T16:35:15.392Z", "updatedInSumire": "2026-10-30T14:33:59.726Z", "legacyRecordedAt": "2026-10-07T16:35:15.392Z", "legacyUpdatedAt": "2026-10-30T14:33:59.726Z" }, "language": null, "projectLanguage": "ja", "languageSource": "not_recorded", "legacyStage": "conjecture", "classification": "unclassified", "classificationStatus": "unclassified", "importance": "unspecified", "section": "unassigned", "problemState": "solved", "resolution": "\\(\\mathbf{Game}\\) は topos でない（[[2610.00003/16]]）ので前層圏と同値でなく，Conway 和を \\(\\mathbf{Game}\\) 自身の上の Day convolution として書くことはできない．一方，有限ゲームの同型類の代表からなる小さな充満部分圏 \\(\\mathcal A\\)（finitely presentable な対象．論文 v3 Lemma B.8．稠密で，1点ゲームを含み Conway 和で閉じる）上の Day convolution \\(\\star\\)，制限 Yoneda \\(N\\)，その左随伴 \\(L\\) について \\(X+Y\\cong L(NX\\star NY)\\) が成り立つ（[[2610.00003/37]]）．閉構造もここから出る：\\(a\\in\\mathcal A\\) について \\(N([X,Y]_+)(a)\\cong\\mathbf{Game}(a+X,Y)\\cong[NX,NY]_{\\mathrm{Day}}(a)\\) なので，Conway 和の internal hom（明示形は [[2610.00003/4]]）は Day convolution の internal hom の（\\(N\\) の像への）制限である．人間による検証は未了．", "verificationEvidence": { "ai": "「2026-08-26 AI研究監査：reflected Day convolution」のcoend計算と結論。", "lean": "", "human": "" }, "aiClaimModel": "", "formalization": "unknown", "humanReview": "unreviewed", "verification": "unverified", "ai": { "use": "assisted", "tools": "Codex, Claude", "additions": "" }, "originalQuestion": "", "references": [ { "url": "https://arxiv.org/abs/2103.00938", "verification": "not_checked", "service": "arxiv", "identifier": "arXiv:2103.00938", "version": null, "pinned": false, "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/" } }